disi-unibo-nlp/MathGames · Datasets at Hugging Face

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1913	1999	semifinal	no	L1 L2 GP	Pattern Recognition	49	The smallest integer whose digits sum to 1 is the number 1. The smallest integer whose digits sum to 2 is the number 2. The smallest integer whose digits sum to 3 is the number 3... The smallest integer whose digits sum to 10 is the number 19. The smallest integer whose digits sum to 11 is the number 29, etc. If we repeat this procedure and write down the sequence of numbers thus obtained, we get: 1, 2, 3, ... 19, 29, ... What is the largest number in this sequence that is the square of an integer? Answer 0 (zero) if you think this number does not exist.	Not supported with pagination yet
1914	1999	semifinal	no	L1 L2 GP	Geometry	45000 cubic centimeters	An aquarium placed on a table has the shape of a rectangular prism with a height of 30 cm. It is filled with water to the brim and then tilted around one of the edges of its base so that the bottom forms an angle of 45° with the plane of the table. In this way, one third of its content spills onto the table. Now it is filled again to the brim and tilted around the other edge of its base so as to form an angle of 45° again with the plane of the table. In this way, 4/5 of the content now spills onto the table. What is the volume of the aquarium in cm³?	Not supported with pagination yet
1916	1999	semifinal	no	L2 GP	Geometry	3 solutions: 24° -72° -168°	Two discs A and B with centers O and P, respectively, externally tangent, perform the following two-step dance movement: A begins to rotate around B, clockwise, such that its center forms an angle α. Then, it is B's turn to rotate around A, always clockwise, and form an angle α/2 around point O. The dancing discs perform 10 complete movements of this dance, after which they return to their starting position for the first time. Give the value of the angle α in degrees.	Not supported with pagination yet
1926	1999	international final day 1	no	C2 L1 GP L2 HC	Geometry	3 solutions (3;4;4), (3.6;8), (6; 6.7)	Matteo measured the lengths of the sides of the triangle he drew. He sums the three measurements, which are all integer centimeters, and obtains a perimeter of 15 cm. However, the teacher points out that the result is inaccurate. Matteo made no calculation errors and correctly used the ruler, taking all measurements starting from the zero mark of the scale. But he realizes that the ruler's scale has a small defect. What are the (exact) lengths of the three sides of Matteo's triangle? (express the three lengths in cm, in increasing order)	Not supported with pagination yet
1928	1999	international final day 1	no	L1 GP L2 HC	Combinatorics	165x125x300+181x25x1=6192,025 kg	An architect has decided, to decorate a city, to create a 5-story pyramid, formed by identical wooden cubes. The architect knows that it will need to be coated with paint to protect it from the elements. Therefore, they had a 1/5 scale sketch made of one cube from the pyramid. They thus realize that the reduced cube weighs 300 g and, when painted, 306 g. What will be the weight of the painted pyramid? N.B. Only the visible parts of the pyramid will be painted. Disregard paint on the edges. Give the result in kg, rounded to the nearest gram.	Not supported with pagination yet
1930	1999	international final day 1	no	L1 GP L2 HC	Arithmetic	2/5	Giuliano encounters some difficulty with fractions. He replaces the fraction bar with a comma and, instead of multiplying, he divides. So, for example, to multiply 12 x 6/25 he divides 12 by 6.25. Today, proceeding in the same way, by multiplying a non-zero number by an irreducible fraction, he obtained a correct result. By what fraction did Giuliano multiply the number?	Not supported with pagination yet
1931	1999	international final day 1	no	L2 HC	Geometry	3 solutions (7;24;25), (8;15;17), (9;12;15)	The side lengths of a right triangle are all integers in cm. An inscribed circle, tangent to the three sides, has a radius of 3 cm. What are the dimensions of the triangle?	Not supported with pagination yet
1932	1999	international final day 1	no	L2 HC	Arithmetic	4 solutions: 9, 18, 54, 774 pearls	A pharaoh had a certain number of magnificent pearls hidden. This number, known to everyone, was considered magical, as were all its divisors, which were the only other magical numbers. After the pharaoh's sudden death, 5 thieves, who had discovered the hiding place, decided to take advantage of it. They arrived at night, and, one after another, each stole the maximum possible number of pearls, this number being magical. The last thief was surprised to find only one! In the morning, the scribe responsible for the treasury, finding the hiding place empty, was pleased to have taken for himself, before the five thieves' visit, a number of pearls (not magical) equivalent to 3/100 of the treasure! How many pearls had the indelicate scribe stolen?	Not supported with pagination yet
1817	2000	team games	no	C2 L1 L2	Geometry	12100 m2	A magnificent willow is planted inside a square plot of land. The sum of its distances from two sides of the square is 100 m, while the sum of the distances from the other two sides is equal to 120 m. What is the area (in square meters) of the land?	Not supported with pagination yet
1818	2000	team games	no	C2 L1 L2	Algebra	10 dividers	The product of all divisors of a natural number (greater than 1) is equal to the fifth power of this number. How many divisors does the number in question have?	Not supported with pagination yet
1819	2000	team games	no	C2 L1 L2	Algebra	42, 53, 76	Three two-digit numbers are formed using the digits 2, 3, 4, 5, 6, 7 (not repeated). The sum of the three numbers is 171; the difference between the two smallest is 11. Find the three numbers (giving them in increasing order).	Not supported with pagination yet
1820	2000	team games	no	C2 L1 L2	Combinatorics	2 solutions: 2, 6	On a circumference of length 24 cm, some points are placed such that the arcs of the circumference between two consecutive points measure either 2 or 3 cm. Furthermore, by connecting any two points, a diameter of the circumference is never formed. How many 3 cm long arcs of the circumference are formed by the points in question?	Not supported with pagination yet
1821	2000	team games	no	C2 L1 L2	Geometry	72 cm	Divide a rectangle into five similar right triangles. When two of these five triangles are not disjoint, they share either a vertex or an entire side, which is then the hypotenuse of one of the two triangles and a leg of the other. The area of the smallest triangle is 2 square cm. What is the area of the rectangle?	Not supported with pagination yet
1822	2000	team games	no	C2 L1 L2	Logic	8 km	Mr. Dubbio, who is a sales representative, lives on State Road No. 7. Today, starting from home to see a client, he made a first trip of 53 km. Subsequently, he made other trips, of 79, 27, and 9 km respectively. Unfortunately, he doesn't remember in which direction he made each of these journeys. "Anyway," he thinks, "I am at most 168 km away from home." What is his minimum distance from home, instead? (give the answer in km)	Not supported with pagination yet
1823	2000	team games	no	C2 L1 L2	Combinatorics	8	Giulia and Bernardo's game involves writing a multi-digit number. The player who starts writes the first digit (which must be non-zero); subsequently and alternately, each player writes a digit to the right of the digit or digits already written. However, players must follow these rules: * after a 9, any digit can be written * after a digit less than 9, a larger digit must be written * each digit can appear (in the final number) at most three times The player who, first, cannot write any digit loses. Giulia starts. What digit must she write to be sure of winning, regardless of Bernardo's play? (answer 0 if you think such a winning strategy does not exist)	Not supported with pagination yet
1825	2000	team games	no	C2 L1 L2	Combinatorics	12	Anna likes to write all possible decimal numbers, using only a 1, a 2, a 3, and a decimal point. How many different numbers can Anna write?	Not supported with pagination yet
1826	2000	team games	no	C2 L1 L2	Geometry	60	Emi has 120 identical small cubes: 80 blue and 40 white. She wants to use them to build a large rectangular parallelepiped using glue. Its surface will be formed entirely by the faces of the small cubes. What is the minimum number of visible faces of the small cubes that will be blue?	Not supported with pagination yet
1827	2000	team games	no	C2 L1 L2	Arithmetic	17	In a "normal" die, the sum of the points on two opposite faces is always equal to 7. Enrico placed three "normal" dice on a table, one on top of the other to form a tower. On the top face of the die at the top of the tower, there is a 4. What is the sum of the points on the five hidden faces (including those between two dice or the table)?	Not supported with pagination yet
1828	2000	team games	no	C2 L1 L2	Combinatorics	9	Eight students (from a first and a second high school class) participated in a competition, all recording distinct scores. The winner scored 8 points; the last placed student scored 1 point. The second-year students collectively scored 18 points. In the ranking, between any two second-year students, there is always at least one first-year student. Gianni and Gianna are the only first-year students who are not separated (in the ranking) by a second-year student. How many points did Gianni and Gianna score together?	Not supported with pagination yet
1833	2000	team games	no	C2 L1 L2	Geometry	522	For his birthday, Mauro received a radio-controlled car model, which can only move forward, either in a straight line or by describing circular arcs with a radius of 63 cm. Currently, the car is at point A, facing North. What is the minimum distance Mauro must make his toy travel so that it returns to the same point A, but oriented South? (let pi = 22/7).	Not supported with pagination yet
1834	2000	semifinal	no	C1	Logic	4 children	In a family, each of the children can state that they have at least one brother and one sister. What is the minimum number of children in this family?	Not supported with pagination yet
1836	2000	semifinal	no	C1 C2 L1 L2 GP	Logic	89690	Ennio didn't want to give me the postal code of his city. Here's how he answered my questions about it: - Like every Italian postal code, it consists of five digits. - The sum of the first digit and the second is 17. - The sum of the second and the third is 15, as is the sum of the third and the fourth. - The sum of the last two digits is 9. - The sum of the last and the first is 8. What is the postal code number of Ennio's city?	Not supported with pagination yet
1837	2000	semifinal	no	C1 C2 L1 L2 GP	Combinatorics	31 fruit	Angelo and Rosi have invited seven friends to dinner this evening. At the end of the meal, they would like to offer fresh fruit, apples and pears, which they will pick from their orchard. However, it is quite far from the house, and Angelo and Rosi, being advanced in years, know that together they cannot carry more than 7 kilograms of fruit. On the other hand, they want each of their guests to be able to choose which fruits to eat. An apple weighs 300 g; a pear weighs 200 g. What is the maximum number of fruits they can pick by going to the orchard only once?	Not supported with pagination yet
1838	2000	semifinal	no	C1 C2 L1 L2 GP	Combinatorics	3 shapes	For Jacob's twelfth birthday, his parents ordered some very special cakes from the pastry chef... in the shape of a triangle with a perimeter of 12 cm. All sides of the triangles have a measurement in cm corresponding to an integer. How many different shapes can the pastry chef make?	Not supported with pagination yet
1839	2000	semifinal	no	C1 C2 L1 L2 GP	Algebra	108 pages	To number the pages of a large notebook, Peter had to write a number of digits double the number of pages in this notebook. How many pages does Peter's notebook have?	Not supported with pagination yet
1841	2000	semifinal	no	C2 L1 L2 GP	Logic	Monday	My four cousins arrive at our house on Sunday morning at breakfast time and are staying for twelve days of vacation. They have a big sweet tooth, just like us! Fortunately, Mom, thinking ahead, has bought 168 chocolate bars so that everyone can, during the twelve days, eat one for breakfast and one for a snack. Unfortunately, on the evening of the ninth day, our cousins have to cut short their stay and return home. We continue, despite their absence, to enjoy the chocolate bars with the same frequency. On what day of the week will we eat the last bar?	Not supported with pagination yet
1842	2000	semifinal	no	C2 L1 L2 GP	Combinatorics	44	During the last games before the basketball cup final, we saw a scout from our future opposing team in the stands. They were taking notes about our usual tactics. At this point, we must disrupt our opponents' expectations. We have thus decided to redistribute our five numbered jerseys in such a way that none of us five wears their usual jersey. In how many ways can we perform this redistribution?	Not supported with pagination yet
1844	2000	semifinal	no	L1 L2 GP	Combinatorics	3 solutions 132 • 264 • 396	On Mauro's calculator, of the keys from 1 to 9, only three of them still work. Mauro sums the six numbers (of two distinct digits) that he can form using only these three keys. Amazingly! The sum that appears still uses the same three digits. What is this sum?	Not supported with pagination yet
1849	2000	final	no	C1	Combinatorics	Eventually visualizing ladders and steps, it occurs that the frog, in order to reach the top of the scale, must make 12 jumps.	A small frog is at the base of a staircase consisting of 21 steps. With its first jump, it reaches the second step; then it continues to climb the stairs with jumps of two steps. Unfortunately, the steps numbered 5, 10, 15, and 20 are slippery and, when the frog lands on one of these, it slips and goes down one step. How many jumps does the frog have to make to reach the twenty-first step?	Not supported with pagination yet
1850	2000	final	no	C1 C2 L1 L2 GP	Logic	It occurs in other words that the text (as it was formulated in the Italian translation) admitted as a solution all the natural numbers between 39 and 49	Jacob can build at most 9 identical towers using all the chocolates from a box, which contains a number of chocolates between 39 and 49. He then realizes that he has as many chocolates remaining as the number of towers he built. How many chocolates did the box contain exactly?	Not supported with pagination yet
1853	2000	final	no	C1 C2 L1 L2 GP	Geometry	Each of the following responses was considered valid: 8, 9, 10, 11, 12	Fabrizio draws five straight roads on the map of the holiday village in such a way that three of the five roads intersect at the same point and three of the five roads are parallel. Into how many regions do these five roads divide the village?	Not supported with pagination yet
1857	2000	final	no	L1 L2 GP	Pattern Recognition	2024	What will be the next year in which the month of February will have 5 Thursdays?	Not supported with pagination yet
1858	2000	final	no	L1 L2 GP	Combinatorics	It's 18 miles that can't be patrolled.	Police officers Donato, Michele, Angelo, Renato, and Pietro patrol a 120 km road. Each one covers exactly one segment of this road. Donato and Michele patrol two different halves. Donato, Michele, and Angelo patrol three different thirds. Donato, Michele, Angelo, and Renato patrol four different quarters. Donato, Michele, Angelo, Renato, and Pietro patrol five different fifths. What is the minimum total length of the segments of this road not patrolled by any police officer?	Not supported with pagination yet
1859	2000	final	no	L2 GP	Logic	In the worst case, to turn on the lights, Emy will have to make a maximum of 5 attempts	Ermy enters an unlit room equipped with 3 switches numbered 1, 2 and 3, whose state (on or off) she does not know. Each switch can be on or off, and for the room to be lit, all three must be on. Ermy presses switch 1, and the room does not light up. She then presses switch 2, but the room still does not light up. She then seeks a strategy that allows her to light the room in a minimum number of attempts. Applying this strategy, what sequence of switches must Ermy operate in the worst-case scenario?	Not supported with pagination yet
1860	2000	final	no	L2 GP	Combinatorics	0	Angelo and Rosi play with the numbers from the set {2, 3, 4, 5, 6, 7, 8, 9}. Angelo starts by writing a number. Then it's Rosi's turn; she chooses another number (always from the set), multiplies it by the previously written number, and writes the product. Subsequently, one at a time, each player picks an unchosen number from the set, multiplies it by the last written number, and writes the product. The first player who writes a number greater than 1000 loses. What number must Angelo choose to be sure to win, regardless of Rosi's moves?	Not supported with pagination yet
1861	2000	international final day 1	no	C1	Algebra	Number of lianas: 8	In the forest, Tarzan moves in a straight line from liana to liana. There are two types of lianas: short ones that allow for jumps of 4 m, and long ones that allow for jumps of 7 m. Tarzan wants to reach a boulder located 41 m from the edge of a pond. How many lianas must he use?	Not supported with pagination yet
1863	2000	international final day 1	no	C1	Algebra	1st dice: 1 2nd dice: 2 3rd dice: 4	While I was playing with three standard dice (whose faces are numbered from 1 to 6), Dedé arrived and told me: "Roll the three dice without letting me see them. Multiply the number rolled on die number 1 by 30, add 5 to the result. Add the number rolled on die number 2, then multiply the obtained result by 10. Finally, add the number rolled on die number 3. What result do you get?" I answered 374 and Dedé guessed the three numbers rolled on my dice. State what the three numbers rolled on each die were.	Not supported with pagination yet
1865	2000	international final day 1	no	C1 C2 L1 GP L2 HC	Algebra	Age of daughter: 26 age of father: 52	My age is double that of my daughter. The current year is 2000. In 2011, our combined ages will total a century. What are our respective ages today?	Not supported with pagination yet
1866	2000	international final day 1	no	C1 C2 L1 GP L2 HC	Algebra	78 bicherees and 39 daily	At the beginning of the century, in the Isère region, two units for measuring area existed: - the "bicberée", equal to 1600 square meters - the "giornale", equal to 1800 square meters Robert Vassel owned a property that measured twice as many "bicberées" as "giornali", for a total area of 195000 square meters. What is the number of "bicberées" and "giornali" for this property?	Not supported with pagination yet
1867	2000	international final day 1	no	C1 C2 L1 GP L2 HC	Algebra	Number of vegetarian dishes: 504	Thomas is the director of a fast-food chain (with three outlets) that offers three dishes on the menu every day: a couscous dish, a fish dish, and a vegetarian dish. Each of the chain's three restaurants has just sent a pie chart illustrating the breakdown of sales for the three dishes offered. Strangely, all three diagrams show a 120-degree angle, and for all three restaurants, the data reads: 222 couscous dishes and 114 fish dishes. Nevertheless, the number of vegetarian dishes sold is different. How many vegetarian dishes did each of the three restaurants sell?	Not supported with pagination yet
1871	2000	international final day 1	no	L1 GP L2 HC	Geometry	1 Solution 12 km	Matilde, José, and Mattia live in the same region. As the crow flies, the houses of Matilde and José are an integer number of kilometers apart, and those of Matilde and Mattia are an integer number of kilometers apart. Furthermore, again as the crow flies, there are exactly 10 more kilometers between José's house and Mattia's house than between José's house and Matilde's house. Finally, the triangle formed by the houses of the three friends is a right-angled triangle. What is the minimum distance between Matilde's house and Mattia's house?	Not supported with pagination yet
1874	2000	international final day 1	no	L2 HC	Geometry	Perimeter: 1394 m	Father XXL's field has the shape of a quadrilateral formed by two right triangles. All the side lengths of the quadrilateral are expressed in an integer number of meters. The perimeter of this quadrilateral is less than 2000 m. What is the maximum possible value for this perimeter?	Not supported with pagination yet
1875	2000	international final day 1	no	L2 HC	Combinatorics	Number: 165	Mattia, who has time to spare today, has decided to fill a large 100 by 200 grid with numbers. In the first row, he wrote, in order, 1, 2, 3, ..., 100. In the first column, he wrote, in order, 1, 2, 3, ..., 200. He continues to fill the grid according to the following rule: each cell contains the smallest strictly positive integer that is different from all numbers written in the portion of the row to its left and from all numbers written in the portion of the column above it. What number is in the last cell of the last column?	Not supported with pagination yet
1876	2000	international final day 2	no	C1	Pattern Recognition	PARIS code: 33 3 37 19 39	I managed to decode the following message: Coded word: 27 3 37 39 11 19 25 25 11 - 1 26 2 36 38 10 18 24 24 10 : 2 13 1 18 19 5 9 12 12 5 Decoded word: M A R S E I L L E Which coded word would correspond to PARIS?	Not supported with pagination yet
1878	2000	international final day 2	no	C1 C2 L1 GP L2 HC	Logic	Duration 7 hours and 15 minutes	This summer I'm going on vacation to Sildavia. Here are the flight times, for the outbound and return journeys (expressed in local time): departure from Paris at 23:30, arrival in Sildaville at 9:45 the next day. Departure from Sildaville: 11:00, arrival in Paris at 15:15 on the same day. The flight duration is the same for both the outbound and return journeys. What is this duration?	Not supported with pagination yet
1879	2000	international final day 2	no	C1 C2 L1 GP L2 HC	Combinatorics	Order of departure: 7 - 1 - 5 - 10 - 2 - 9 - 3 - 6 - 8 - 4	I have a deck of 10 playing cards. I place the top card at the bottom of the deck, then I turn over the next card onto the table: it's an ace. I place the top card at the bottom of the deck for two consecutive times, then I turn over the next one: it's a 2. I put the next card at the bottom of the deck, then I turn over a card: it's a 3. I continue in this manner, alternatively placing a card at the bottom of the deck once or twice, then turning over the next one. The cards turned over, in order, are: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10. What was the initial order of the cards?	Not supported with pagination yet
1884	2000	international final day 2	no	C2 L1 GP L2 HC	Combinatorics	3 solutions: 6, 8, 10	They arrived discreetly, all dressed in black. I observed them closely. Each of them shook hands with exactly three others, except for one of them, who shook hands with only one person. I didn't count them, but there were fewer of them than a complete football team. How many could there have been? Note: A complete football team consists of 11 players.	Not supported with pagination yet
1885	2000	international final day 2	no	L1 GP L2 HC	Algebra	3 solutions. Age of major: 6, 15, 18	On their trip to Gori, Russia, the Pitt family, who have four children, discovered the following fact. The sum of the ages of the oldest and the youngest is equal to that of the other two. However, the product of the ages of the oldest and the youngest is half the product of the ages of the other two. The oldest is less than 20 years old. What is his age?	Not supported with pagination yet
1886	2000	international final day 2	no	L1 GP L2 HC	Algebra	Each one will pay an extra FF 5.5	Romain and Julien are not great mathematicians: to do their homework, they invited Sophie to the high school cafeteria. They ordered coffee, croissants, and chocolate bars. In this order, for Romain: 1, 3, 7; for Julien: 1, 4, 10 (what an eater!!); for Sophie: 1, 1, 1 (she had to work!). Romain, for his consumption, must pay 29 FF and Julien 38 FF. They wish to thank Sophie by splitting what she has to pay in half. How much more will each of them pay?	Not supported with pagination yet
1888	2000	international final day 2	no	L2 HC	Algebra	2 solutions: 570, 580	Audrey is spending her holidays in Syldavia. In this country, coins are out of circulation, and there are only three types of banknotes, worth respectively 57, 62, and 72 crowns. Yesterday, Audrey bought some croissants from the baker, for a cost of 4 crowns. She gave the baker a certain number of banknotes, for an amount less than 600 crowns, and the baker gave her the exact change for her purchase. What sum did Audrey give to the vendor?	Not supported with pagination yet
1889	2000	international final day 2	no	L2 HC	Combinatorics	The probability is 25/256	The magician has 13 cards, which he fans out. He has a spectator choose two consecutive cards. It is assumed that the spectator chooses randomly, and that all choices are equiprobable. The magician re-forms the fan, without changing the order of the cards, and has another spectator choose two consecutive cards, and so on, until only one card remains in his hand. Before beginning, he placed the Ace of Hearts in the center of the deck. What is the probability that he can triumphantly brandish the Ace of Hearts at the end of the game?	Not supported with pagination yet
1753	2001	autumn games	no	C1	Logic	Pinocchio took.13	In France – as you may know – grades are given out of twenty. Read this dialogue between Pinocchio and his father about his school results, but be careful: Pinocchio has developed the bad habit of never telling the truth! Geppetto: - Did you get more than 12 on the math test? Pinocchio: - No, Dad. G.: - Then you obviously got less than 14? P.: - No, Dad. G.: - I don't understand. But is your grade an even number? P.: - Yes, Dad. What grade did Pinocchio get? (It is an integer).	Not supported with pagination yet
1756	2001	autumn games	no	C1 C2	Algebra	There are two solutions: 3; 7	On a large sheet of paper, Jacob has drawn several squares and several triangles. He counts the sides of the drawn figures (all separate) and finds a total of 29. How many triangles does Jacob have?	Not supported with pagination yet
1757	2001	autumn games	no	C1 C2	Combinatorics	20	The number 2002 is a four-digit number whose sum is 4: 2+0+0+2=4. Including this example, how many four-digit numbers are there for which the sum of their digits is equal to 4? Note: No number can begin with zero.	Not supported with pagination yet
1760	2001	autumn games	no	C2 L1	Algebra	20 cm	To determine if an individual is overweight, the World Health Organization has established a "Body Mass Index", which is calculated by dividing a person's weight (in kg) by the square of their height (in m). With his 81 kg, Renato has a BMI equal to 25, while Mauro, with his 80 kg, has a BMI of 20. What is the difference in height between Mauro and Renato, in cm?	Not supported with pagination yet
1761	2001	autumn games	no	C2 L1	Algebra	205 liters	There are two barrels that have the same capacity. Currently, the two barrels contain a total of 350 liters of Brunello di Montalcino. Then, after having drawn off 20 liters of the famous Tuscan wine from the first barrel and 80 liters from the second, it is noted that the remaining wine is exactly at the same level in the two barrels. What is the minimum capacity of each barrel?	Not supported with pagination yet
1762	2001	autumn games	no	C2 L1	Arithmetic	3	The bunch of black grapes consists of 183 individual grapes, while the bunch of white grapes comprises 252. My friends and I started eating the first bunch, sharing all its black grapes equally among us. We then ate the entire bunch of white grapes, and here too, each of us had an equal number of grapes. How many of us were there?	Not supported with pagination yet
1763	2001	autumn games	no	L1 L2	Arithmetic	0	Carla states that when she arranges the cards in piles of twelve, five are left over; when she arranges the same cards in piles of fifteen, four remain. What is the minimum number of cards that Carla must organize? Note: Reply zero if you think Carla miscounted and the problem as stated has no solution.	Not supported with pagination yet
1764	2001	autumn games	no	L1 L2	Algebra	7h 30min	Guido Piano often takes very long trips. He is very cautious and meticulous in organizing his itineraries. Now he has to travel 1695 km in three stages, which he completes at constant speeds of 50, 60, and 70 km per hour, respectively. The second stage lasts five quarters of the first, and the third lasts as long as the first two stages combined. What is the duration of the second stage, in hours and minutes?	Not supported with pagination yet
1765	2001	autumn games	no	L1 L2	Logic	Mr Ruffini	Which of the three following mathematicians and men of science lived (at least partially) in the 19th century: Isaac Newton, René Descartes, Paolo Ruffini?	Not supported with pagination yet
1767	2001	autumn games	no	L2	Combinatorics	People can line up in 100800 ways	There is a group composed of 4 men and 5 women. In how many different ways can the people in the group line up, knowing that an internal rule of the group prohibits men from occupying the first and the last position?	Not supported with pagination yet
1769	2001	autumn games	no	L2	Geometry	The area is 42 cm2	Enrico built a triangle whose sides measure 8√13, 30, and 34 cm, respectively. Then draw the three segments connecting the midpoints of the triangle's sides and fold the resulting model along the drawn segments, so as to form a triangular-based pyramid. What is the volume of this pyramid?	Not supported with pagination yet
1770	2001	autumn games	no	L2	Geometry	The volume of the pyramid is 408 cm3	A bulldog, a poodle, and a dalmatian are tied to a lamppost. We know that: • 13 m separate the bulldog from the poodle, 14 m the poodle from the dalmatian, and 15 m the dalmatian from the bulldog; • the bulldog views the poodle's leash at the same angle as the poodle views the dalmatian's leash; • the poodle views the dalmatian's leash at the same angle as the dalmatian views the bulldog's leash; • the dalmatian views the bulldog's leash at the same angle as the bulldog views the lamppost. The three leashes, perfectly taut, are horizontal and fixed at the base of the lamppost. The bulldog's leash measures 2.35 m. What is the height of the lamppost?	Not supported with pagination yet
1773	2001	team games	no	C2 L1 L2	Algebra	The sum is equal to 10	Let x and y be two positive integers such that the sum of their sum and their product equals 34. What is x+y equal to?	Not supported with pagination yet
1774	2001	team games	no	C2 L1 L2	Geometry	Number of points: 7	What is the largest number of points that can be placed in a circle with a radius of 100 m (including the circumference) such that the distance between any two of them is at least 100 m?	Not supported with pagination yet
1775	2001	team games	no	C2 L1 L2	Logic	Number of points: 5	Solve the previous problem, where it is now required that the distance between any two points is greater than 100 m.	Not supported with pagination yet
1776	2001	team games	no	C2 L1 L2	Pattern Recognition	Number one: 152076	Peter has nothing to do and, to kill time, writes -in words! -all the first 2001 multiples of 2001: two thousand one, four thousand two, six thousand three, .... Angelo also has the problem of how to pass the time, and so he decides to copy Peter's 2001 numbers in alphabetical order. Which one will be the first?	Not supported with pagination yet
1777	2001	team games	no	C2 L1 L2	Geometry	Number of faces: 5	There are two pyramids, one with a triangular base and the other with a square base. The edges of the two pyramids have the same length. Glue the two pyramids together along one of their triangular faces (and the triangles that come into contact in this way perfectly coincide). What is the number of faces of the solid thus obtained?	Not supported with pagination yet
1778	2001	team games	no	C2 L1 L2	Geometry	Minimum number of cuts: 9	There is a cube, formed by gluing together 125 small identical cubes. Now, one wants to have these 125 cubes separated again. What is the minimum number of cuts required to obtain all 125 individual cubes, given that between one cut and the next, the obtained pieces can be moved and rearranged as desired?	Not supported with pagination yet
1779	2001	team games	no	C2 L1 L2	Combinatorics	Maximum number of islands: 10	The islands of the Playful Archipelago have this characteristic: each is connected, by a bridge, to at most three others; furthermore, it is always possible to go from any island to any other, with at most one intermediate stop. What is the maximum number of islands in the archipelago?	Not supported with pagination yet
1781	2001	team games	no	C2 L1 L2	Logic	The three solutions are: 268 x 4 = 1072; 394 x 4 = 1576; 398 x 9 = 3582	In each of the following square brackets, erase one digit so that the remaining digits form an exact multiplication: [32] [96] [84] x [ 94 ] = [31] [50] [87] [62] The problem has three solutions: find them all.	Not supported with pagination yet
1784	2001	team games	no	C2 L1 L2	Combinatorics	120	The product of two consecutive (positive) integers is always divisible by 2; the product of 3 consecutive numbers is always divisible by 2, by 3, and by 6; the product of 5 consecutive numbers is .... Write the largest number by which the product of 5 consecutive numbers is always divisible.	Not supported with pagination yet
1785	2001	team games	no	C2 L1 L2	Logic	DAVIDE	Davide, Enrico and Matteo meet up, after quite some time, at the bar and exchange some confidences. Davide: "I haven't found my soulmate yet." Enrico: "Neither have I, I've found her." Matteo: "Enrico is lying." Davide: "Matteo is telling the truth." In reality, only one of the three friends is lying. Which of the three surely has not found their soulmate?	Not supported with pagination yet
1786	2001	team games	no	C2 L1 L2	Logic	27 years	When my daughter was born, I don't remember how old I was; however, I was definitely over 20 years old and not yet 30. Now, if I swap the digits of my age, I get my daughter's age. How old was I when my daughter was born?	Not supported with pagination yet
1787	2001	team games	no	C2 L1 L2	Algebra	The trip to Paris costs 440000 lire	Three friends each won a prize. The total prize money is 1,200,000 lire. The first one says: "What I won would allow me to take a trip to Paris; however, I'm missing 20,000 lire." The second one says: "I would also like to go; however, I'm missing 40,000 lire." The third one says: "I'm missing 60,000 lire." How much does the trip to Paris cost?	Not supported with pagination yet
1788	2001	team games	no	C2 L1 L2	Logic	876	To open a safe, one must know the three digits (from 1 to 9) of the code in order. Here are the attempts made by someone: * 1 2 3 -> all the digits are wrong * 4 5 6 -> only one digit is correct and is in the right place * 6 1 2 -> only one digit is correct but is not in the right place * 5 4 7 -> only one digit is correct but is not in the right place * 8 4 9 -> only one digit is correct and is in the right place. Can you open the safe?	Not supported with pagination yet
1792	2001	semifinal	no	C1 C2	Geometry	The perimeter of Mr. Penterba's land therefore measures 250 meters.	Mr. Penterba, a famous nurseryman, has decided to divide his square plot of land into five rectangular plots in order to have magnificent rows of carnations, roses, dahlias, asters, and zinnias. The perimeter of each plot is 150 meters. What is the perimeter of Mr. Penterba's land?	Not supported with pagination yet
1793	2001	semifinal	no	C1 C2	Arithmetic	Our book has 96 pages.	When our printer wanted to number the pages of his latest book, from page 1 to the last, he made an error. He indeed printed "6"s instead of all "9"s (while correctly printing all other digits). The numbering of all pages required 36 "6" digits. How many pages does the book have?	Not supported with pagination yet
1794	2001	semifinal	no	C1 C2 L1 GP	Logic	It's Emy the frog who finds Prince Charming.	Erny, Valentina, and Susaima are conversing, more or less amicably, on a beautiful water lily. V.: "I have not found Prince Charming." E.: "Neither have I, I found him." S.: "Erny is lying." V.: "Susaima is telling the truth." In reality, only one of the three frogs is lying. Which of the three has truly found Prince Charming?	Not supported with pagination yet
1796	2001	semifinal	no	C2 L1 L2 GP	Logic	That happens exactly 91 times.	My deskmate has a watch that works perfectly, but it's a bit strange: the hand that marks the seconds turns backward (while those that mark the minutes and hours turn normally). At the beginning of our Math test, exactly at 2 PM, his watch showed the correct time, without even a second of error. My deskmate handed in the test exactly at 2:45 PM. How many times, during the test, did his watch show the correct time (including the initial and final instants)?	Not supported with pagination yet
1797	2001	semifinal	no	C2 L1 L2 GP	Arithmetic	283,5 gold sovereigns (price paid by Be.Ta).	In a faraway land, sales taxes vary by province. In Algebraville, all sales are taxed at 15%. In Geometryburg, sales are initially taxed at 8%; then, a second tax of 5% is added to the price that includes the initial 8% tax. Alexander Fa and Benedict Ta bought the same book (thus with the same price, excluding taxes) in the two different provinces. Al. Fa paid 287.5 gold sovereigns for it in Algebraville. How much did Be. Ta pay for it, who bought his book in Geometryburg?	Not supported with pagination yet
1798	2001	semifinal	no	C2 L1 L2 GP	Algebra	at the time of departure, Annapesava's backpack was 2400 grams.	Chiara and Anna have organized a trek in Valnontey (a beautiful - non-virtual - site in the heart of Gran Paradiso National Park). At the start, they have two backpacks of the same weight. In the evening, after having eaten all the supplies that were in Chiara's backpack, they realize that it weighs 2/3 of Anna's (whose contents have remained unchanged from the start). Chiara and Anna then rebalance the backpacks, to give them the same weight, by transferring only some clothing items from Anna's backpack to Chiara's. Arriving at the end of their trip, they have run out of all provisions, and Anna's backpack has a weight that is 3/4 of Chiara's, which is 500 grams less. What was the weight of Anna's backpack at the moment of departure?	Not supported with pagination yet
1799	2001	semifinal	no	L1 L2 GP	Pattern Recognition	The smallest positive integer required is 249 (its square is in fact 62001).	What is the smallest positive integer whose square ends with the digits 2001?	Not supported with pagination yet
1800	2001	semifinal	no	L1 L2 GP	Algebra	The question has three solutions: 15, 18, 21	Angelo, Rosi, and Renato are all celebrating their birthday today. They have three different ages, and Angelo, the youngest of the three, is four years younger than Renato, who is the oldest. Passionate about numbers, they enjoy calculating all possible sums of two or three numbers chosen from their ages. By adding up the numbers obtained from these sums, they get a first result. They then calculate all positive differences between two of their ages, and by adding up the results of these differences, they obtain a second result. They then divide the first result by the second and... surprise!... they get Renato's age. What is this age?	Not supported with pagination yet
1802	2001	semifinal	no	L2 GP	Combinatorics	3712/6561	Omelia, after peeling her tangerine, observes that it is formed by eight identical segments. The greengrocer told her that the probability that each of the eight segments contains one or more seeds is equal to 1/3. What is the probability that there exists (at least) half a tangerine, consisting of four consecutive segments, that contains no seeds?	Not supported with pagination yet
1803	2001	final	no	C1	Arithmetic	Division 250: 8 gives the rest 2: On the last page there will be 2 photos.	Atma's parents plan to arrange the 250 most beautiful photos of their wedding in some albums. Each album consists of 12 pages, and each page can hold 8 photographs. How many photographs will there be on the last page used (keeping in mind that Atma's parents utilize all available spaces progressively)?	Not supported with pagination yet
1805	2001	final	no	C1 C2 L1 L2 GP	Algebra	The question admits two solutions: Michele spent 14 or 17 francs.	Donato and Michele spent the day at Mathlandia park in Paris. At noon, each of them had a sandwich and a drink. At the bar, there was a choice of: ham sandwiches (7 francs), cheese sandwiches (11 francs), or salmon sandwiches (14 francs); for drinks there was milk (6 francs), orange juice (7 francs), or an exotic fruit beverage (9 francs). Donato spent 6 francs more than Michele. How much did Michele spend? Write a solution.	Not supported with pagination yet
1809	2001	final	no	C2 L1 L2 GP	Arithmetic	23:46	It's 8 PM. Chiara's VCR shows 4 AM. Its clock is faulty and runs 15% faster than a normal clock. What time should Chiara program to record her favorite show, which starts tomorrow at 4:40 PM?	Not supported with pagination yet
1810	2001	final	no	L1 L2 GP	Geometry	This way, it is (in correspondence to A=4) that the smallest possible perimeter of the rectangle is 10 cm.	With a ruler and set square always at hand, Angelo enjoys drawing a rectangle and a rhombus. He then notices that his two quadrilaterals have not only the same perimeter, but also the same area. Furthermore, the lengths of the sides of the rectangle and the diagonals of the rhombus are expressed as integers (in centimeters). What is the smallest possible perimeter of Angelo's rectangle?	Not supported with pagination yet
1811	2001	final	no	L1 L2 GP	Combinatorics	13	How many different trapezoids exist that have a perimeter equal to 11 cm and all sides expressed as integers (in cm)? (Two trapezoids will be considered identical if one can be obtained from the other by rotation or by symmetry).	Not supported with pagination yet
1812	2001	final	no	L1 L2 GP	Pattern Recognition	The department code is 18.	Agent 007 must carry out a mission in a French department identified by a two-digit code. To find the first digit, one must consider the sequence 2587193460 7x; for the second digit, the sequence 0491329000 00. Each of these two sequences consists of 10 digits (chosen from 0 to 9 inclusive), followed by a two-digit key belonging to the set {0; 1; 2; 3; 4; 5; 6; 7; 8; 9; x} where x corresponds to the number 10. The first digit of the key is the remainder of the division by 11 of the sum of the first ten digits; the second digit of the key is equal to the remainder of the division by 11 of the sum of the products of the first ten digits by their position in the sequence. Each of the two sequences reported above contains a deliberately incorrect symbol. The two correct symbols (which replace the two incorrect symbols) give, in order, the code of the French department. What is this code?	Not supported with pagination yet
1672	2002	autumn games	no	C1	Logic	The number designed by Michele is 21	Donato must guess an integer that Michele has chosen in great secret. Here is the information that Donato collects, little by little and in a disordered way. The number to find: is less than 32; greater than 18; less than 22; greater than 16; less than 24; greater than 20 and less than 28. What is the number Michele thought of?	Not supported with pagination yet
1673	2002	autumn games	no	C1	Arithmetic	The goal was achieved by pressing 7 keys	On the calculator, today, only three keys are working: the "C" key, which resets the number displayed on the screen to 0; the "+2" key, which adds 2 to the displayed number; the "+3" key, which adds 3 to the displayed number. Initially, the number "1987" (the year of the first "International Mathematical Games Championship") appears on the screen, and Rosi wants to make "17" appear instead. What is the minimum number of times Rosi must press a key on the calculator?	Not supported with pagination yet
1677	2002	autumn games	no	C1 C2	Combinatorics	81	Carla wrote the number 12 as a sum of several natural numbers. Then she multiplied all the terms of the sum together. What is the maximum value of the result of her product?	Not supported with pagination yet
1679	2002	autumn games	no	C1 C2	Combinatorics	Renato, in all, can write 64 numbers.	Renato writes two-digit numbers (from 10 to 99, inclusive) such that: - the tens digit and the units digit are never equal; - the tens digit and the units digit are never consecutive (for example, he will not write the numbers 45 or 87, as they do not satisfy this condition). What is the maximum number of different numbers Renato can write while respecting these conditions?	Not supported with pagination yet
1680	2002	autumn games	no	C2 L1	Combinatorics	In total, we will have, at least, 25	Marco places four standard dice (where the sum of opposite faces is 7) on a table, in close contact with each other, such that their top faces form a square and the numbers on the four top faces are all different. He then calculates the sum of all the numbers written on the visible faces of the dice. What is the minimum value of the sum obtained by Marco?	Not supported with pagination yet
1681	2002	autumn games	no	C2 L1	Geometry	The perimeter of Milena's mosaic measures 100 cm.	Milena begins to create a mosaic (without gaps) using three squares and six equilateral triangles, all of which have sides of 10 cm. The joints between the tiles of the mosaic (assembled edge-to-edge) have a total length of 1 meter. What is the perimeter of Milena's mosaic?	Not supported with pagination yet
1682	2002	autumn games	no	C2 L1	Geometry	664; 329; 128; 61	A trapezoid has an area equal to 335 cm² and its minor base measures 6 cm. The lengths of its height and major base are both expressed as integer numbers of centimeters. What is the length of the trapezoid's major base?	Not supported with pagination yet
1683	2002	autumn games	no	C2 L1	Algebra	There are two solutions: Jacob's savings amount to 109 Euro or 229 Euro.	With his savings, which are under 250 Euros, Jacob wants to buy CDs and books as gifts for his friends. CDs cost 15 Euros each; books cost 8 Euros each. If Jacob were to buy only CDs, he would be 11 Euros short of buying one more; if instead he were to buy only books, he would have 5 Euros left over. How much are Jacob's savings?	Not supported with pagination yet
1684	2002	autumn games	no	L1 L2	Algebra	The system admits as solutions (whole) m = 11, n = 5 The volume of the parallelepiped is 440 cm3	A rectangular parallelepiped has one dimension that is the arithmetic mean of the other two. The sum of the three dimensions is 24 cm, the three dimensions are integers (in centimeters), and the total surface area of the solid is 366 cm². Calculate the volume of the parallelepiped.	Not supported with pagination yet

1913

1999

semifinal

no

L1 L2 GP

Pattern Recognition

49

The smallest integer whose digits sum to 1 is the number 1. The smallest integer whose digits sum to 2 is the number 2. The smallest integer whose digits sum to 3 is the number 3... The smallest integer whose digits sum to 10 is the number 19. The smallest integer whose digits sum to 11 is the number 29, etc. If we repeat this procedure and write down the sequence of numbers thus obtained, we get: 1, 2, 3, ... 19, 29, ... What is the largest number in this sequence that is the square of an integer? Answer 0 (zero) if you think this number does not exist.

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1914

1999

semifinal

no

L1 L2 GP

Geometry

45000 cubic centimeters

An aquarium placed on a table has the shape of a rectangular prism with a height of 30 cm. It is filled with water to the brim and then tilted around one of the edges of its base so that the bottom forms an angle of 45° with the plane of the table. In this way, one third of its content spills onto the table. Now it is filled again to the brim and tilted around the other edge of its base so as to form an angle of 45° again with the plane of the table. In this way, 4/5 of the content now spills onto the table. What is the volume of the aquarium in cm³?

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1916

1999

semifinal

no

L2 GP

Geometry

3 solutions: 24° -72° -168°

Two discs A and B with centers O and P, respectively, externally tangent, perform the following two-step dance movement: A begins to rotate around B, clockwise, such that its center forms an angle α. Then, it is B's turn to rotate around A, always clockwise, and form an angle α/2 around point O. The dancing discs perform 10 complete movements of this dance, after which they return to their starting position for the first time. Give the value of the angle α in degrees.

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1926

1999

international final day 1

no

C2 L1 GP L2 HC

Geometry

3 solutions (3;4;4), (3.6;8), (6; 6.7)

Matteo measured the lengths of the sides of the triangle he drew. He sums the three measurements, which are all integer centimeters, and obtains a perimeter of 15 cm. However, the teacher points out that the result is inaccurate. Matteo made no calculation errors and correctly used the ruler, taking all measurements starting from the zero mark of the scale. But he realizes that the ruler's scale has a small defect. What are the (exact) lengths of the three sides of Matteo's triangle? (express the three lengths in cm, in increasing order)

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1928

1999

international final day 1

no

L1 GP L2 HC

Combinatorics

165x125x300+181x25x1=6192,025 kg

An architect has decided, to decorate a city, to create a 5-story pyramid, formed by identical wooden cubes. The architect knows that it will need to be coated with paint to protect it from the elements. Therefore, they had a 1/5 scale sketch made of one cube from the pyramid. They thus realize that the reduced cube weighs 300 g and, when painted, 306 g. What will be the weight of the painted pyramid? N.B. Only the visible parts of the pyramid will be painted. Disregard paint on the edges. Give the result in kg, rounded to the nearest gram.

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1930

1999

international final day 1

no

L1 GP L2 HC

Arithmetic

2/5

Giuliano encounters some difficulty with fractions. He replaces the fraction bar with a comma and, instead of multiplying, he divides. So, for example, to multiply 12 x 6/25 he divides 12 by 6.25. Today, proceeding in the same way, by multiplying a non-zero number by an irreducible fraction, he obtained a correct result. By what fraction did Giuliano multiply the number?

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1931

1999

international final day 1

no

L2 HC

Geometry

3 solutions (7;24;25), (8;15;17), (9;12;15)

The side lengths of a right triangle are all integers in cm. An inscribed circle, tangent to the three sides, has a radius of 3 cm. What are the dimensions of the triangle?

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1932

1999

international final day 1

no

L2 HC

Arithmetic

4 solutions: 9, 18, 54, 774 pearls

A pharaoh had a certain number of magnificent pearls hidden. This number, known to everyone, was considered magical, as were all its divisors, which were the only other magical numbers. After the pharaoh's sudden death, 5 thieves, who had discovered the hiding place, decided to take advantage of it. They arrived at night, and, one after another, each stole the maximum possible number of pearls, this number being magical. The last thief was surprised to find only one! In the morning, the scribe responsible for the treasury, finding the hiding place empty, was pleased to have taken for himself, before the five thieves' visit, a number of pearls (not magical) equivalent to 3/100 of the treasure! How many pearls had the indelicate scribe stolen?

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1817

2000

team games

no

C2 L1 L2

Geometry

12100 m2

A magnificent willow is planted inside a square plot of land. The sum of its distances from two sides of the square is 100 m, while the sum of the distances from the other two sides is equal to 120 m. What is the area (in square meters) of the land?

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1818

2000

team games

no

C2 L1 L2

Algebra

10 dividers

The product of all divisors of a natural number (greater than 1) is equal to the fifth power of this number. How many divisors does the number in question have?

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1819

2000

team games

no

C2 L1 L2

Algebra

42, 53, 76

Three two-digit numbers are formed using the digits 2, 3, 4, 5, 6, 7 (not repeated). The sum of the three numbers is 171; the difference between the two smallest is 11. Find the three numbers (giving them in increasing order).

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1820

2000

team games

no

C2 L1 L2

Combinatorics

2 solutions: 2, 6

On a circumference of length 24 cm, some points are placed such that the arcs of the circumference between two consecutive points measure either 2 or 3 cm. Furthermore, by connecting any two points, a diameter of the circumference is never formed. How many 3 cm long arcs of the circumference are formed by the points in question?

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1821

2000

team games

no

C2 L1 L2

Geometry

72 cm

Divide a rectangle into five similar right triangles. When two of these five triangles are not disjoint, they share either a vertex or an entire side, which is then the hypotenuse of one of the two triangles and a leg of the other. The area of the smallest triangle is 2 square cm. What is the area of the rectangle?

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1822

2000

team games

no

C2 L1 L2

Logic

8 km

Mr. Dubbio, who is a sales representative, lives on State Road No. 7. Today, starting from home to see a client, he made a first trip of 53 km. Subsequently, he made other trips, of 79, 27, and 9 km respectively. Unfortunately, he doesn't remember in which direction he made each of these journeys. "Anyway," he thinks, "I am at most 168 km away from home." What is his minimum distance from home, instead? (give the answer in km)

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1823

2000

team games

no

C2 L1 L2

Combinatorics

8

Giulia and Bernardo's game involves writing a multi-digit number. The player who starts writes the first digit (which must be non-zero); subsequently and alternately, each player writes a digit to the right of the digit or digits already written. However, players must follow these rules: * after a 9, any digit can be written * after a digit less than 9, a larger digit must be written * each digit can appear (in the final number) at most three times The player who, first, cannot write any digit loses. Giulia starts. What digit must she write to be sure of winning, regardless of Bernardo's play? (answer 0 if you think such a winning strategy does not exist)

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1825

2000

team games

no

C2 L1 L2

Combinatorics

12

Anna likes to write all possible decimal numbers, using only a 1, a 2, a 3, and a decimal point. How many different numbers can Anna write?

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1826

2000

team games

no

C2 L1 L2

Geometry

60

Emi has 120 identical small cubes: 80 blue and 40 white. She wants to use them to build a large rectangular parallelepiped using glue. Its surface will be formed entirely by the faces of the small cubes. What is the minimum number of visible faces of the small cubes that will be blue?

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1827

2000

team games

no

C2 L1 L2

Arithmetic

17

In a "normal" die, the sum of the points on two opposite faces is always equal to 7. Enrico placed three "normal" dice on a table, one on top of the other to form a tower. On the top face of the die at the top of the tower, there is a 4. What is the sum of the points on the five hidden faces (including those between two dice or the table)?

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1828

2000

team games

no

C2 L1 L2

Combinatorics

9

Eight students (from a first and a second high school class) participated in a competition, all recording distinct scores. The winner scored 8 points; the last placed student scored 1 point. The second-year students collectively scored 18 points. In the ranking, between any two second-year students, there is always at least one first-year student. Gianni and Gianna are the only first-year students who are not separated (in the ranking) by a second-year student. How many points did Gianni and Gianna score together?

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1833

2000

team games

no

C2 L1 L2

Geometry

522

For his birthday, Mauro received a radio-controlled car model, which can only move forward, either in a straight line or by describing circular arcs with a radius of 63 cm. Currently, the car is at point A, facing North. What is the minimum distance Mauro must make his toy travel so that it returns to the same point A, but oriented South? (let pi = 22/7).

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1834

2000

semifinal

no

C1

Logic

4 children

In a family, each of the children can state that they have at least one brother and one sister. What is the minimum number of children in this family?

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1836

2000

semifinal

no

C1 C2 L1 L2 GP

Logic

89690

Ennio didn't want to give me the postal code of his city. Here's how he answered my questions about it: - Like every Italian postal code, it consists of five digits. - The sum of the first digit and the second is 17. - The sum of the second and the third is 15, as is the sum of the third and the fourth. - The sum of the last two digits is 9. - The sum of the last and the first is 8. What is the postal code number of Ennio's city?

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1837

2000

semifinal

no

C1 C2 L1 L2 GP

Combinatorics

31 fruit

Angelo and Rosi have invited seven friends to dinner this evening. At the end of the meal, they would like to offer fresh fruit, apples and pears, which they will pick from their orchard. However, it is quite far from the house, and Angelo and Rosi, being advanced in years, know that together they cannot carry more than 7 kilograms of fruit. On the other hand, they want each of their guests to be able to choose which fruits to eat. An apple weighs 300 g; a pear weighs 200 g. What is the maximum number of fruits they can pick by going to the orchard only once?

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1838

2000

semifinal

no

C1 C2 L1 L2 GP

Combinatorics

3 shapes

For Jacob's twelfth birthday, his parents ordered some very special cakes from the pastry chef... in the shape of a triangle with a perimeter of 12 cm. All sides of the triangles have a measurement in cm corresponding to an integer. How many different shapes can the pastry chef make?

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1839

2000

semifinal

no

C1 C2 L1 L2 GP

Algebra

108 pages

To number the pages of a large notebook, Peter had to write a number of digits double the number of pages in this notebook. How many pages does Peter's notebook have?

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1841

2000

semifinal

no

C2 L1 L2 GP

Logic

Monday

My four cousins arrive at our house on Sunday morning at breakfast time and are staying for twelve days of vacation. They have a big sweet tooth, just like us! Fortunately, Mom, thinking ahead, has bought 168 chocolate bars so that everyone can, during the twelve days, eat one for breakfast and one for a snack. Unfortunately, on the evening of the ninth day, our cousins have to cut short their stay and return home. We continue, despite their absence, to enjoy the chocolate bars with the same frequency. On what day of the week will we eat the last bar?

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1842

2000

semifinal

no

C2 L1 L2 GP

Combinatorics

44

During the last games before the basketball cup final, we saw a scout from our future opposing team in the stands. They were taking notes about our usual tactics. At this point, we must disrupt our opponents' expectations. We have thus decided to redistribute our five numbered jerseys in such a way that none of us five wears their usual jersey. In how many ways can we perform this redistribution?

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1844

2000

semifinal

no

L1 L2 GP

Combinatorics

3 solutions 132 • 264 • 396

On Mauro's calculator, of the keys from 1 to 9, only three of them still work. Mauro sums the six numbers (of two distinct digits) that he can form using only these three keys. Amazingly! The sum that appears still uses the same three digits. What is this sum?

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1849

2000

final

no

C1

Combinatorics

Eventually visualizing ladders and steps, it occurs that the frog, in order to reach the top of the scale, must make 12 jumps.

A small frog is at the base of a staircase consisting of 21 steps. With its first jump, it reaches the second step; then it continues to climb the stairs with jumps of two steps. Unfortunately, the steps numbered 5, 10, 15, and 20 are slippery and, when the frog lands on one of these, it slips and goes down one step. How many jumps does the frog have to make to reach the twenty-first step?

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1850

2000

final

no

C1 C2 L1 L2 GP

Logic

It occurs in other words that the text (as it was formulated in the Italian translation) admitted as a solution all the natural numbers between 39 and 49

Jacob can build at most 9 identical towers using all the chocolates from a box, which contains a number of chocolates between 39 and 49. He then realizes that he has as many chocolates remaining as the number of towers he built. How many chocolates did the box contain exactly?

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1853

2000

final

no

C1 C2 L1 L2 GP

Geometry

Each of the following responses was considered valid: 8, 9, 10, 11, 12

Fabrizio draws five straight roads on the map of the holiday village in such a way that three of the five roads intersect at the same point and three of the five roads are parallel. Into how many regions do these five roads divide the village?

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1857

2000

final

no

L1 L2 GP

Pattern Recognition

2024

What will be the next year in which the month of February will have 5 Thursdays?

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1858

2000

final

no

L1 L2 GP

Combinatorics

It's 18 miles that can't be patrolled.

Police officers Donato, Michele, Angelo, Renato, and Pietro patrol a 120 km road. Each one covers exactly one segment of this road. Donato and Michele patrol two different halves. Donato, Michele, and Angelo patrol three different thirds. Donato, Michele, Angelo, and Renato patrol four different quarters. Donato, Michele, Angelo, Renato, and Pietro patrol five different fifths. What is the minimum total length of the segments of this road not patrolled by any police officer?

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1859

2000

final

no

L2 GP

Logic

In the worst case, to turn on the lights, Emy will have to make a maximum of 5 attempts

Ermy enters an unlit room equipped with 3 switches numbered 1, 2 and 3, whose state (on or off) she does not know. Each switch can be on or off, and for the room to be lit, all three must be on. Ermy presses switch 1, and the room does not light up. She then presses switch 2, but the room still does not light up. She then seeks a strategy that allows her to light the room in a minimum number of attempts. Applying this strategy, what sequence of switches must Ermy operate in the worst-case scenario?

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1860

2000

final

no

L2 GP

Combinatorics

0

Angelo and Rosi play with the numbers from the set {2, 3, 4, 5, 6, 7, 8, 9}. Angelo starts by writing a number. Then it's Rosi's turn; she chooses another number (always from the set), multiplies it by the previously written number, and writes the product. Subsequently, one at a time, each player picks an unchosen number from the set, multiplies it by the last written number, and writes the product. The first player who writes a number greater than 1000 loses. What number must Angelo choose to be sure to win, regardless of Rosi's moves?

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1861

2000

international final day 1

no

C1

Algebra

Number of lianas: 8

In the forest, Tarzan moves in a straight line from liana to liana. There are two types of lianas: short ones that allow for jumps of 4 m, and long ones that allow for jumps of 7 m. Tarzan wants to reach a boulder located 41 m from the edge of a pond. How many lianas must he use?

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1863

2000

international final day 1

no

C1

Algebra

1st dice: 1 2nd dice: 2 3rd dice: 4

While I was playing with three standard dice (whose faces are numbered from 1 to 6), Dedé arrived and told me: "Roll the three dice without letting me see them. Multiply the number rolled on die number 1 by 30, add 5 to the result. Add the number rolled on die number 2, then multiply the obtained result by 10. Finally, add the number rolled on die number 3. What result do you get?" I answered 374 and Dedé guessed the three numbers rolled on my dice. State what the three numbers rolled on each die were.

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1865

2000

international final day 1

no

C1 C2 L1 GP L2 HC

Algebra

Age of daughter: 26 age of father: 52

My age is double that of my daughter. The current year is 2000. In 2011, our combined ages will total a century. What are our respective ages today?

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1866

2000

international final day 1

no

C1 C2 L1 GP L2 HC

Algebra

78 bicherees and 39 daily

At the beginning of the century, in the Isère region, two units for measuring area existed: - the "bicberée", equal to 1600 square meters - the "giornale", equal to 1800 square meters Robert Vassel owned a property that measured twice as many "bicberées" as "giornali", for a total area of 195000 square meters. What is the number of "bicberées" and "giornali" for this property?

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1867

2000

international final day 1

no

C1 C2 L1 GP L2 HC

Algebra

Number of vegetarian dishes: 504

Thomas is the director of a fast-food chain (with three outlets) that offers three dishes on the menu every day: a couscous dish, a fish dish, and a vegetarian dish. Each of the chain's three restaurants has just sent a pie chart illustrating the breakdown of sales for the three dishes offered. Strangely, all three diagrams show a 120-degree angle, and for all three restaurants, the data reads: 222 couscous dishes and 114 fish dishes. Nevertheless, the number of vegetarian dishes sold is different. How many vegetarian dishes did each of the three restaurants sell?

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1871

2000

international final day 1

no

L1 GP L2 HC

Geometry

1 Solution 12 km

Matilde, José, and Mattia live in the same region. As the crow flies, the houses of Matilde and José are an integer number of kilometers apart, and those of Matilde and Mattia are an integer number of kilometers apart. Furthermore, again as the crow flies, there are exactly 10 more kilometers between José's house and Mattia's house than between José's house and Matilde's house. Finally, the triangle formed by the houses of the three friends is a right-angled triangle. What is the minimum distance between Matilde's house and Mattia's house?

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1874

2000

international final day 1

no

L2 HC

Geometry

Perimeter: 1394 m

Father XXL's field has the shape of a quadrilateral formed by two right triangles. All the side lengths of the quadrilateral are expressed in an integer number of meters. The perimeter of this quadrilateral is less than 2000 m. What is the maximum possible value for this perimeter?

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1875

2000

international final day 1

no

L2 HC

Combinatorics

Number: 165

Mattia, who has time to spare today, has decided to fill a large 100 by 200 grid with numbers. In the first row, he wrote, in order, 1, 2, 3, ..., 100. In the first column, he wrote, in order, 1, 2, 3, ..., 200. He continues to fill the grid according to the following rule: each cell contains the smallest strictly positive integer that is different from all numbers written in the portion of the row to its left and from all numbers written in the portion of the column above it. What number is in the last cell of the last column?

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1876

2000

international final day 2

no

C1

Pattern Recognition

PARIS code: 33 3 37 19 39

I managed to decode the following message: Coded word: 27 3 37 39 11 19 25 25 11 - 1 26 2 36 38 10 18 24 24 10 : 2 13 1 18 19 5 9 12 12 5 Decoded word: M A R S E I L L E Which coded word would correspond to PARIS?

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1878

2000

international final day 2

no

C1 C2 L1 GP L2 HC

Logic

Duration 7 hours and 15 minutes

This summer I'm going on vacation to Sildavia. Here are the flight times, for the outbound and return journeys (expressed in local time): departure from Paris at 23:30, arrival in Sildaville at 9:45 the next day. Departure from Sildaville: 11:00, arrival in Paris at 15:15 on the same day. The flight duration is the same for both the outbound and return journeys. What is this duration?

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1879

2000

international final day 2

no

C1 C2 L1 GP L2 HC

Combinatorics

Order of departure: 7 - 1 - 5 - 10 - 2 - 9 - 3 - 6 - 8 - 4

I have a deck of 10 playing cards. I place the top card at the bottom of the deck, then I turn over the next card onto the table: it's an ace. I place the top card at the bottom of the deck for two consecutive times, then I turn over the next one: it's a 2. I put the next card at the bottom of the deck, then I turn over a card: it's a 3. I continue in this manner, alternatively placing a card at the bottom of the deck once or twice, then turning over the next one. The cards turned over, in order, are: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10. What was the initial order of the cards?

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1884

2000

international final day 2

no

C2 L1 GP L2 HC

Combinatorics

3 solutions: 6, 8, 10

They arrived discreetly, all dressed in black. I observed them closely. Each of them shook hands with exactly three others, except for one of them, who shook hands with only one person. I didn't count them, but there were fewer of them than a complete football team. How many could there have been? Note: A complete football team consists of 11 players.

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1885

2000

international final day 2

no

L1 GP L2 HC

Algebra

3 solutions. Age of major: 6, 15, 18

On their trip to Gori, Russia, the Pitt family, who have four children, discovered the following fact. The sum of the ages of the oldest and the youngest is equal to that of the other two. However, the product of the ages of the oldest and the youngest is half the product of the ages of the other two. The oldest is less than 20 years old. What is his age?

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1886

2000

international final day 2

no

L1 GP L2 HC

Algebra

Each one will pay an extra FF 5.5

Romain and Julien are not great mathematicians: to do their homework, they invited Sophie to the high school cafeteria. They ordered coffee, croissants, and chocolate bars. In this order, for Romain: 1, 3, 7; for Julien: 1, 4, 10 (what an eater!!); for Sophie: 1, 1, 1 (she had to work!). Romain, for his consumption, must pay 29 FF and Julien 38 FF. They wish to thank Sophie by splitting what she has to pay in half. How much more will each of them pay?

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1888

2000

international final day 2

no

L2 HC

Algebra

2 solutions: 570, 580

Audrey is spending her holidays in Syldavia. In this country, coins are out of circulation, and there are only three types of banknotes, worth respectively 57, 62, and 72 crowns. Yesterday, Audrey bought some croissants from the baker, for a cost of 4 crowns. She gave the baker a certain number of banknotes, for an amount less than 600 crowns, and the baker gave her the exact change for her purchase. What sum did Audrey give to the vendor?

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1889

2000

international final day 2

no

L2 HC

Combinatorics

The probability is 25/256

The magician has 13 cards, which he fans out. He has a spectator choose two consecutive cards. It is assumed that the spectator chooses randomly, and that all choices are equiprobable. The magician re-forms the fan, without changing the order of the cards, and has another spectator choose two consecutive cards, and so on, until only one card remains in his hand. Before beginning, he placed the Ace of Hearts in the center of the deck. What is the probability that he can triumphantly brandish the Ace of Hearts at the end of the game?

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1753

2001

autumn games

no

C1

Logic

Pinocchio took.13

In France – as you may know – grades are given out of twenty. Read this dialogue between Pinocchio and his father about his school results, but be careful: Pinocchio has developed the bad habit of never telling the truth! Geppetto: - Did you get more than 12 on the math test? Pinocchio: - No, Dad. G.: - Then you obviously got less than 14? P.: - No, Dad. G.: - I don't understand. But is your grade an even number? P.: - Yes, Dad. What grade did Pinocchio get? (It is an integer).

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1756

2001

autumn games

no

C1 C2

Algebra

There are two solutions: 3; 7

On a large sheet of paper, Jacob has drawn several squares and several triangles. He counts the sides of the drawn figures (all separate) and finds a total of 29. How many triangles does Jacob have?

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1757

2001

autumn games

no

C1 C2

Combinatorics

20

The number 2002 is a four-digit number whose sum is 4: 2+0+0+2=4. Including this example, how many four-digit numbers are there for which the sum of their digits is equal to 4? Note: No number can begin with zero.

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1760

2001

autumn games

no

C2 L1

Algebra

20 cm

To determine if an individual is overweight, the World Health Organization has established a "Body Mass Index", which is calculated by dividing a person's weight (in kg) by the square of their height (in m). With his 81 kg, Renato has a BMI equal to 25, while Mauro, with his 80 kg, has a BMI of 20. What is the difference in height between Mauro and Renato, in cm?

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1761

2001

autumn games

no

C2 L1

Algebra

205 liters

There are two barrels that have the same capacity. Currently, the two barrels contain a total of 350 liters of Brunello di Montalcino. Then, after having drawn off 20 liters of the famous Tuscan wine from the first barrel and 80 liters from the second, it is noted that the remaining wine is exactly at the same level in the two barrels. What is the minimum capacity of each barrel?

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1762

2001

autumn games

no

C2 L1

Arithmetic

3

The bunch of black grapes consists of 183 individual grapes, while the bunch of white grapes comprises 252. My friends and I started eating the first bunch, sharing all its black grapes equally among us. We then ate the entire bunch of white grapes, and here too, each of us had an equal number of grapes. How many of us were there?

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1763

2001

autumn games

no

L1 L2

Arithmetic

0

Carla states that when she arranges the cards in piles of twelve, five are left over; when she arranges the same cards in piles of fifteen, four remain. What is the minimum number of cards that Carla must organize? Note: Reply zero if you think Carla miscounted and the problem as stated has no solution.

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1764

2001

autumn games

no

L1 L2

Algebra

7h 30min

Guido Piano often takes very long trips. He is very cautious and meticulous in organizing his itineraries. Now he has to travel 1695 km in three stages, which he completes at constant speeds of 50, 60, and 70 km per hour, respectively. The second stage lasts five quarters of the first, and the third lasts as long as the first two stages combined. What is the duration of the second stage, in hours and minutes?

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1765

2001

autumn games

no

L1 L2

Logic

Mr Ruffini

Which of the three following mathematicians and men of science lived (at least partially) in the 19th century: Isaac Newton, René Descartes, Paolo Ruffini?

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1767

2001

autumn games

no

L2

Combinatorics

People can line up in 100800 ways

There is a group composed of 4 men and 5 women. In how many different ways can the people in the group line up, knowing that an internal rule of the group prohibits men from occupying the first and the last position?

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1769

2001

autumn games

no

L2

Geometry

The area is 42 cm2

Enrico built a triangle whose sides measure 8√13, 30, and 34 cm, respectively. Then draw the three segments connecting the midpoints of the triangle's sides and fold the resulting model along the drawn segments, so as to form a triangular-based pyramid. What is the volume of this pyramid?

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1770

2001

autumn games

no

L2

Geometry

The volume of the pyramid is 408 cm3

A bulldog, a poodle, and a dalmatian are tied to a lamppost. We know that: • 13 m separate the bulldog from the poodle, 14 m the poodle from the dalmatian, and 15 m the dalmatian from the bulldog; • the bulldog views the poodle's leash at the same angle as the poodle views the dalmatian's leash; • the poodle views the dalmatian's leash at the same angle as the dalmatian views the bulldog's leash; • the dalmatian views the bulldog's leash at the same angle as the bulldog views the lamppost. The three leashes, perfectly taut, are horizontal and fixed at the base of the lamppost. The bulldog's leash measures 2.35 m. What is the height of the lamppost?

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1773

2001

team games

no

C2 L1 L2

Algebra

The sum is equal to 10

Let x and y be two positive integers such that the sum of their sum and their product equals 34. What is x+y equal to?

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1774

2001

team games

no

C2 L1 L2

Geometry

Number of points: 7

What is the largest number of points that can be placed in a circle with a radius of 100 m (including the circumference) such that the distance between any two of them is at least 100 m?

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1775

2001

team games

no

C2 L1 L2

Logic

Number of points: 5

Solve the previous problem, where it is now required that the distance between any two points is greater than 100 m.

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1776

2001

team games

no

C2 L1 L2

Pattern Recognition

Number one: 152076

Peter has nothing to do and, to kill time, writes -in words! -all the first 2001 multiples of 2001: two thousand one, four thousand two, six thousand three, .... Angelo also has the problem of how to pass the time, and so he decides to copy Peter's 2001 numbers in alphabetical order. Which one will be the first?

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1777

2001

team games

no

C2 L1 L2

Geometry

Number of faces: 5

There are two pyramids, one with a triangular base and the other with a square base. The edges of the two pyramids have the same length. Glue the two pyramids together along one of their triangular faces (and the triangles that come into contact in this way perfectly coincide). What is the number of faces of the solid thus obtained?

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1778

2001

team games

no

C2 L1 L2

Geometry

Minimum number of cuts: 9

There is a cube, formed by gluing together 125 small identical cubes. Now, one wants to have these 125 cubes separated again. What is the minimum number of cuts required to obtain all 125 individual cubes, given that between one cut and the next, the obtained pieces can be moved and rearranged as desired?

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1779

2001

team games

no

C2 L1 L2

Combinatorics

Maximum number of islands: 10

The islands of the Playful Archipelago have this characteristic: each is connected, by a bridge, to at most three others; furthermore, it is always possible to go from any island to any other, with at most one intermediate stop. What is the maximum number of islands in the archipelago?

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1781

2001

team games

no

C2 L1 L2

Logic

The three solutions are: 268 x 4 = 1072; 394 x 4 = 1576; 398 x 9 = 3582

In each of the following square brackets, erase one digit so that the remaining digits form an exact multiplication: [32] [96] [84] x [ 94 ] = [31] [50] [87] [62] The problem has three solutions: find them all.

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1784

2001

team games

no

C2 L1 L2

Combinatorics

120

The product of two consecutive (positive) integers is always divisible by 2; the product of 3 consecutive numbers is always divisible by 2, by 3, and by 6; the product of 5 consecutive numbers is .... Write the largest number by which the product of 5 consecutive numbers is always divisible.

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1785

2001

team games

no

C2 L1 L2

Logic

DAVIDE

Davide, Enrico and Matteo meet up, after quite some time, at the bar and exchange some confidences. Davide: "I haven't found my soulmate yet." Enrico: "Neither have I, I've found her." Matteo: "Enrico is lying." Davide: "Matteo is telling the truth." In reality, only one of the three friends is lying. Which of the three surely has not found their soulmate?

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1786

2001

team games

no

C2 L1 L2

Logic

27 years

When my daughter was born, I don't remember how old I was; however, I was definitely over 20 years old and not yet 30. Now, if I swap the digits of my age, I get my daughter's age. How old was I when my daughter was born?

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1787

2001

team games

no

C2 L1 L2

Algebra

The trip to Paris costs 440000 lire

Three friends each won a prize. The total prize money is 1,200,000 lire. The first one says: "What I won would allow me to take a trip to Paris; however, I'm missing 20,000 lire." The second one says: "I would also like to go; however, I'm missing 40,000 lire." The third one says: "I'm missing 60,000 lire." How much does the trip to Paris cost?

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1788

2001

team games

no

C2 L1 L2

Logic

876

To open a safe, one must know the three digits (from 1 to 9) of the code in order. Here are the attempts made by someone: * 1 2 3 -> all the digits are wrong * 4 5 6 -> only one digit is correct and is in the right place * 6 1 2 -> only one digit is correct but is not in the right place * 5 4 7 -> only one digit is correct but is not in the right place * 8 4 9 -> only one digit is correct and is in the right place. Can you open the safe?

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1792

2001

semifinal

no

C1 C2

Geometry

The perimeter of Mr. Penterba's land therefore measures 250 meters.

Mr. Penterba, a famous nurseryman, has decided to divide his square plot of land into five rectangular plots in order to have magnificent rows of carnations, roses, dahlias, asters, and zinnias. The perimeter of each plot is 150 meters. What is the perimeter of Mr. Penterba's land?

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1793

2001

semifinal

no

C1 C2

Arithmetic

Our book has 96 pages.

When our printer wanted to number the pages of his latest book, from page 1 to the last, he made an error. He indeed printed "6"s instead of all "9"s (while correctly printing all other digits). The numbering of all pages required 36 "6" digits. How many pages does the book have?

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1794

2001

semifinal

no

C1 C2 L1 GP

Logic

It's Emy the frog who finds Prince Charming.

Erny, Valentina, and Susaima are conversing, more or less amicably, on a beautiful water lily. V.: "I have not found Prince Charming." E.: "Neither have I, I found him." S.: "Erny is lying." V.: "Susaima is telling the truth." In reality, only one of the three frogs is lying. Which of the three has truly found Prince Charming?

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1796

2001

semifinal

no

C2 L1 L2 GP

Logic

That happens exactly 91 times.

My deskmate has a watch that works perfectly, but it's a bit strange: the hand that marks the seconds turns backward (while those that mark the minutes and hours turn normally). At the beginning of our Math test, exactly at 2 PM, his watch showed the correct time, without even a second of error. My deskmate handed in the test exactly at 2:45 PM. How many times, during the test, did his watch show the correct time (including the initial and final instants)?

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1797

2001

semifinal

no

C2 L1 L2 GP

Arithmetic

283,5 gold sovereigns (price paid by Be.Ta).

In a faraway land, sales taxes vary by province. In Algebraville, all sales are taxed at 15%. In Geometryburg, sales are initially taxed at 8%; then, a second tax of 5% is added to the price that includes the initial 8% tax. Alexander Fa and Benedict Ta bought the same book (thus with the same price, excluding taxes) in the two different provinces. Al. Fa paid 287.5 gold sovereigns for it in Algebraville. How much did Be. Ta pay for it, who bought his book in Geometryburg?

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1798

2001

semifinal

no

C2 L1 L2 GP

Algebra

at the time of departure, Annapesava's backpack was 2400 grams.

Chiara and Anna have organized a trek in Valnontey (a beautiful - non-virtual - site in the heart of Gran Paradiso National Park). At the start, they have two backpacks of the same weight. In the evening, after having eaten all the supplies that were in Chiara's backpack, they realize that it weighs 2/3 of Anna's (whose contents have remained unchanged from the start). Chiara and Anna then rebalance the backpacks, to give them the same weight, by transferring only some clothing items from Anna's backpack to Chiara's. Arriving at the end of their trip, they have run out of all provisions, and Anna's backpack has a weight that is 3/4 of Chiara's, which is 500 grams less. What was the weight of Anna's backpack at the moment of departure?

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1799

2001

semifinal

no

L1 L2 GP

Pattern Recognition

The smallest positive integer required is 249 (its square is in fact 62001).

What is the smallest positive integer whose square ends with the digits 2001?

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1800

2001

semifinal

no

L1 L2 GP

Algebra

The question has three solutions: 15, 18, 21

Angelo, Rosi, and Renato are all celebrating their birthday today. They have three different ages, and Angelo, the youngest of the three, is four years younger than Renato, who is the oldest. Passionate about numbers, they enjoy calculating all possible sums of two or three numbers chosen from their ages. By adding up the numbers obtained from these sums, they get a first result. They then calculate all positive differences between two of their ages, and by adding up the results of these differences, they obtain a second result. They then divide the first result by the second and... surprise!... they get Renato's age. What is this age?

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1802

2001

semifinal

no

L2 GP

Combinatorics

3712/6561

Omelia, after peeling her tangerine, observes that it is formed by eight identical segments. The greengrocer told her that the probability that each of the eight segments contains one or more seeds is equal to 1/3. What is the probability that there exists (at least) half a tangerine, consisting of four consecutive segments, that contains no seeds?

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1803

2001

final

no

C1

Arithmetic

Division 250: 8 gives the rest 2: On the last page there will be 2 photos.

Atma's parents plan to arrange the 250 most beautiful photos of their wedding in some albums. Each album consists of 12 pages, and each page can hold 8 photographs. How many photographs will there be on the last page used (keeping in mind that Atma's parents utilize all available spaces progressively)?

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1805

2001

final

no

C1 C2 L1 L2 GP

Algebra

The question admits two solutions: Michele spent 14 or 17 francs.

Donato and Michele spent the day at Mathlandia park in Paris. At noon, each of them had a sandwich and a drink. At the bar, there was a choice of: ham sandwiches (7 francs), cheese sandwiches (11 francs), or salmon sandwiches (14 francs); for drinks there was milk (6 francs), orange juice (7 francs), or an exotic fruit beverage (9 francs). Donato spent 6 francs more than Michele. How much did Michele spend? Write a solution.

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1809

2001

final

no

C2 L1 L2 GP

Arithmetic

23:46

It's 8 PM. Chiara's VCR shows 4 AM. Its clock is faulty and runs 15% faster than a normal clock. What time should Chiara program to record her favorite show, which starts tomorrow at 4:40 PM?

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1810

2001

final

no

L1 L2 GP

Geometry

This way, it is (in correspondence to A=4) that the smallest possible perimeter of the rectangle is 10 cm.

With a ruler and set square always at hand, Angelo enjoys drawing a rectangle and a rhombus. He then notices that his two quadrilaterals have not only the same perimeter, but also the same area. Furthermore, the lengths of the sides of the rectangle and the diagonals of the rhombus are expressed as integers (in centimeters). What is the smallest possible perimeter of Angelo's rectangle?

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1811

2001

final

no

L1 L2 GP

Combinatorics

13

How many different trapezoids exist that have a perimeter equal to 11 cm and all sides expressed as integers (in cm)? (Two trapezoids will be considered identical if one can be obtained from the other by rotation or by symmetry).

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1812

2001

final

no

L1 L2 GP

Pattern Recognition

The department code is 18.

Agent 007 must carry out a mission in a French department identified by a two-digit code. To find the first digit, one must consider the sequence 2587193460 7x; for the second digit, the sequence 0491329000 00. Each of these two sequences consists of 10 digits (chosen from 0 to 9 inclusive), followed by a two-digit key belonging to the set {0; 1; 2; 3; 4; 5; 6; 7; 8; 9; x} where x corresponds to the number 10. The first digit of the key is the remainder of the division by 11 of the sum of the first ten digits; the second digit of the key is equal to the remainder of the division by 11 of the sum of the products of the first ten digits by their position in the sequence. Each of the two sequences reported above contains a deliberately incorrect symbol. The two correct symbols (which replace the two incorrect symbols) give, in order, the code of the French department. What is this code?

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1672

2002

autumn games

no

C1

Logic

The number designed by Michele is 21

Donato must guess an integer that Michele has chosen in great secret. Here is the information that Donato collects, little by little and in a disordered way. The number to find: is less than 32; greater than 18; less than 22; greater than 16; less than 24; greater than 20 and less than 28. What is the number Michele thought of?

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1673

2002

autumn games

no

C1

Arithmetic

The goal was achieved by pressing 7 keys

On the calculator, today, only three keys are working: the "C" key, which resets the number displayed on the screen to 0; the "+2" key, which adds 2 to the displayed number; the "+3" key, which adds 3 to the displayed number. Initially, the number "1987" (the year of the first "International Mathematical Games Championship") appears on the screen, and Rosi wants to make "17" appear instead. What is the minimum number of times Rosi must press a key on the calculator?

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1677

2002

autumn games

no

C1 C2

Combinatorics

81

Carla wrote the number 12 as a sum of several natural numbers. Then she multiplied all the terms of the sum together. What is the maximum value of the result of her product?

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1679

2002

autumn games

no

C1 C2

Combinatorics

Renato, in all, can write 64 numbers.

Renato writes two-digit numbers (from 10 to 99, inclusive) such that: - the tens digit and the units digit are never equal; - the tens digit and the units digit are never consecutive (for example, he will not write the numbers 45 or 87, as they do not satisfy this condition). What is the maximum number of different numbers Renato can write while respecting these conditions?

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1680

2002

autumn games

no

C2 L1

Combinatorics

In total, we will have, at least, 25

Marco places four standard dice (where the sum of opposite faces is 7) on a table, in close contact with each other, such that their top faces form a square and the numbers on the four top faces are all different. He then calculates the sum of all the numbers written on the visible faces of the dice. What is the minimum value of the sum obtained by Marco?

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1681

2002

autumn games

no

C2 L1

Geometry

The perimeter of Milena's mosaic measures 100 cm.

Milena begins to create a mosaic (without gaps) using three squares and six equilateral triangles, all of which have sides of 10 cm. The joints between the tiles of the mosaic (assembled edge-to-edge) have a total length of 1 meter. What is the perimeter of Milena's mosaic?

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1682

2002

autumn games

no

C2 L1

Geometry

664; 329; 128; 61

A trapezoid has an area equal to 335 cm² and its minor base measures 6 cm. The lengths of its height and major base are both expressed as integer numbers of centimeters. What is the length of the trapezoid's major base?

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1683

2002

autumn games

no

C2 L1

Algebra

There are two solutions: Jacob's savings amount to 109 Euro or 229 Euro.

With his savings, which are under 250 Euros, Jacob wants to buy CDs and books as gifts for his friends. CDs cost 15 Euros each; books cost 8 Euros each. If Jacob were to buy only CDs, he would be 11 Euros short of buying one more; if instead he were to buy only books, he would have 5 Euros left over. How much are Jacob's savings?

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1684

2002

autumn games

no

L1 L2

Algebra

The system admits as solutions (whole) m = 11, n = 5 The volume of the parallelepiped is 440 cm3

A rectangular parallelepiped has one dimension that is the arithmetic mean of the other two. The sum of the three dimensions is 24 cm, the three dimensions are integers (in centimeters), and the total surface area of the solid is 366 cm². Calculate the volume of the parallelepiped.

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