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

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1687	2002	autumn games	no	L2	Combinatorics	The muzzian language has a maximum of 81 different words.	A new inhabited planet, Planet Muzz, has recently been discovered. Its inhabitants... the Muzzians... have a very simple alphabet, which comprises only four letters: A, M, U, Z. The words that can be formed with this alphabet obey the following rules: - a word contains at least two letters (of which at least one is a vowel), and is formed by a maximum of four letters; - two vowels are always separated by at least one consonant; - two different consonants are always separated by at least one vowel; - no word contains three identical letters. How many different words does the Muzzian language comprise, at most?	Not supported with pagination yet
1688	2002	autumn games	no	L2	Arithmetic	it ends with 14 zeros	All integers between 50 and 100 (inclusive) are multiplied together. How many zeros does the number representing this product end with?	Not supported with pagination yet
1690	2002	autumn games	no	L2	Combinatorics	81	On his computer screen, Desiderio has written the digits 12345. He then enjoys inserting between two consecutive digits present on his screen (i.e., between 1 and 2, between 2 and 3, between 3 and 4, between 4 and 5) a "+" sign, a "-" sign, or... nothing at all. He can thus obtain, for example, the expression 1+2-34+5 or 123-45. How many possible "expressions" are there? Note: do not forget to count the examples included in the text.	Not supported with pagination yet
1692	2002	team games	no	C2 L1 L2	Combinatorics	The cake can be divided, at most, into 16 pages.	Into how many parts, at most, can a cake be divided with 5 cuts?	Not supported with pagination yet
1693	2002	team games	no	C2 L1 L2	Combinatorics	The cake can be divided, at most, into 5051 parts.	Into how many pieces can a cake be divided, at most, with 100 cuts?	Not supported with pagination yet
1695	2002	team games	no	C2 L1 L2	Geometry	MN = 5 km	Grandpa Desiderio has a plot of land shaped like a trapezoid ABCD, in which the bases AB and CD measure 1 km and 7 km, respectively. He wants to divide it into two equivalent parts for his two grandchildren by drawing a straight line MN parallel to the bases. How long will the segment MN be (in km)?	Not supported with pagination yet
1697	2002	team games	no	C2 L1 L2	Logic	Vincenzo knows 10 guests.	At a party, attended by a total of 20 young people, Alice is bored to death: she only knows one other person. For the second guest - Bruno - things are a bit better: he knows two guests. The third (Carolina) is a bit luckier, because she knows three people. The fourth knows four guests and so on.... each guest knows one more person than the previous one, up to the nineteenth - Ugo - who knows everyone. How many people does Vincenzo, the last of the 20 guests, know?	Not supported with pagination yet
1702	2002	team games	no	C2 L1 L2	Arithmetic	4 years of the last century had the same propriety.	Anna noticed that by dividing 2001 by the sum of its digits, a number is obtained which is the product of two distinct prime numbers (different from 1). For how many years of the last century (from 1901 to 2000) does the same property hold?	Not supported with pagination yet
1703	2002	team games	no	C2 L1 L2	Combinatorics	The largest group shall consist of at least 52 persons.	The 2002 inhabitants of Europolis are very superstitious and always keep in their pocket, as a good luck charm, one or at most two coins (among those currently in circulation). One day the local administrators ask each of the inhabitants: "What is the value of the coin (or the sum of the two coins) you have in your pocket?" Based on the answers, they then form different groups, placing people with the same amount in the same group. At least how many people are in the largest group?	Not supported with pagination yet
1704	2002	team games	no	C2 L1 L2	Arithmetic	1999= 2^4 * 5^3 - 1 = (4^2 *5^3 -1).	Try to write the number 1999 using only the digits 1, 2, 3, 4, 5 (each exactly once) and the elementary operations of addition, subtraction, multiplication, division, and exponentiation.	Not supported with pagination yet
1705	2002	team games	no	C2 L1 L2	Algebra	The first page "jumped" was 162	When a book bores him, Enrico enjoys summing its page numbers (from 1 up to the last one). He did this with his latest book and found the sum to be 19575. However, he then realized he had "skipped" two pages because they were stuck together. What was the first "skipped" page?	Not supported with pagination yet
1706	2002	team games	no	C2 L1 L2	Logic	Donato's grandfather was born in 1921	It's December 31, 2001. Before the toast, Michele asks Donato: "How old is your grandfather?" Donato is not always direct in his answers. In this case, he replies: "Take the day and the month he was born; if you multiply them, you get his grandfather's age, while if you sum them, you get the year he was born minus 1900." In what year was Donato's grandfather born?	Not supported with pagination yet
1707	2002	team games	no	C2 L1 L2	Geometry	Jacob may have painted, at most, 2,000 cubes.	Jacob has 2001 identical Lego bricks, which he joins to form a large cuboid. Then he paints the outer faces green (including the bottom one) and removes the bricks with at least one green face. With the remaining ones, he repeats the operation and so on... until he realizes that, if he were to paint the last cuboid he formed, no entirely white bricks would remain. At this point, he stops. At most, how many bricks could Jacob have painted green (on at least one face)?	Not supported with pagination yet
1708	2002	team games	no	C2 L1 L2	Geometry	The fourth edge measures 2 m.	Angelo is a somewhat peculiar architect. The roofs of his houses all have a pyramidal shape. In the last building he designed, the "base" has a rectangular shape, with two lateral edges of the pyramid (which originate from two non-consecutive vertices of the rectangle) measuring, respectively, 7 m and 6 m. A third edge measures 9 m. How long (in m) is the fourth edge?	Not supported with pagination yet
1709	2002	team games	no	C2 L1 L2	Pattern Recognition	Next year with the same property is 2012	The situation is the one described in game 11, but now we want to know what the next year will be for which the same property holds.	Not supported with pagination yet
1711	2002	Rosi's games	no	C2 L1 L2	Combinatorics	You must take at least 13 popsicles	In the ice cream freezer there are 84 popsicles of 7 different flavors, specifically a dozen of each flavor. What is the minimum number of popsicles that must be taken, without looking, to be sure of having taken two of different flavors?	Not supported with pagination yet
1713	2002	Rosi's games	no	C2 L1 L2	Arithmetic	Each boy will receive 13 slices	Each of the twelve salamis on the table is sliced with twelve cuts. Twelve friends receive the same number of slices. How many slices will each boy receive?	Not supported with pagination yet
1714	2002	Rosi's games	no	C2 L1 L2	Geometry	The triangle is obtusangle	The sides of a triangle measure 24 cm, 10 cm, and 27 cm. What type of triangle is it? Obtuse-angled, right-angled, or acute-angled?	Not supported with pagination yet
1715	2002	Rosi's games	no	C2 L1 L2	Combinatorics	The figure "9" repeats itself 600 times	Write every integer number, from 1 to 2002. How many times did you use the digit “9”?	Not supported with pagination yet
1717	2002	Rosi's games	no	C2 L1 L2	Logic	289	What is the smallest number greater than 15 that is not prime and not divisible by any of the prime numbers less than 15?	Not supported with pagination yet
1718	2002	Rosi's games	no	C2 L1 L2	Logic	83 people	Our gondola lift system consists of a total of 110 gondolas. All the gondolas that are ascending are occupied: if two people are in one gondola, only one person is in the next one (and vice versa). What is the maximum number of people simultaneously in the ascending gondolas?	Not supported with pagination yet
1719	2002	Rosi's games	no	C2 L1 L2	Arithmetic	22 x 91 = 2002 or 26 x 77 = 2002	Complete the following operations. (The products are equal, but the factors must be different). _ _ x _ _ = ------- 2 0 0 2 _ _ x _ _ = ------- 2 0 0 2	Not supported with pagination yet
1720	2002	Rosi's games	no	C2 L1 L2	Arithmetic	2 hours	Piero is a good painter: to paint a room, he takes 3 hours. His assistant, Paolo, is much slower and takes 6 hours to complete an identical one. Working together, how long would Piero and Paolo take to paint a room?	Not supported with pagination yet
1722	2002	Rosi's games	no	C2 L1 L2	Arithmetic	1000 Euro	Three friends, playing Super Enalotto, won a total of 39,000 Euros. According to prior agreements, they divide the winnings into parts directly proportional to 1/2, 1/4, and 1/5. However, wanting to be generous, they decide that each will keep the amount rounded down to the nearest whole thousand Euros and donate the remaining amount to charity. Andrea, their friend, exclaims: "How stingy!" What is the total amount the three lucky friends donate to charity?	Not supported with pagination yet
1724	2002	Rosi's games	no	C2 L1 L2	Combinatorics	The cake can be divided at most with 16 cuts	How many pieces can a cake be divided into, at most, with 5 cuts?	Not supported with pagination yet
1726	2002	Rosi's games	no	C2 L1 L2	Arithmetic	2^4 x 5^3 + 1 (4^2 x 5^3 + 1)	Try to write the number 2001, using only the digits 1,2,3,4,5 (each once and only once) and the elementary operations of addition, subtraction, multiplication, division, and exponentiation.	Not supported with pagination yet
1732	2002	semifinal	no	C1 C2	Logic	Solution 13 movements	Desiderio owns a magnificent padlock. He can change the current combination: 3 9 4 8 5 with two types of movements: either he decreases a single digit by 1 unit, or he decreases, always by 1 unit, multiple digits, but only if they are adjacent and equal. For example, he can go from 14442 to 13332 (but also, if he wants, to 14332). What is the minimum number of movements Desiderio needs to reach the combination 20002?	Not supported with pagination yet
1734	2002	semifinal	no	C1 C2 L1 L2 GP	Arithmetic	Solution 1982	The number 2000, increased by the sum of its digits, becomes 2002. However, Angelo has found another number that, increased by the sum of its digits, gives 2002. What is this number?	Not supported with pagination yet
1735	2002	semifinal	no	C2 L1 L2 GP	Combinatorics	Solution 1429 seconds	Adult millipedes take one second to remove a shoe, while their children take two seconds. A millipede family consists of a father, a mother, and three children. The parents, once barefoot, can help their children. Each millipede can remove only one shoe at a time, either for themselves or for a child (who meanwhile continues their own operations). What is the minimum time our millipede family will take to remove all their shoes? Note: it is assumed that each millipede actually has... 1000 legs! Note: the time must be given in seconds and possibly rounded up.	Not supported with pagination yet
1737	2002	semifinal	no	L1 L2 GP	Logic	Solution 250	The Mathlandia Congress has gathered 2000 delegates who belong to two categories of people: liars who always lie and loyal ones who always tell the truth. Each delegate specializes in Algebra or Geometry or Probability, and no one has more than one specialization. Each delegate is successively asked: "Are you an algebraist? Do you specialize in Geometry? Are you a probabilist?" The number of "yes" answers for each of the three questions is, respectively, 100, 540, 1610. How many liars are there at the Congress?	Not supported with pagination yet
1741	2002	final	no	C1	Algebra	The sum of age is 32 years	Roberto, Riccardo, and Renato are three brothers who live in Maremma, in a town not far from the sea. In four years, their combined age will be 44 years. Their father, Mario, is over 40 years old. Grandma Fernanda - their mother's mother - is between 70 and 80 years old. Roberto and Riccardo are fans of Fiorentina; Renato, on the other hand, supports Juventus. What is the sum of the current ages of Roberto, Riccardo, and Renato?	Not supported with pagination yet
1745	2002	final	no	C1 C2 L1 L2 GP	Geometry	The length of the rectangle is 7 units	On a squared sheet of paper (each square with a side of 1 cm), following the grid lines, Elena draws a rectangle with a base length of 6 cm. Then she draws the diagonal of the rectangle and notices that it crosses 12 squares but does not cross any grid intersection point (with the exception of its two endpoints). How many cm is the height of the rectangle?	Not supported with pagination yet
1746	2002	final	no	C1 C2 L1 L2 GP	Algebra	Carla owns 40 euros	Carla wants to organize her 1-euro coins. She then forms piles of 9 coins and notices that the number of remaining coins is equal to the number of piles she made. She then decides to redistribute all her coins into piles of seven: again, she observes that the number of remaining coins is equal to the number of piles formed. How many 1-euro coins does Carla have?	Not supported with pagination yet
1747	2002	final	no	C2 L1 L2 GP	Algebra	The golden weight is 11 grams	A balance scale has 17 weights, respectively 1 gram, 2 grams, 3 grams, ... 17 grams. Ten weights are black, six are silver, and only one is gold. The silver weights total 32 grams more than the black ones. What is the weight in grams of the gold weight?	Not supported with pagination yet
1748	2002	final	no	C2 L1 L2 GP	Algebra	Rosetta has 11 tickets of 50 euros in cash	At the end of the day, Rosetta, the most famous baker in Milan, tallies her cash and finds herself with 870 euros, consisting of 10, 20, and 50 euro notes. The number of notes of each denomination are consecutive numbers. How many 50 euro notes does Rosetta have in the till?	Not supported with pagination yet
1749	2002	final	no	L1 L2 GP	Algebra	The three numbers are 3, 5, 7	Desiderio wrote three prime numbers and noticed that their product is equal to 7 times their sum. What are these three numbers, in increasing order?	Not supported with pagination yet
1750	2002	final	no	L1 L2 GP	Combinatorics	The number of nuts is 13	To pass the long winter evenings, Rosi and Angelo have invented a game with 20 walnuts they collected in the garden. Rosi starts. She divides the 20 walnuts on the table into several equal piles. (The number of walnuts in each pile must be either 1 or a prime number). Then she takes one pile, removes it from the table, and pockets it. Now it's Angelo's turn, who repeats the same operations: he regroups the walnuts remaining on the table, divides them following the same criteria, takes a pile, and pockets it. Then it's Rosi's turn and then Angelo's, and so on, with the proviso that each player must make at least two piles, unless only one walnut remains on the table (in which case the player will take it). The goal of the game is to pocket the maximum possible number of walnuts. How many walnuts is Rosi guaranteed to be able to take, regardless of Angelo's play?	Not supported with pagination yet
1751	2002	final	no	L2 GP	Combinatorics	The product is 140 (20x7)	Jacob often goes to the beach, where he has a beautiful collection of marbles that he runs on fantastic sand tracks. For him, when it rains, his dad invented another game: he arranges 21 marbles into 6 boxes, in such a way that no box remains empty and that all boxes contain a different number of marbles. The game begins: with each “move”, Jacob can take a certain number of marbles (of his choice) from one box and put them into another, on the condition, however, that he thereby doubles the contents of this latter box. Following this rule, Jacob puts the maximum possible number of marbles into one box. Give the product of the maximum number of marbles in one box by the minimum number of “moves” to achieve it.	Not supported with pagination yet
1752	2002	final	no	L2 GP	Combinatorics	At most, the total is 24	The performances of 18 figure skaters are evaluated by 9 judges. After the performance, each judge assigns a rank (from 1st to 18th place) to each competitor, without ties. For any given competitor, the ranks assigned by the 9 judges do not differ from each other by more than 3 ranks. For each competitor, the sum of the 9 assigned ranks is then calculated. What is the maximum possible value of the smallest total score obtained by any skater?	Not supported with pagination yet
1571	2003	autumn games	no	C1	Arithmetic	The largest whole number of trs and even digits all different is 864	What is the largest integer formed by three distinct even digits?	Not supported with pagination yet
1573	2003	autumn games	no	C1	Arithmetic	The diabolical number is 120 : 3 = 40, The satanic number is 2 x 133 = 266, The magical number is 10 x ( 266 = 40) = 2260 The magical number of the witch is 2260	Our witch specializes in mathematical potions. Here are the ingredients for her concoction: 3 133 38 42 2 56 9 120 6 “I divide the largest even number by the smallest odd number and get the diabolical number. Then I multiply the smallest even number by the largest odd number and get the satanic number. Finally, I multiply by 10 the difference between the satanic number and the diabolical number and get the ... magic number!”. What is the witch's magic number?	Not supported with pagination yet
1574	2003	autumn games	no	C1 C2	Algebra	9	I have 65 candies. I eat one and distribute the rest equally among all my teammates. Each teammate receives a number of candies equal to the number of my teammates. How many of us are on the team?	Not supported with pagination yet
1577	2003	autumn games	no	C2 L1	Combinatorics	On the planet Numerus you can register at most 90 x 441 = 39690 vessels	On the planet Numerus, spacecraft have license plates, just like cars on Earth. These license plates, however, consist of only two digits, followed by two letters. For example: 95 LV. Note: no license plate starts with 0, and the alphabet in use on Numerus consists of 21 letters. How many spacecraft can be registered at most on the planet Numerus?	Not supported with pagination yet
1578	2003	autumn games	no	C2 L1	Combinatorics	97586	What is the largest number of five different digits that I can write following the rule that two adjacent digits can never be consecutive (like, for example, 1 and 2 or 8 and 7)?	Not supported with pagination yet
1581	2003	autumn games	no	L1 L2	Arithmetic	18 days.	Carla bought 72 mice to feed her cats for 12 days. Each cat eats the same (whole) number of mice per day, and this number remains the same, day after day. If Carla decides to go on vacation with two cats, for how many days will her other cats have food, sharing the 72 mice?	Not supported with pagination yet
1582	2003	autumn games	no	L1 L2	Combinatorics	The family consists of 240 members	I belong to a family of 4-digit numbers. Our first digit, which is non-zero, is strictly less than our second digit, which is equal to the third; finally, our third digit is strictly greater than the fourth. How many members does my family consist of?	Not supported with pagination yet
1583	2003	autumn games	no	L1 L2	Algebra	The age of the son of Nando is 15	Nando is a puzzle enthusiast and explains his son's age as follows: "To get his current age, you must take three times the age he will be in two years and subtract three times his age from three years ago." What is Nando's son's age?	Not supported with pagination yet
1584	2003	autumn games	no	L1 L2	Algebra	3 days, 22 hours 30 minutes	A wild duck takes 9 days to go from Norway to Morocco and 7 days to go from Morocco to Norway. Assume that all ducks fly in a straight line and at a constant speed. Two ducks depart together, one from Norway and the other from Morocco. How long after will they meet in flight? (Give the answer in days, hours, and minutes).	Not supported with pagination yet
1585	2003	autumn games	no	L2	Geometry	The distance between their ends is 5,292 cm	The hands of a clock are respectively 4 cm and 6 cm long. What is the distance between their ends when the clock shows two o'clock? (If necessary, use 1.414 for √2; 1.732 for √3; 2.23 for √5; 2.646 for √7).	Not supported with pagination yet
1586	2003	autumn games	no	L2	Logic	In the family we have 6 red fishes	My family consists of Dad, Mom, my brothers, my sisters, me and... my goldfish. There are a total of 14 hands and 13 mouths. Fortunately, only the goldfish are without hands. How many goldfish do we have in the family?	Not supported with pagination yet
1588	2003	autumn games	no	L2	Combinatorics	The required number is 8/69	The fraction machine accepts, as input, irreducible fractions strictly between 0 and 1. Each fraction f = n/d gives as output, as a result, the reciprocal of the number f+d. These results are then arranged in decreasing order. What is the twentieth result? (Give this result as an irreducible fraction.)	Not supported with pagination yet
1590	2003	team games	no	C2 L1 L2	Arithmetic	the corridor can measure 3 m, 6 me 9 m	To tile the corridor of his house, Michele can use square tiles with side lengths of 20 cm, 25 cm, or 30 cm. The corridor is not very long – less than 10 m – but it has the property that in all three cases (using small, medium, or large tiles), it can be exactly covered with tiles of uniform size. What is its length?	Not supported with pagination yet
1591	2003	team games	no	C2 L1 L2	Combinatorics	0	Nine tokens are aligned: each of them has a white face and a black face. Initially, all faces are white. In one move, two tokens can be flipped, provided they are adjacent. What is the minimum number of moves to obtain an alignment of 9 tokens, all black?	Not supported with pagination yet
1592	2003	team games	no	C2 L1 L2	Algebra	Desire sold its 9 Soviet stamps and received 15 Cuban stamps.	Desiderio has 45 stamps, some Soviet and others Cuban. Now, however, he wants to specialize in collecting Cuban stamps and asks for help from his friend Amerigo, who is also a passionate philatelist. The agreement is: 3 Soviet stamps for 5 Cuban stamps. At the end of the exchange, Desiderio has 51 stamps, all of which are Cuban. How many Soviet stamps did he initially have?	Not supported with pagination yet
1593	2003	team games	no	C2 L1 L2	Combinatorics	The five friends can be arranged in 36 different ways.	Carla, Liliana, Milena, Rosi, and Silvia are posing for a souvenir photo: three of them can sit in the first row; the other two, standing, in the second row. How many are the possible positions they can occupy (and therefore the possible different souvenir photos) considering that Carla and Liliana always want to stay close, one next to the other?	Not supported with pagination yet
1594	2003	team games	no	C2 L1 L2	Algebra	1*4= 6	At the last International Congress, mathematicians defined a new (non-commutative) operation on integers, denoted a * b. For now, it is only known that: 0 * a = a + 1 a * 0 = (a – 1) * 1 (a + 1) * (b + 1) = a * [ (a + 1) * b ] What is the value of 1 * 4?	Not supported with pagination yet
1595	2003	team games	no	C2 L1 L2	Algebra	The amount of the cheque is 5,11 €	Pietro goes to the bank to cash a check (in Euros) because he has run out of money. The distracted teller, when giving him the money, accidentally swaps the number of euros with the number of cents. Pietro doesn't notice anything and, with the money received, goes to buy the local newspaper (which that day costs only 83 cents). It is at that point, after buying the newspaper, that he realizes he has exactly twice the amount he should have received from the check in his pocket. What was the original amount of the check?	Not supported with pagination yet
1596	2003	team games	no	C2 L1 L2	Arithmetic	The answer to be given and 0	Nando has an irresistible attraction to betting. So, one day, he starts by betting 1 Euro. He loses and then bets 2 Euros. He loses again and bets 4 Euros. He continues this way – always losing and always doubling the amount bet – until after the tenth bet he realizes he has lost his entire initial sum of Euros. After how many bets had Nando lost exactly half of his initial sum of Euros?	Not supported with pagination yet
1598	2003	team games	no	C2 L1 L2	Pattern Recognition	666	Find (and discover...!) the natural number that satisfies the following conditions: * it is greater than 100 and less than 1000; * it is written using a single digit (repeated); * it is the sum of the first n natural numbers (for some n).	Not supported with pagination yet
1599	2003	team games	no	C2 L1 L2	Algebra	48	When on the escalator, Renato doesn't stand still and climbs step by step at a constant speed of one step per second. He thus takes 30 seconds to reach the top. One day, distracted, to go up he takes the escalator going down (at the same speed it goes up) and takes 2 minutes to reach the top. How many steps are there on the escalator (when it is still)?	Not supported with pagination yet
1602	2003	team games	no	C2 L1 L2	Combinatorics	14	A person who writes a number on a slip of paper (and pays 5 Euros) is entitled to participate in the drawing of a ticket, which displays another number between 100 and 999. If the sum of the digits of this drawn number is equal to the number previously written on the slip of paper, 50 Euros are won. Knowing that the 900 numbers between 100 and 999 have the same probability of being drawn, what is the number to write on the slip of paper that gives the greatest probability of winning?	Not supported with pagination yet
1604	2003	team games	no	C2 L1 L2	Algebra	147 grams	A, B, C, D, E, are five (good!) chocolate eggs. Some are also filled with cream; they have the same weight but are heavier than those without filling (which also all have the same weight). A and E, together, weigh 252 grams; A, B and C, together, weigh 420 grams; B, C, D and E, together, weigh 567 grams. How much does each of the cream-filled eggs weigh?	Not supported with pagination yet
1605	2003	team games	no	C2 L1 L2	Algebra	The required difference is 97.07 kg	Three of the horses participating in the Grand Prix, when weighed (together with their jockeys), registered a total of 1000 kg. The three jockeys, altogether, weigh 181 kg. The first horse weighs five times its jockey; the second four and a half times its jockey; the third four times the weight of its jockey. What is the minimum difference that can exist between the heaviest horse and the lightest horse? (The answer can be rounded to the nearest Kg.)	Not supported with pagination yet
1606	2003	team games	no	C2 L1 L2	Arithmetic	you will get a number with 999 times the number 1	Calculate the sum N = 9 + 99 + 999 + .... + 99 ... 9 (where the last number is composed of the digit 9, repeated 999 times). How many times does the digit 1 appear in N?	Not supported with pagination yet
1607	2003	team games	no	C2 L1 L2	Algebra	September and 53 years	Dr. Settembre's nine children were born, all spaced out by the same number of years. In 1998, the square of Dr. Settembre's age was equal to the sum of the squares of the ages of his nine children. What is Dr. Settembre's age today, in 2003?	Not supported with pagination yet
1608	2003	team games	no	C2 L1 L2	Pattern Recognition	The condition is satisfied only if X is worth 7	Angelo, the deli owner, finds an incredibly long code number on a package (approximately 50 digits long). He then decides to slice it into two-digit "chunks", starting from the right, and then summing the various "chunks" (for example, if the number ends with … 367523, he writes 23 + 75 + 36 + …). This way he obtains the number 1998. The next day, not satisfied with the procedure, he applies another method to the incredibly long number: he writes its first digit (always starting from the right), subtracts the second, adds the third, subtracts the fourth… and so on (in the previous example, he would have written 3 – 2 + 5 – 7 + 6 – 3 + …). He thus obtains a (positive) single-digit number. Which one?	Not supported with pagination yet
1609	2003	Rosi's games	no	C2 L1 L2	Arithmetic	Vasco returned 3.80 Euro	Vasco buys a gift, which costs 18.90 Euros, for his best friend. He pays with a 50 Euro banknote, but the cashier, distracted, gives him 34.90 Euros as change. After leaving the shop, Vasco notices the error and goes back in to return the amount that does not belong to him. How much did he return?	Not supported with pagination yet
1610	2003	Rosi's games	no	C2 L1 L2	Algebra	Federico will have 33 anJli	On January 1, 1991, Federico was twice Leone's age; on January 1, 1999, Federico was 10 years older than Leone. How old will Federico be on January 1, 2004?	Not supported with pagination yet
1611	2003	Rosi's games	no	C2 L1 L2	Combinatorics	The required numbers are: 8	How many three-digit numbers are there, greater than 600, in which the units digit is half the hundreds digit, and the tens digit is different from both the units digit and the hundreds digit?	Not supported with pagination yet
1612	2003	Rosi's games	no	C2 L1 L2	Arithmetic	The film lasted: 87 minutes	The movie started at 18:45 and finished at 20:12. How long did it last?	Not supported with pagination yet
1615	2003	Rosi's games	no	C2 L1 L2	Algebra	The girls are: 16	In a class of 27 students, there are five more girls than boys. How many girls are there?	Not supported with pagination yet
1616	2003	Rosi's games	no	C2 L1 L2	Combinatorics	The minimum number of gtiaccioli e: 6	In the ice cream freezer there are 60 popsicles of 5 different flavors, a dozen for each flavor. What is the minimum number of popsicles that must be taken (without looking ...) to be sure of having taken two of the same flavor?	Not supported with pagination yet
1617	2003	Rosi's games	no	C2 L1 L2	Logic	George's intruder is 3	Giorgio, using nine of the following ten digits, has written three consecutive numbers: 0-0-0-1-1-2-2-3-9-9. What is the unused digit?	Not supported with pagination yet
1618	2003	Rosi's games	no	C2 L1 L2	Algebra	The five friends hanno caught, altogether, 40 pesci	Five friends (Alberto, Carlo, Davide, Edoardo, Filip) went fishing together. Alberto and Carlo, together, caught 14 fish; Carlo and Davide, together, caught 20; Davide and Edoardo, together, 18; Edoardo and Filip, 12; finally, Alberto and Filip caught 16. How many fish did the five friends catch in total?	Not supported with pagination yet
1620	2003	semifinal	no	C1	Logic	48 eggs	Rosina brings a basket full of eggs to the market. First, Carla buys half of the eggs, then Milena buys half of the remaining eggs. Finally, Maddalena pays for 10 eggs and Rosina tells her: 'Miss, I'll give you my last two eggs; this way I will return to the farm with an empty basket.' How many eggs did Rosina arrive at the market with?	Not supported with pagination yet
1622	2003	semifinal	no	C1 C2	Algebra	360 grams	In the Borgio Verezzi fishing competition, scores are awarded by giving competitors 50 points for each fish, plus 1 point for each gram of fish caught. Jacob caught 19 fish for a total weight of 2430 grams. Mirko, on the other hand, had caught 14 fish for a total weight of 1860 grams but, just a moment before the final whistle, he manages to catch 2 fish of the same weight and ends up with the same score as Jacob. What is the weight in grams of one of the two last fish caught by Mirko?	Not supported with pagination yet
1625	2003	semifinal	no	C1 C2 L1 L2 GP	Algebra	6 guys	At a Mathematics competition, the number of girls was double the number of boys. Each participant scored 8, 9, or 10 points, and altogether they scored a total of 156 points. How many boys participated in the competition?	Not supported with pagination yet
1630	2003	semifinal	no	L1 L2 GP	Algebra	4 Solutions (20;4) (10;6) (8;8) (6;18)	Chiara and Anna have each chosen a positive integer. The product of one-third of Chiara's number and one-fifth of Anna's number is equal to the sum of one-fifth of Chiara's number and one-third of Anna's number. What are the two numbers?	Not supported with pagination yet
1631	2003	semifinal	no	L2 GP	Geometry	2 solutions: 4 cm; 18 cm	In the plane, Nando has drawn three lines that converge at a point, forming angles of 60 degrees. He then marks a point in the plane and measures its distance to two of the three lines. He finds these distances to be 7 cm and 11 cm. What is the distance from this point to the third line? Give this distance in centimeters, rounded to the nearest hundredth if necessary.	Not supported with pagination yet
1632	2003	semifinal	no	L2 GP	Arithmetic	80	Multiplying a (very large) number by 5, a forty-digit number is obtained, composed exactly of thirty '5's and ten '7's. What is the sum of the digits of the initial number?	Not supported with pagination yet
1636	2003	final	no	C1 C2 L1 GP	Arithmetic	(7+7)x7 x7+7+7= 707	Mirko, the prankster, enjoyed deleting the symbols (parentheses), +, and x from his friend Jacob's calculation. Put them back in the correct places, so that the equation written below is satisfied: 7 7 7 7 7 7 7 = 707	Not supported with pagination yet
1637	2003	final	no	C1 C2 L1 L2 GP	Geometry	The minimum number of intersections is: 4	Consider 4 circles, all with the same radius and never tangent to each other. Arrange them in the plane such that the figure formed is connected (i.e., forms a single "piece"). What is the minimum total number of intersection points the four circles will have?	Not supported with pagination yet
1641	2003	final	no	L1 L2 GP	Geometry	Oscar will travel 150 m	The great pyramid of Pharaoh Matemankhamon has a square base with a side length of 100 m; its four faces are equilateral triangles. Oscar - the scarab of the Nile - is at the foot of the pyramid, at the midpoint of the base of the South face. As part of his systematic exploration of the valley, Oscar wants to go to the point diametrically opposite to his current location (i.e., at the midpoint of the base of the North face), following the shortest possible path and climbing the pyramid, if necessary. What distance will Oscar travel?	Not supported with pagination yet
1643	2003	final	no	L2 GP	Algebra	The banana is long: 21 cm	A rope is placed along the top edge of a fence and is as long as each side of the fence itself. The rope weighs 300 grams per meter. At one end of the rope, there is a small monkey holding a banana, while at the other end there is a counterweight with a weight equal to that of the small monkey. The banana weighs 10 grams per cm. The total length of the rope, in meters, is equal to 1/3 of the monkey's age in years, and the monkey's weight, in grams, is equal to 200 times the age of the monkey's mother. The sum of the monkey's age and its mother's age is 30 years. Adding twice the monkey's weight and 40 times the banana's weight, the same total is obtained as would be obtained by adding 10 times the rope's weight and the counterweight's weight. The monkey's current age is half of the age its mother will be when the monkey reaches the age its mother is today. How long is the banana?	Not supported with pagination yet
1646	2003	international final day 1	no	CE	Logic	Friday	I was born on June 13th. My friend Peggy was born on May 7th. This year her birthday is on a Wednesday. Find the day of my birthday.	Not supported with pagination yet
1649	2003	international final day 1	no	C1 C2 L1 GP L2 HC	Logic	Nicolas	The president of FIGM has been kidnapped. The police have three suspects: 2 always lie and only one always tells the truth. Here is an excerpt from the interrogation: Nicola: "I did not kidnap the president." Matteo: "Nicola is not a liar." Maria: "Matteo did not kidnap the president." Who kidnapped the president?	Not supported with pagination yet
1650	2003	international final day 1	no	C1 C2 L1 GP L2 HC	Combinatorics	12 sauts	A chess knight moves on a 20x20 chessboard. Recall that a chess knight moves along the diagonal of a 2x3 rectangle (it is considered that the knight is always positioned at the center of the square). After tracing a triangular path, the knight returned to its starting square, always respecting the rules of movement. What is the minimum number of jumps it made?	Not supported with pagination yet
1651	2003	international final day 1	no	C1 C2 L1 GP L2 HC	Algebra	14 enfants réponses admises : 12 ou 13 enfants	Matilde and Mattia have invited their family and some friends to a birthday lunch. Mattia has two large identical rectangular tables. He makes the following observation: "If I join them along one side, so as to form a large rectangle, only 20 people will be able to sit around them (allowing the same space for each guest). If I join them along the other side, I gain two seats, but it's still not enough!" Matilde observes: "That's true, but we could use two separate tables because there are as many adults as children, and by doing so there would be enough room for everyone." How many children will attend this lunch?	Not supported with pagination yet
1654	2003	international final day 1	no	L1 GP L2 HC	Logic	7 solutions: 10125; 2025; 30375; 405; 50625; 6075; 70875	The FIGM locks the solutions of its games in a safe, which opens with a code. This code, considered as an integer, does not end with a 0 and has the following particularity. If a certain digit is removed, the new number obtained is equal to 1/9 of the initial number. It is then possible to remove another digit from this second number and obtain exactly 1/81 of the initial number. What is the code number of the FIGM's safe?	Not supported with pagination yet
1658	2003	international final day 1	no	L2 HC	Combinatorics	1 solution: 1949 et 1997	A distracted professor gets locked in a Museum. A guard notices his presence and points out to him that it is forbidden to enter the museum at that hour. He offers to let him out if he solves the following enigma. « Here is a number: 3892153. A computer will quickly establish that it is equal to 1752^2 + 907 X 907, but also to 1172^2 + 1587^2. Can you find two integers greater than 1 whose product it is? » What are these numbers?	Not supported with pagination yet
1660	2003	international final day 2	no	CE	Arithmetic	304	Domenico counts aloud « 1, 2, 3, 4 ». Then he continues mentally and finishes aloud with « 309, 310 ». How many numbers did he count mentally?	Not supported with pagination yet
1661	2003	international final day 2	no	CE	Logic	9 h 07	Eliana, Melina, and Geraldina agreed to meet at 8:50 AM at the park. Eliana arrived fifteen minutes late. Melina arrived three minutes early, and Geraldina arrived two minutes after Eliana. At what time did the three friends meet up?	Not supported with pagination yet
1665	2003	international final day 2	no	C1 C2 L1 GP L2 HC	Combinatorics	12 façons	This morning I don't know what to wear. I have a choice between a dress, a skirt, trousers, a shirt and four pairs of shoes. In how many different ways can I get dressed? Observation: to get dressed, I wear only one top garment, only one bottom garment and only one pair of shoes. If I wear the dress, I only have to put on my shoes.	Not supported with pagination yet
1666	2003	international final day 2	no	C1 C2 L1 GP L2 HC	Algebra	20 %	On a jar of chestnut cream, the following information can be read: • prepared with 50g of fruit per 100g of product • sugar content: 60g per 100g. It is also stated that, apart from a vanilla flavoring (of negligible mass), chestnut pulp and sugar are the only ingredients of this artisanal product. What is the percentage of sugar contained in the chestnut pulp?	Not supported with pagination yet
1667	2003	international final day 2	no	C2 L1 GP L2 HC	Algebra	40 euros	Catherine has a 30% discount voucher, valid for the simultaneous purchase of three items. When she is about to pay her bill of 168 euros, the sales assistant informs her that she can get a 40% discount if she buys a fourth item. But Catherine only has 168 euros in her pocket. After a quick calculation, she realizes that, despite this, she can buy a fourth item, in addition to those she has already chosen and still wants to buy. What is the maximum price of the fourth item she can buy?	Not supported with pagination yet
1669	2003	international final day 2	no	C2 L1 GP L2 HC	Algebra	1 solution : 4999	In the equations CI + CI = LY and CIJM + CIJM = JMLY, the same letter always replaces the same digit, but a digit can be replaced by different letters. No number begins with a 0. Find the value of CIJM.	Not supported with pagination yet
1670	2003	international final day 2	no	C2 L1 GP L2 HC	Combinatorics	3 solutions: 4624 ; 6084 ; 8464	The FIGM safe's code changes every day. Today it must be a four-digit number (where the first and last digits are not zero), the square of an integer, and all its digits must have the same parity. What is today's code?	Not supported with pagination yet
1463	2004	autumn games	no	CE C1	Arithmetic	Guido will eat 9 slices	Jacob is a generous boy and wants to share his round cake, divided into slices, with his friends. He gives half of it to Guido, who in turn gives half of what he receives to Giacomo, who isn't very hungry and then returns half of what he received to Guido. How many slices of the cake will Guido eat?	Not supported with pagination yet
1464	2004	autumn games	no	CE C1	Arithmetic	Angelo has to give Rosi 9 €.	Rosi and Angelo always divide their expenses in equal parts. Yesterday Rosi went to the supermarket and spent 35 Euros. Today Angelo made other purchases for 17 Euros. How much money must Angelo give to Rosi to settle the accounts?	Not supported with pagination yet
1469	2004	autumn games	no	C2 L1	Arithmetic	There are 210 houses on the same side of my house.	On my street, houses on one side are numbered with even numbers, and on the other side with odd numbers. My house is number 98. If they had started the numbering from the other end of the street, it would be number 324. How many houses are there on the same side as my house?	Not supported with pagination yet

1687

2002

autumn games

no

L2

Combinatorics

The muzzian language has a maximum of 81 different words.

A new inhabited planet, Planet Muzz, has recently been discovered. Its inhabitants... the Muzzians... have a very simple alphabet, which comprises only four letters: A, M, U, Z. The words that can be formed with this alphabet obey the following rules: - a word contains at least two letters (of which at least one is a vowel), and is formed by a maximum of four letters; - two vowels are always separated by at least one consonant; - two different consonants are always separated by at least one vowel; - no word contains three identical letters. How many different words does the Muzzian language comprise, at most?

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1688

2002

autumn games

no

L2

Arithmetic

it ends with 14 zeros

All integers between 50 and 100 (inclusive) are multiplied together. How many zeros does the number representing this product end with?

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1690

2002

autumn games

no

L2

Combinatorics

81

On his computer screen, Desiderio has written the digits 12345. He then enjoys inserting between two consecutive digits present on his screen (i.e., between 1 and 2, between 2 and 3, between 3 and 4, between 4 and 5) a "+" sign, a "-" sign, or... nothing at all. He can thus obtain, for example, the expression 1+2-34+5 or 123-45. How many possible "expressions" are there? Note: do not forget to count the examples included in the text.

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1692

2002

team games

no

C2 L1 L2

Combinatorics

The cake can be divided, at most, into 16 pages.

Into how many parts, at most, can a cake be divided with 5 cuts?

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1693

2002

team games

no

C2 L1 L2

Combinatorics

The cake can be divided, at most, into 5051 parts.

Into how many pieces can a cake be divided, at most, with 100 cuts?

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1695

2002

team games

no

C2 L1 L2

Geometry

MN = 5 km

Grandpa Desiderio has a plot of land shaped like a trapezoid ABCD, in which the bases AB and CD measure 1 km and 7 km, respectively. He wants to divide it into two equivalent parts for his two grandchildren by drawing a straight line MN parallel to the bases. How long will the segment MN be (in km)?

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1697

2002

team games

no

C2 L1 L2

Logic

Vincenzo knows 10 guests.

At a party, attended by a total of 20 young people, Alice is bored to death: she only knows one other person. For the second guest - Bruno - things are a bit better: he knows two guests. The third (Carolina) is a bit luckier, because she knows three people. The fourth knows four guests and so on.... each guest knows one more person than the previous one, up to the nineteenth - Ugo - who knows everyone. How many people does Vincenzo, the last of the 20 guests, know?

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1702

2002

team games

no

C2 L1 L2

Arithmetic

4 years of the last century had the same propriety.

Anna noticed that by dividing 2001 by the sum of its digits, a number is obtained which is the product of two distinct prime numbers (different from 1). For how many years of the last century (from 1901 to 2000) does the same property hold?

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1703

2002

team games

no

C2 L1 L2

Combinatorics

The largest group shall consist of at least 52 persons.

The 2002 inhabitants of Europolis are very superstitious and always keep in their pocket, as a good luck charm, one or at most two coins (among those currently in circulation). One day the local administrators ask each of the inhabitants: "What is the value of the coin (or the sum of the two coins) you have in your pocket?" Based on the answers, they then form different groups, placing people with the same amount in the same group. At least how many people are in the largest group?

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1704

2002

team games

no

C2 L1 L2

Arithmetic

1999= 2^4 * 5^3 - 1 = (4^2 *5^3 -1).

Try to write the number 1999 using only the digits 1, 2, 3, 4, 5 (each exactly once) and the elementary operations of addition, subtraction, multiplication, division, and exponentiation.

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1705

2002

team games

no

C2 L1 L2

Algebra

The first page "jumped" was 162

When a book bores him, Enrico enjoys summing its page numbers (from 1 up to the last one). He did this with his latest book and found the sum to be 19575. However, he then realized he had "skipped" two pages because they were stuck together. What was the first "skipped" page?

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1706

2002

team games

no

C2 L1 L2

Logic

Donato's grandfather was born in 1921

It's December 31, 2001. Before the toast, Michele asks Donato: "How old is your grandfather?" Donato is not always direct in his answers. In this case, he replies: "Take the day and the month he was born; if you multiply them, you get his grandfather's age, while if you sum them, you get the year he was born minus 1900." In what year was Donato's grandfather born?

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1707

2002

team games

no

C2 L1 L2

Geometry

Jacob may have painted, at most, 2,000 cubes.

Jacob has 2001 identical Lego bricks, which he joins to form a large cuboid. Then he paints the outer faces green (including the bottom one) and removes the bricks with at least one green face. With the remaining ones, he repeats the operation and so on... until he realizes that, if he were to paint the last cuboid he formed, no entirely white bricks would remain. At this point, he stops. At most, how many bricks could Jacob have painted green (on at least one face)?

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1708

2002

team games

no

C2 L1 L2

Geometry

The fourth edge measures 2 m.

Angelo is a somewhat peculiar architect. The roofs of his houses all have a pyramidal shape. In the last building he designed, the "base" has a rectangular shape, with two lateral edges of the pyramid (which originate from two non-consecutive vertices of the rectangle) measuring, respectively, 7 m and 6 m. A third edge measures 9 m. How long (in m) is the fourth edge?

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1709

2002

team games

no

C2 L1 L2

Pattern Recognition

Next year with the same property is 2012

The situation is the one described in game 11, but now we want to know what the next year will be for which the same property holds.

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1711

2002

Rosi's games

no

C2 L1 L2

Combinatorics

You must take at least 13 popsicles

In the ice cream freezer there are 84 popsicles of 7 different flavors, specifically a dozen of each flavor. What is the minimum number of popsicles that must be taken, without looking, to be sure of having taken two of different flavors?

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1713

2002

Rosi's games

no

C2 L1 L2

Arithmetic

Each boy will receive 13 slices

Each of the twelve salamis on the table is sliced with twelve cuts. Twelve friends receive the same number of slices. How many slices will each boy receive?

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1714

2002

Rosi's games

no

C2 L1 L2

Geometry

The triangle is obtusangle

The sides of a triangle measure 24 cm, 10 cm, and 27 cm. What type of triangle is it? Obtuse-angled, right-angled, or acute-angled?

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1715

2002

Rosi's games

no

C2 L1 L2

Combinatorics

The figure "9" repeats itself 600 times

Write every integer number, from 1 to 2002. How many times did you use the digit “9”?

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1717

2002

Rosi's games

no

C2 L1 L2

Logic

289

What is the smallest number greater than 15 that is not prime and not divisible by any of the prime numbers less than 15?

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1718

2002

Rosi's games

no

C2 L1 L2

Logic

83 people

Our gondola lift system consists of a total of 110 gondolas. All the gondolas that are ascending are occupied: if two people are in one gondola, only one person is in the next one (and vice versa). What is the maximum number of people simultaneously in the ascending gondolas?

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1719

2002

Rosi's games

no

C2 L1 L2

Arithmetic

22 x 91 = 2002 or 26 x 77 = 2002

Complete the following operations. (The products are equal, but the factors must be different). _ _ x _ _ = ------- 2 0 0 2 _ _ x _ _ = ------- 2 0 0 2

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1720

2002

Rosi's games

no

C2 L1 L2

Arithmetic

2 hours

Piero is a good painter: to paint a room, he takes 3 hours. His assistant, Paolo, is much slower and takes 6 hours to complete an identical one. Working together, how long would Piero and Paolo take to paint a room?

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1722

2002

Rosi's games

no

C2 L1 L2

Arithmetic

1000 Euro

Three friends, playing Super Enalotto, won a total of 39,000 Euros. According to prior agreements, they divide the winnings into parts directly proportional to 1/2, 1/4, and 1/5. However, wanting to be generous, they decide that each will keep the amount rounded down to the nearest whole thousand Euros and donate the remaining amount to charity. Andrea, their friend, exclaims: "How stingy!" What is the total amount the three lucky friends donate to charity?

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1724

2002

Rosi's games

no

C2 L1 L2

Combinatorics

The cake can be divided at most with 16 cuts

How many pieces can a cake be divided into, at most, with 5 cuts?

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1726

2002

Rosi's games

no

C2 L1 L2

Arithmetic

2^4 x 5^3 + 1 (4^2 x 5^3 + 1)

Try to write the number 2001, using only the digits 1,2,3,4,5 (each once and only once) and the elementary operations of addition, subtraction, multiplication, division, and exponentiation.

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1732

2002

semifinal

no

C1 C2

Logic

Solution 13 movements

Desiderio owns a magnificent padlock. He can change the current combination: 3 9 4 8 5 with two types of movements: either he decreases a single digit by 1 unit, or he decreases, always by 1 unit, multiple digits, but only if they are adjacent and equal. For example, he can go from 14442 to 13332 (but also, if he wants, to 14332). What is the minimum number of movements Desiderio needs to reach the combination 20002?

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1734

2002

semifinal

no

C1 C2 L1 L2 GP

Arithmetic

Solution 1982

The number 2000, increased by the sum of its digits, becomes 2002. However, Angelo has found another number that, increased by the sum of its digits, gives 2002. What is this number?

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1735

2002

semifinal

no

C2 L1 L2 GP

Combinatorics

Solution 1429 seconds

Adult millipedes take one second to remove a shoe, while their children take two seconds. A millipede family consists of a father, a mother, and three children. The parents, once barefoot, can help their children. Each millipede can remove only one shoe at a time, either for themselves or for a child (who meanwhile continues their own operations). What is the minimum time our millipede family will take to remove all their shoes? Note: it is assumed that each millipede actually has... 1000 legs! Note: the time must be given in seconds and possibly rounded up.

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1737

2002

semifinal

no

L1 L2 GP

Logic

Solution 250

The Mathlandia Congress has gathered 2000 delegates who belong to two categories of people: liars who always lie and loyal ones who always tell the truth. Each delegate specializes in Algebra or Geometry or Probability, and no one has more than one specialization. Each delegate is successively asked: "Are you an algebraist? Do you specialize in Geometry? Are you a probabilist?" The number of "yes" answers for each of the three questions is, respectively, 100, 540, 1610. How many liars are there at the Congress?

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1741

2002

final

no

C1

Algebra

The sum of age is 32 years

Roberto, Riccardo, and Renato are three brothers who live in Maremma, in a town not far from the sea. In four years, their combined age will be 44 years. Their father, Mario, is over 40 years old. Grandma Fernanda - their mother's mother - is between 70 and 80 years old. Roberto and Riccardo are fans of Fiorentina; Renato, on the other hand, supports Juventus. What is the sum of the current ages of Roberto, Riccardo, and Renato?

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1745

2002

final

no

C1 C2 L1 L2 GP

Geometry

The length of the rectangle is 7 units

On a squared sheet of paper (each square with a side of 1 cm), following the grid lines, Elena draws a rectangle with a base length of 6 cm. Then she draws the diagonal of the rectangle and notices that it crosses 12 squares but does not cross any grid intersection point (with the exception of its two endpoints). How many cm is the height of the rectangle?

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1746

2002

final

no

C1 C2 L1 L2 GP

Algebra

Carla owns 40 euros

Carla wants to organize her 1-euro coins. She then forms piles of 9 coins and notices that the number of remaining coins is equal to the number of piles she made. She then decides to redistribute all her coins into piles of seven: again, she observes that the number of remaining coins is equal to the number of piles formed. How many 1-euro coins does Carla have?

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1747

2002

final

no

C2 L1 L2 GP

Algebra

The golden weight is 11 grams

A balance scale has 17 weights, respectively 1 gram, 2 grams, 3 grams, ... 17 grams. Ten weights are black, six are silver, and only one is gold. The silver weights total 32 grams more than the black ones. What is the weight in grams of the gold weight?

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1748

2002

final

no

C2 L1 L2 GP

Algebra

Rosetta has 11 tickets of 50 euros in cash

At the end of the day, Rosetta, the most famous baker in Milan, tallies her cash and finds herself with 870 euros, consisting of 10, 20, and 50 euro notes. The number of notes of each denomination are consecutive numbers. How many 50 euro notes does Rosetta have in the till?

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1749

2002

final

no

L1 L2 GP

Algebra

The three numbers are 3, 5, 7

Desiderio wrote three prime numbers and noticed that their product is equal to 7 times their sum. What are these three numbers, in increasing order?

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1750

2002

final

no

L1 L2 GP

Combinatorics

The number of nuts is 13

To pass the long winter evenings, Rosi and Angelo have invented a game with 20 walnuts they collected in the garden. Rosi starts. She divides the 20 walnuts on the table into several equal piles. (The number of walnuts in each pile must be either 1 or a prime number). Then she takes one pile, removes it from the table, and pockets it. Now it's Angelo's turn, who repeats the same operations: he regroups the walnuts remaining on the table, divides them following the same criteria, takes a pile, and pockets it. Then it's Rosi's turn and then Angelo's, and so on, with the proviso that each player must make at least two piles, unless only one walnut remains on the table (in which case the player will take it). The goal of the game is to pocket the maximum possible number of walnuts. How many walnuts is Rosi guaranteed to be able to take, regardless of Angelo's play?

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1751

2002

final

no

L2 GP

Combinatorics

The product is 140 (20x7)

Jacob often goes to the beach, where he has a beautiful collection of marbles that he runs on fantastic sand tracks. For him, when it rains, his dad invented another game: he arranges 21 marbles into 6 boxes, in such a way that no box remains empty and that all boxes contain a different number of marbles. The game begins: with each “move”, Jacob can take a certain number of marbles (of his choice) from one box and put them into another, on the condition, however, that he thereby doubles the contents of this latter box. Following this rule, Jacob puts the maximum possible number of marbles into one box. Give the product of the maximum number of marbles in one box by the minimum number of “moves” to achieve it.

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1752

2002

final

no

L2 GP

Combinatorics

At most, the total is 24

The performances of 18 figure skaters are evaluated by 9 judges. After the performance, each judge assigns a rank (from 1st to 18th place) to each competitor, without ties. For any given competitor, the ranks assigned by the 9 judges do not differ from each other by more than 3 ranks. For each competitor, the sum of the 9 assigned ranks is then calculated. What is the maximum possible value of the smallest total score obtained by any skater?

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1571

2003

autumn games

no

C1

Arithmetic

The largest whole number of trs and even digits all different is 864

What is the largest integer formed by three distinct even digits?

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1573

2003

autumn games

no

C1

Arithmetic

The diabolical number is 120 : 3 = 40, The satanic number is 2 x 133 = 266, The magical number is 10 x ( 266 = 40) = 2260 The magical number of the witch is 2260

Our witch specializes in mathematical potions. Here are the ingredients for her concoction: 3 133 38 42 2 56 9 120 6 “I divide the largest even number by the smallest odd number and get the diabolical number. Then I multiply the smallest even number by the largest odd number and get the satanic number. Finally, I multiply by 10 the difference between the satanic number and the diabolical number and get the ... magic number!”. What is the witch's magic number?

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1574

2003

autumn games

no

C1 C2

Algebra

9

I have 65 candies. I eat one and distribute the rest equally among all my teammates. Each teammate receives a number of candies equal to the number of my teammates. How many of us are on the team?

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1577

2003

autumn games

no

C2 L1

Combinatorics

On the planet Numerus you can register at most 90 x 441 = 39690 vessels

On the planet Numerus, spacecraft have license plates, just like cars on Earth. These license plates, however, consist of only two digits, followed by two letters. For example: 95 LV. Note: no license plate starts with 0, and the alphabet in use on Numerus consists of 21 letters. How many spacecraft can be registered at most on the planet Numerus?

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1578

2003

autumn games

no

C2 L1

Combinatorics

97586

What is the largest number of five different digits that I can write following the rule that two adjacent digits can never be consecutive (like, for example, 1 and 2 or 8 and 7)?

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1581

2003

autumn games

no

L1 L2

Arithmetic

18 days.

Carla bought 72 mice to feed her cats for 12 days. Each cat eats the same (whole) number of mice per day, and this number remains the same, day after day. If Carla decides to go on vacation with two cats, for how many days will her other cats have food, sharing the 72 mice?

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1582

2003

autumn games

no

L1 L2

Combinatorics

The family consists of 240 members

I belong to a family of 4-digit numbers. Our first digit, which is non-zero, is strictly less than our second digit, which is equal to the third; finally, our third digit is strictly greater than the fourth. How many members does my family consist of?

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1583

2003

autumn games

no

L1 L2

Algebra

The age of the son of Nando is 15

Nando is a puzzle enthusiast and explains his son's age as follows: "To get his current age, you must take three times the age he will be in two years and subtract three times his age from three years ago." What is Nando's son's age?

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1584

2003

autumn games

no

L1 L2

Algebra

3 days, 22 hours 30 minutes

A wild duck takes 9 days to go from Norway to Morocco and 7 days to go from Morocco to Norway. Assume that all ducks fly in a straight line and at a constant speed. Two ducks depart together, one from Norway and the other from Morocco. How long after will they meet in flight? (Give the answer in days, hours, and minutes).

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1585

2003

autumn games

no

L2

Geometry

The distance between their ends is 5,292 cm

The hands of a clock are respectively 4 cm and 6 cm long. What is the distance between their ends when the clock shows two o'clock? (If necessary, use 1.414 for √2; 1.732 for √3; 2.23 for √5; 2.646 for √7).

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1586

2003

autumn games

no

L2

Logic

In the family we have 6 red fishes

My family consists of Dad, Mom, my brothers, my sisters, me and... my goldfish. There are a total of 14 hands and 13 mouths. Fortunately, only the goldfish are without hands. How many goldfish do we have in the family?

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1588

2003

autumn games

no

L2

Combinatorics

The required number is 8/69

The fraction machine accepts, as input, irreducible fractions strictly between 0 and 1. Each fraction f = n/d gives as output, as a result, the reciprocal of the number f+d. These results are then arranged in decreasing order. What is the twentieth result? (Give this result as an irreducible fraction.)

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1590

2003

team games

no

C2 L1 L2

Arithmetic

the corridor can measure 3 m, 6 me 9 m

To tile the corridor of his house, Michele can use square tiles with side lengths of 20 cm, 25 cm, or 30 cm. The corridor is not very long – less than 10 m – but it has the property that in all three cases (using small, medium, or large tiles), it can be exactly covered with tiles of uniform size. What is its length?

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1591

2003

team games

no

C2 L1 L2

Combinatorics

0

Nine tokens are aligned: each of them has a white face and a black face. Initially, all faces are white. In one move, two tokens can be flipped, provided they are adjacent. What is the minimum number of moves to obtain an alignment of 9 tokens, all black?

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1592

2003

team games

no

C2 L1 L2

Algebra

Desire sold its 9 Soviet stamps and received 15 Cuban stamps.

Desiderio has 45 stamps, some Soviet and others Cuban. Now, however, he wants to specialize in collecting Cuban stamps and asks for help from his friend Amerigo, who is also a passionate philatelist. The agreement is: 3 Soviet stamps for 5 Cuban stamps. At the end of the exchange, Desiderio has 51 stamps, all of which are Cuban. How many Soviet stamps did he initially have?

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1593

2003

team games

no

C2 L1 L2

Combinatorics

The five friends can be arranged in 36 different ways.

Carla, Liliana, Milena, Rosi, and Silvia are posing for a souvenir photo: three of them can sit in the first row; the other two, standing, in the second row. How many are the possible positions they can occupy (and therefore the possible different souvenir photos) considering that Carla and Liliana always want to stay close, one next to the other?

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1594

2003

team games

no

C2 L1 L2

Algebra

1*4= 6

At the last International Congress, mathematicians defined a new (non-commutative) operation on integers, denoted a * b. For now, it is only known that: 0 * a = a + 1 a * 0 = (a – 1) * 1 (a + 1) * (b + 1) = a * [ (a + 1) * b ] What is the value of 1 * 4?

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1595

2003

team games

no

C2 L1 L2

Algebra

The amount of the cheque is 5,11 €

Pietro goes to the bank to cash a check (in Euros) because he has run out of money. The distracted teller, when giving him the money, accidentally swaps the number of euros with the number of cents. Pietro doesn't notice anything and, with the money received, goes to buy the local newspaper (which that day costs only 83 cents). It is at that point, after buying the newspaper, that he realizes he has exactly twice the amount he should have received from the check in his pocket. What was the original amount of the check?

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1596

2003

team games

no

C2 L1 L2

Arithmetic

The answer to be given and 0

Nando has an irresistible attraction to betting. So, one day, he starts by betting 1 Euro. He loses and then bets 2 Euros. He loses again and bets 4 Euros. He continues this way – always losing and always doubling the amount bet – until after the tenth bet he realizes he has lost his entire initial sum of Euros. After how many bets had Nando lost exactly half of his initial sum of Euros?

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1598

2003

team games

no

C2 L1 L2

Pattern Recognition

666

Find (and discover...!) the natural number that satisfies the following conditions: * it is greater than 100 and less than 1000; * it is written using a single digit (repeated); * it is the sum of the first n natural numbers (for some n).

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1599

2003

team games

no

C2 L1 L2

Algebra

48

When on the escalator, Renato doesn't stand still and climbs step by step at a constant speed of one step per second. He thus takes 30 seconds to reach the top. One day, distracted, to go up he takes the escalator going down (at the same speed it goes up) and takes 2 minutes to reach the top. How many steps are there on the escalator (when it is still)?

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1602

2003

team games

no

C2 L1 L2

Combinatorics

14

A person who writes a number on a slip of paper (and pays 5 Euros) is entitled to participate in the drawing of a ticket, which displays another number between 100 and 999. If the sum of the digits of this drawn number is equal to the number previously written on the slip of paper, 50 Euros are won. Knowing that the 900 numbers between 100 and 999 have the same probability of being drawn, what is the number to write on the slip of paper that gives the greatest probability of winning?

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1604

2003

team games

no

C2 L1 L2

Algebra

147 grams

A, B, C, D, E, are five (good!) chocolate eggs. Some are also filled with cream; they have the same weight but are heavier than those without filling (which also all have the same weight). A and E, together, weigh 252 grams; A, B and C, together, weigh 420 grams; B, C, D and E, together, weigh 567 grams. How much does each of the cream-filled eggs weigh?

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1605

2003

team games

no

C2 L1 L2

Algebra

The required difference is 97.07 kg

Three of the horses participating in the Grand Prix, when weighed (together with their jockeys), registered a total of 1000 kg. The three jockeys, altogether, weigh 181 kg. The first horse weighs five times its jockey; the second four and a half times its jockey; the third four times the weight of its jockey. What is the minimum difference that can exist between the heaviest horse and the lightest horse? (The answer can be rounded to the nearest Kg.)

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1606

2003

team games

no

C2 L1 L2

Arithmetic

you will get a number with 999 times the number 1

Calculate the sum N = 9 + 99 + 999 + .... + 99 ... 9 (where the last number is composed of the digit 9, repeated 999 times). How many times does the digit 1 appear in N?

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1607

2003

team games

no

C2 L1 L2

Algebra

September and 53 years

Dr. Settembre's nine children were born, all spaced out by the same number of years. In 1998, the square of Dr. Settembre's age was equal to the sum of the squares of the ages of his nine children. What is Dr. Settembre's age today, in 2003?

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1608

2003

team games

no

C2 L1 L2

Pattern Recognition

The condition is satisfied only if X is worth 7

Angelo, the deli owner, finds an incredibly long code number on a package (approximately 50 digits long). He then decides to slice it into two-digit "chunks", starting from the right, and then summing the various "chunks" (for example, if the number ends with … 367523, he writes 23 + 75 + 36 + …). This way he obtains the number 1998. The next day, not satisfied with the procedure, he applies another method to the incredibly long number: he writes its first digit (always starting from the right), subtracts the second, adds the third, subtracts the fourth… and so on (in the previous example, he would have written 3 – 2 + 5 – 7 + 6 – 3 + …). He thus obtains a (positive) single-digit number. Which one?

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1609

2003

Rosi's games

no

C2 L1 L2

Arithmetic

Vasco returned 3.80 Euro

Vasco buys a gift, which costs 18.90 Euros, for his best friend. He pays with a 50 Euro banknote, but the cashier, distracted, gives him 34.90 Euros as change. After leaving the shop, Vasco notices the error and goes back in to return the amount that does not belong to him. How much did he return?

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1610

2003

Rosi's games

no

C2 L1 L2

Algebra

Federico will have 33 anJli

On January 1, 1991, Federico was twice Leone's age; on January 1, 1999, Federico was 10 years older than Leone. How old will Federico be on January 1, 2004?

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1611

2003

Rosi's games

no

C2 L1 L2

Combinatorics

The required numbers are: 8

How many three-digit numbers are there, greater than 600, in which the units digit is half the hundreds digit, and the tens digit is different from both the units digit and the hundreds digit?

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1612

2003

Rosi's games

no

C2 L1 L2

Arithmetic

The film lasted: 87 minutes

The movie started at 18:45 and finished at 20:12. How long did it last?

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1615

2003

Rosi's games

no

C2 L1 L2

Algebra

The girls are: 16

In a class of 27 students, there are five more girls than boys. How many girls are there?

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1616

2003

Rosi's games

no

C2 L1 L2

Combinatorics

The minimum number of gtiaccioli e: 6

In the ice cream freezer there are 60 popsicles of 5 different flavors, a dozen for each flavor. What is the minimum number of popsicles that must be taken (without looking ...) to be sure of having taken two of the same flavor?

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1617

2003

Rosi's games

no

C2 L1 L2

Logic

George's intruder is 3

Giorgio, using nine of the following ten digits, has written three consecutive numbers: 0-0-0-1-1-2-2-3-9-9. What is the unused digit?

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1618

2003

Rosi's games

no

C2 L1 L2

Algebra

The five friends hanno caught, altogether, 40 pesci

Five friends (Alberto, Carlo, Davide, Edoardo, Filip) went fishing together. Alberto and Carlo, together, caught 14 fish; Carlo and Davide, together, caught 20; Davide and Edoardo, together, 18; Edoardo and Filip, 12; finally, Alberto and Filip caught 16. How many fish did the five friends catch in total?

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1620

2003

semifinal

no

C1

Logic

48 eggs

Rosina brings a basket full of eggs to the market. First, Carla buys half of the eggs, then Milena buys half of the remaining eggs. Finally, Maddalena pays for 10 eggs and Rosina tells her: 'Miss, I'll give you my last two eggs; this way I will return to the farm with an empty basket.' How many eggs did Rosina arrive at the market with?

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1622

2003

semifinal

no

C1 C2

Algebra

360 grams

In the Borgio Verezzi fishing competition, scores are awarded by giving competitors 50 points for each fish, plus 1 point for each gram of fish caught. Jacob caught 19 fish for a total weight of 2430 grams. Mirko, on the other hand, had caught 14 fish for a total weight of 1860 grams but, just a moment before the final whistle, he manages to catch 2 fish of the same weight and ends up with the same score as Jacob. What is the weight in grams of one of the two last fish caught by Mirko?

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1625

2003

semifinal

no

C1 C2 L1 L2 GP

Algebra

6 guys

At a Mathematics competition, the number of girls was double the number of boys. Each participant scored 8, 9, or 10 points, and altogether they scored a total of 156 points. How many boys participated in the competition?

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1630

2003

semifinal

no

L1 L2 GP

Algebra

4 Solutions (20;4) (10;6) (8;8) (6;18)

Chiara and Anna have each chosen a positive integer. The product of one-third of Chiara's number and one-fifth of Anna's number is equal to the sum of one-fifth of Chiara's number and one-third of Anna's number. What are the two numbers?

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1631

2003

semifinal

no

L2 GP

Geometry

2 solutions: 4 cm; 18 cm

In the plane, Nando has drawn three lines that converge at a point, forming angles of 60 degrees. He then marks a point in the plane and measures its distance to two of the three lines. He finds these distances to be 7 cm and 11 cm. What is the distance from this point to the third line? Give this distance in centimeters, rounded to the nearest hundredth if necessary.

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1632

2003

semifinal

no

L2 GP

Arithmetic

80

Multiplying a (very large) number by 5, a forty-digit number is obtained, composed exactly of thirty '5's and ten '7's. What is the sum of the digits of the initial number?

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1636

2003

final

no

C1 C2 L1 GP

Arithmetic

(7+7)x7 x7+7+7= 707

Mirko, the prankster, enjoyed deleting the symbols (parentheses), +, and x from his friend Jacob's calculation. Put them back in the correct places, so that the equation written below is satisfied: 7 7 7 7 7 7 7 = 707

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1637

2003

final

no

C1 C2 L1 L2 GP

Geometry

The minimum number of intersections is: 4

Consider 4 circles, all with the same radius and never tangent to each other. Arrange them in the plane such that the figure formed is connected (i.e., forms a single "piece"). What is the minimum total number of intersection points the four circles will have?

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1641

2003

final

no

L1 L2 GP

Geometry

Oscar will travel 150 m

The great pyramid of Pharaoh Matemankhamon has a square base with a side length of 100 m; its four faces are equilateral triangles. Oscar - the scarab of the Nile - is at the foot of the pyramid, at the midpoint of the base of the South face. As part of his systematic exploration of the valley, Oscar wants to go to the point diametrically opposite to his current location (i.e., at the midpoint of the base of the North face), following the shortest possible path and climbing the pyramid, if necessary. What distance will Oscar travel?

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1643

2003

final

no

L2 GP

Algebra

The banana is long: 21 cm

A rope is placed along the top edge of a fence and is as long as each side of the fence itself. The rope weighs 300 grams per meter. At one end of the rope, there is a small monkey holding a banana, while at the other end there is a counterweight with a weight equal to that of the small monkey. The banana weighs 10 grams per cm. The total length of the rope, in meters, is equal to 1/3 of the monkey's age in years, and the monkey's weight, in grams, is equal to 200 times the age of the monkey's mother. The sum of the monkey's age and its mother's age is 30 years. Adding twice the monkey's weight and 40 times the banana's weight, the same total is obtained as would be obtained by adding 10 times the rope's weight and the counterweight's weight. The monkey's current age is half of the age its mother will be when the monkey reaches the age its mother is today. How long is the banana?

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1646

2003

international final day 1

no

CE

Logic

Friday

I was born on June 13th. My friend Peggy was born on May 7th. This year her birthday is on a Wednesday. Find the day of my birthday.

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1649

2003

international final day 1

no

C1 C2 L1 GP L2 HC

Logic

Nicolas

The president of FIGM has been kidnapped. The police have three suspects: 2 always lie and only one always tells the truth. Here is an excerpt from the interrogation: Nicola: "I did not kidnap the president." Matteo: "Nicola is not a liar." Maria: "Matteo did not kidnap the president." Who kidnapped the president?

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1650

2003

international final day 1

no

C1 C2 L1 GP L2 HC

Combinatorics

12 sauts

A chess knight moves on a 20x20 chessboard. Recall that a chess knight moves along the diagonal of a 2x3 rectangle (it is considered that the knight is always positioned at the center of the square). After tracing a triangular path, the knight returned to its starting square, always respecting the rules of movement. What is the minimum number of jumps it made?

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1651

2003

international final day 1

no

C1 C2 L1 GP L2 HC

Algebra

14 enfants réponses admises : 12 ou 13 enfants

Matilde and Mattia have invited their family and some friends to a birthday lunch. Mattia has two large identical rectangular tables. He makes the following observation: "If I join them along one side, so as to form a large rectangle, only 20 people will be able to sit around them (allowing the same space for each guest). If I join them along the other side, I gain two seats, but it's still not enough!" Matilde observes: "That's true, but we could use two separate tables because there are as many adults as children, and by doing so there would be enough room for everyone." How many children will attend this lunch?

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1654

2003

international final day 1

no

L1 GP L2 HC

Logic

7 solutions: 10125; 2025; 30375; 405; 50625; 6075; 70875

The FIGM locks the solutions of its games in a safe, which opens with a code. This code, considered as an integer, does not end with a 0 and has the following particularity. If a certain digit is removed, the new number obtained is equal to 1/9 of the initial number. It is then possible to remove another digit from this second number and obtain exactly 1/81 of the initial number. What is the code number of the FIGM's safe?

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1658

2003

international final day 1

no

L2 HC

Combinatorics

1 solution: 1949 et 1997

A distracted professor gets locked in a Museum. A guard notices his presence and points out to him that it is forbidden to enter the museum at that hour. He offers to let him out if he solves the following enigma. « Here is a number: 3892153. A computer will quickly establish that it is equal to 1752^2 + 907 X 907, but also to 1172^2 + 1587^2. Can you find two integers greater than 1 whose product it is? » What are these numbers?

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1660

2003

international final day 2

no

CE

Arithmetic

304

Domenico counts aloud « 1, 2, 3, 4 ». Then he continues mentally and finishes aloud with « 309, 310 ». How many numbers did he count mentally?

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1661

2003

international final day 2

no

CE

Logic

9 h 07

Eliana, Melina, and Geraldina agreed to meet at 8:50 AM at the park. Eliana arrived fifteen minutes late. Melina arrived three minutes early, and Geraldina arrived two minutes after Eliana. At what time did the three friends meet up?

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1665

2003

international final day 2

no

C1 C2 L1 GP L2 HC

Combinatorics

12 façons

This morning I don't know what to wear. I have a choice between a dress, a skirt, trousers, a shirt and four pairs of shoes. In how many different ways can I get dressed? Observation: to get dressed, I wear only one top garment, only one bottom garment and only one pair of shoes. If I wear the dress, I only have to put on my shoes.

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1666

2003

international final day 2

no

C1 C2 L1 GP L2 HC

Algebra

20 %

On a jar of chestnut cream, the following information can be read: • prepared with 50g of fruit per 100g of product • sugar content: 60g per 100g. It is also stated that, apart from a vanilla flavoring (of negligible mass), chestnut pulp and sugar are the only ingredients of this artisanal product. What is the percentage of sugar contained in the chestnut pulp?

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1667

2003

international final day 2

no

C2 L1 GP L2 HC

Algebra

40 euros

Catherine has a 30% discount voucher, valid for the simultaneous purchase of three items. When she is about to pay her bill of 168 euros, the sales assistant informs her that she can get a 40% discount if she buys a fourth item. But Catherine only has 168 euros in her pocket. After a quick calculation, she realizes that, despite this, she can buy a fourth item, in addition to those she has already chosen and still wants to buy. What is the maximum price of the fourth item she can buy?

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1669

2003

international final day 2

no

C2 L1 GP L2 HC

Algebra

1 solution : 4999

In the equations CI + CI = LY and CIJM + CIJM = JMLY, the same letter always replaces the same digit, but a digit can be replaced by different letters. No number begins with a 0. Find the value of CIJM.

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1670

2003

international final day 2

no

C2 L1 GP L2 HC

Combinatorics

3 solutions: 4624 ; 6084 ; 8464

The FIGM safe's code changes every day. Today it must be a four-digit number (where the first and last digits are not zero), the square of an integer, and all its digits must have the same parity. What is today's code?

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1463

2004

autumn games

no

CE C1

Arithmetic

Guido will eat 9 slices

Jacob is a generous boy and wants to share his round cake, divided into slices, with his friends. He gives half of it to Guido, who in turn gives half of what he receives to Giacomo, who isn't very hungry and then returns half of what he received to Guido. How many slices of the cake will Guido eat?

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1464

2004

autumn games

no

CE C1

Arithmetic

Angelo has to give Rosi 9 €.

Rosi and Angelo always divide their expenses in equal parts. Yesterday Rosi went to the supermarket and spent 35 Euros. Today Angelo made other purchases for 17 Euros. How much money must Angelo give to Rosi to settle the accounts?

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1469

2004

autumn games

no

C2 L1

Arithmetic

There are 210 houses on the same side of my house.

On my street, houses on one side are numbered with even numbers, and on the other side with odd numbers. My house is number 98. If they had started the numbering from the other end of the street, it would be number 324. How many houses are there on the same side as my house?

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