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

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1407	2005	semifinal	no	C1	Arithmetic	Liliana will pay the coat Euro 48,48	Even during sales, Renato never gives a discount of more than 20 Euros. Incidentally, he has a strange way of labeling discounted merchandise: he always ensures that, in the price, the number of Euros is equal to the number of cents. For example, the t-shirt I like and would like to buy was discounted to 29.29 Euros. My friend Liliana wants to buy a coat which, before the sales, cost 67.99 Euros. How much will she pay for it, if Renato gives her the maximum discount?	Not supported with pagination yet
1408	2005	semifinal	no	C1	Logic	We will finish cooking king at 4:20 p.m.	It's 2:40 PM and my friends have finished preparing some magnificent desserts; now, they just need to be cooked. In my oven, however, only one "piece" can be baked at a time and – as all good cooks know – baking cannot be interrupted. Anna has prepared a cake that bakes in half an hour; Chiara, a dessert that bakes in 20 minutes, and Ingrid a tart that now needs to rest for exactly 35 minutes before being baked for three quarters of an hour. At what time – as soon as possible! – will we have finished baking everything?	Not supported with pagination yet
1409	2005	semifinal	no	C1 C2	Logic	The total points were: 19	During a handball match between the Algebrists' team and the Geometers' team, which ended with a score of 23 to 19 in favor of the Algebrists, there was a moment when the Algebrists had as many points as the Geometers subsequently scored until the end of the match. How many points did the two teams have, together, at that precise moment?	Not supported with pagination yet
1411	2005	semifinal	no	C1 C2 L1	Logic	The number you searched for is: 163	Find a three-digit number, with all digits distinct, such that: - the sum of its digits is equal to 10; - the product of the first two digits is equal to 6; - the tens digit is the largest of the three digits.	Not supported with pagination yet
1412	2005	semifinal	no	C1 C2 L1 L2 GP	Combinatorics	At most the members of the group are: 8	In a group of friends, each has 10 tokens available (numbered from 1 to 10) and must choose three of them whose sum is 13. By doing so, everyone manages to form their own "group" of three tokens such that it is different from those of the others (meaning that it has at least one different "number"). How many, at most, are the members of the group?	Not supported with pagination yet
1413	2005	semifinal	no	C1 C2 L1 L2 GP	Geometry	The dimensions of the fourth rectangle are: 5 x 6	A cardboard square has been divided into four rectangles. Three of these rectangles have dimensions 4x6, 5x9, and 2x11. What are the dimensions of the fourth rectangle?	Not supported with pagination yet
1414	2005	semifinal	no	C1 C2 L1 L2 GP	Arithmetic	There are 18 pieces of chocolate	Seven students received twelve identical chocolate bars, each weighing 91 grams. They then decided to divide them equally among themselves, making the minimum number of pieces. How many pieces of chocolate are there (including whole bars) at the moment of its equal division among the seven students?	Not supported with pagination yet
1415	2005	semifinal	no	C2 L1 L2 GP	Algebra	The captain is 70 years old.	For his birthday, the captain – not yet a centenarian – invited his 3 daughters, his 5 grandchildren, and his 7 great-grandchildren: - the 3 daughters have consecutive ages; - the 5 grandchildren have consecutive ages; - the 7 great-grandchildren have consecutive ages; - the sum of the daughters' ages is equal to the sum of the grandchildren's ages and equal to the sum of the great-grandchildren's ages; - the captain's age is equal to two-thirds of the sum of the daughters' ages. How old is the captain?	Not supported with pagination yet
1417	2005	semifinal	no	L1 L2 GP	Combinatorics	The smallest value of N is 57	Among N arbitrarily chosen positive integers, we want to be sure to find two whose sum or difference is divisible by 111. What is the smallest possible value of N?	Not supported with pagination yet
1420	2005	final	no	C1	Pattern Recognition	Michel had, this month, at most, 9 votes of Mathematics	In Michel's school, as in all French schools, grades range from 0 to 20 (inclusive). In his gradebook, Michel has written down his Math grades for this month. They are all integers. The first grade is an 8; then, each subsequent grade is triple or half of the previous one. What is the maximum number of Math grades Michel had this month?	Not supported with pagination yet
1421	2005	final	no	C1	Arithmetic	Milena, Rosi, Carla, Desire	At the end of the school's cross-country race, the teacher asks his students to check their heart rates. Carla counts 25 beats in 15 seconds; Milena counts 24 beats in 20 seconds; Rosi counts 45 beats in 30 seconds; Desiderio counts 110 beats in one minute. Order the students (indicated by the initial of their name) from the one with the slowest pulse to the one with the fastest pulse.	Not supported with pagination yet
1422	2005	final	no	C1 C2	Logic	Luca is 1 year old	Today, Luca, Chiara, and Anna's combined age is 60 years. When Luca was born – after 1995 – Chiara and Anna were both older than 10. Anna is one year younger than Chiara. Chiara's age is a multiple of 6. What is Luca's age?	Not supported with pagination yet
1423	2005	final	no	C1 C2	Algebra	60 shells	Jacob's shell collection consists of 287 shells, divided into 5 boxes: a blue box, a green box, a yellow box, a red box, and a white box. The blue box and the green box contain the same number of shells. Similarly, the yellow box and the red box have the same number of shells. The box with the fewest shells contains 53; another box contains 57. What is the minimum number of shells in the box that contains the most (or one of the boxes with the most, if there are two)?	Not supported with pagination yet
1427	2005	final	no	C1 C2 L1 L2 GP	Algebra	13	How many three-digit integers are equal to 46 times the sum of their digits?	Not supported with pagination yet
1429	2005	final	no	C2 L1 L2 GP	Algebra	There are 2 solutions: 2012 2026	Marco's birth year is special in that the product of its digits is the square of a strictly positive integer. Today, in 2005, Marco is waiting for the year in which he will have an age equal to the square root of the product of the digits of his birth year. In what year will this occur?	Not supported with pagination yet
1430	2005	final	no	L1 L2 GP	Combinatorics	3 Solutions	We want to write 13 distinct positive integers in ascending order such that their sum is 94. How many different solutions are there?	Not supported with pagination yet
1431	2005	final	no	L1 L2 GP	Algebra	The speed of Angelo is 10 m/sec	Angelo and Renato are training for the upcoming Cycling World Championship, using an oval track of 360 meters. Angelo overtakes Renato every three minutes. If, however, one of them were to ride in the opposite direction, they would meet every twenty seconds. What is Angelo's speed? (Give the answer in meters per second)	Not supported with pagination yet
1436	2005	international final day 1	no	C1	Logic	Nicola, Maximilian, Camilla, Roman, Thomas	The time allowed for the test in category C1 is two hours. It started at 2:35 PM and must end at 4:35 PM. Romano submits three-quarters of an hour before the end. Tommaso submits one and a half hours after the start of the test. Massimo submits halfway through the allotted time. Camilla submits at 3:45 PM. Nicola submits 50 minutes after the start of the test. Rank the five participants according to their submission order (write the initials of their names).	Not supported with pagination yet
1439	2005	international final day 1	no	C1 C2	Logic	mark No 15	Teo and Tommaso play on an escalator that consists of 29 visible steps. Teo starts from the top and, in the time the escalator moves up one step, he descends 6 steps. At the same time, Tommaso starts from the bottom and climbs 4 steps. If they continue to descend and ascend at the same speed, at which step do they meet, counting the visible steps, at the precise moment they meet, starting from the bottom?	Not supported with pagination yet
1444	2005	international final day 1	no	C2 L1 L2	Combinatorics	2 solutions : 4 and 21	In the convent of the City of Mathematics, if you meet two nuns, chosen randomly from the set of nuns residing there, you have exactly a one in two chance that they are both brown. How many nuns reside in the convent?	Not supported with pagination yet
1445	2005	international final day 1	no	L1 L2	Combinatorics	3 solutions : 24, 40, 72	Ludovico takes a portion of the 104 cards from a canasta deck, forming a new deck which we will call "M". He shows us the seventeenth card from the top of this new deck, then reseals it without altering the cards' respective positions, and places it on the table. Finally, he asks Mina to perform the following operations: take the first card of deck M and place it at the bottom; take the next card and discard it; continue this process until only one card remains. Mina does as Ludovico asked and, to her surprise: the last card remaining is exactly the one Ludovico had shown. How many cards were in deck M?	Not supported with pagination yet
1448	2005	international final day 1	no	L2	Geometry	42 carrés.	Four distinct points have been chosen on a plane. Squares are to be drawn, each such that at least one of its sides (possibly extended) passes through each of these points. It is known that the four chosen points are such that it is possible to draw a finite number of squares satisfying this condition. What is the maximum number of squares satisfying this condition that can be drawn?	Not supported with pagination yet
1451	2005	international final day 2	no	C1 C2	Arithmetic	9 h 10 min	In the last four days, Maria has played a lot of tennis. On Saturday, she played from 18:00 to 19:30; on Sunday, from 12:00 to 13:43, then in the evening for 1 hour and 55 minutes; on Monday, her match lasted 2 hours and 5 minutes; and on Tuesday, 117 minutes. Maria, very tired, wonders how much time she has spent on the tennis courts in the last four days. Help her find the answer in hours and minutes.	Not supported with pagination yet
1455	2005	international final day 2	no	C2 L1 L2	Combinatorics	216 marquis	Geraldina enjoys doing favors for her neighbors who live in the same building. Here's what she needs to do this afternoon: visit Francesco, Tiberio, and Margherita to say hello; return Isa's DVD to Stan; pick up the keys from Sergio to collect his mail on the first floor. There are twelve steps between each floor, and the elevator is out of order. What is the minimum number of steps she must climb and descend to do all this (starting from her apartment and returning to it)? Note: Sergio and Stan are not in a hurry, so Geraldina does not have to bring them the mail and the DVD immediately after picking them up. 8th floor Margherita 7th floor Isa 6th floor Francesco 5th floor Sergio 4th floor Tiberio 3rd floor Géraldina 2nd floor Stan 1st floor	Not supported with pagination yet
1458	2005	international final day 2	no	L1 L2	Combinatorics	3 croisements	Randomly place six points on a sheet of paper. Using a pencil, connect every pair of them with lines. Suppose that any two lines - either have no point in common; - or intersect exactly once and thus have one and only one point in common. What is the minimum number of intersections you will have?	Not supported with pagination yet
1227	2006	autumn games CE	no	CE	Pattern Recognition	The next issue is: 18063 which corresponds to 2007x9	Desiderio begins performing the multiplications 2007 x 1; 2007 x 2; 2007 x 3; 2007 x 4; etc. Desiderio then realizes that, unlike the first two results, the result of 2007 x 3 is written with all different digits. What is the next result that possesses the same property (of being written with all different digits)?	Not supported with pagination yet
1229	2006	autumn games CE	no	CE	Arithmetic	The height of the Luke pyramid is 95 cm = (14 + 13 + 12 +.... + 7 + 6 + 5).	Today, at Luca's kindergarten, the 'little ones' are playing with cube-shaped boxes which they place one on top of the other, such that the side of one box always measures one centimeter less than that of the box it rests on. Luca has ten boxes. The side of the largest one measures 14 cm. His pyramid is built using all the boxes, stacked from largest to smallest. What is the height of Luca's pyramid?	Not supported with pagination yet
1230	2006	autumn games CE	no	CE	Logic	Anna will greet her friends in the following order: S-I-G-C Silvia, Ingrid, Giovanna and Claudia.	Before going abroad for six months for her studies, Anna wants to say goodbye to her friends who live in her apartment building. Ingrid is home between 11 AM and 11:25 AM; Giovanna returns from the supermarket at 11:30 AM; Claudia will return home at a quarter to noon; Silvia (who has to go play tennis) told her to visit before noon. Anna wants to stay 20 minutes with each of her friends and leave immediately afterward. In what order should she say goodbye to her friends to be able to leave as soon as possible? Answer by providing (in order) the initials of their names.	Not supported with pagination yet
1233	2006	autumn games	no	C1	Arithmetic	95 cm	Today, at Luca's kindergarten, the "little ones" are playing with cube-shaped boxes, which they place one on top of the other, such that the side of one box always measures one centimeter less than that of the box it rests on. Luca has ten boxes. The side of the largest one measures 14 cm. His pyramid is built using all the boxes, stacked from largest to smallest. What is the height of Luca's pyramid?	Not supported with pagination yet
1234	2006	autumn games	no	C1	Logic	S-I-G-C (Silvia, Ingrid, Giovanna, Claudia)	Before going abroad for six months for study reasons, Anna wants to say goodbye to her friends who live in her apartment building. Ingrid is home between 11:00 AM and 11:25 AM; Giovanna returns from the supermarket at 11:30 AM; Claudia will be home at a quarter to noon; Silvia (who has to go play golf) told her to stop by before noon. Anna wants to stay 20 minutes with each of her friends and leave immediately afterwards. In what order should she greet her friends to be able to leave as soon as possible? Answer by providing (in order) the initials of their names.	Not supported with pagination yet
1236	2006	autumn games	no	C1 C2	Combinatorics	78 €	For the end-of-year show, the physical education teacher needs to buy 49 ribbons. Here are the supplier's offers: One ribbon: 3 Euros. A pack of 2 ribbons: 5 Euros. A pack of 5 ribbons: 10 Euros. Additionally, if you buy three identical packages, the fourth one is free. If she calculates carefully, what is the minimum amount the teacher will have to spend?	Not supported with pagination yet
1237	2006	autumn games	no	C1 C2	Algebra	1917 or 1932	Our grandfather was born before the Second World War, but after the start of the First. When he celebrates his next birthday, the year – 2007 – will be equal to his birth year increased by five times the sum of the digits of his birth year. In what year was the grandfather born? (Provide all possible solutions)	Not supported with pagination yet
1238	2006	autumn games	no	C1 C2	Logic	The number of true sentences is 3	How many simultaneously true sentences can the box below contain? 1. In this box, sentence 2 is false. 2. In this box, sentence 3 is false. 3. In this box, sentence 4 is false. 4. In this box, sentence 5 is false. 5. In this box, sentence 6 is false. 6. In this box, sentence 1 is false.	Not supported with pagination yet
1239	2006	autumn games	no	C1 C2	Algebra	Old price 45 €, new price 54 €	Sara has finally decided to buy a video game, which she has actually been dreaming of for a long time. When the store cashier informs her of the price to pay, Sara is however surprised: "It's not possible! You've swapped the units digit with the tens digit!" "I'm sorry," the cashier replies, "but since yesterday all video game prices have increased by 20%." The new price is expressed as an integer number of Euros, less than 100. What is this price?	Not supported with pagination yet
1241	2006	autumn games	no	C2 L1 L2	Geometry	Angle greater than 92°	The four interior angles of a quadrilateral have measures that are distinct integers (in degrees). What is the minimum possible measure (in degrees) of the largest angle of the quadrilateral?	Not supported with pagination yet
1242	2006	autumn games	no	C2 L1 L2	Logic	Local time 16:40	Nando's watch gains 3 minutes every hour. His partner Giorgio's watch, however, loses 4 minutes in an hour. Although they were set correctly at the same time this morning, this afternoon, one now shows 3:55 PM while the other indicates 5:07 PM. What was the time this morning (according to an "official" clock) when the two watches were set?	Not supported with pagination yet
1244	2006	autumn games	no	L1 L2	Pattern Recognition	Total 765 numbers	Carla lists all numbers whose digits contain at least one repeated digit (up to and including 2007): 11, 22, 33, ..., 99, 100, 101, ..., 2006, 2007. How many numbers does Carla write in total?	Not supported with pagination yet
1246	2006	autumn games	no	L1 L2	Geometry	Angle greater than 90°	In a triangle with sides measuring respectively 6 cm, 8 cm, and 2√7 cm, the midpoints of its three sides are marked; then, the triangle whose vertices are these three points is drawn. What is the measure (in degrees) of the largest angle of this second triangle?	Not supported with pagination yet
1247	2006	autumn games	no	L1 L2	Arithmetic	Sum of numbers 35	Four positive integers are such that the GCD (greatest common divisor) of any two of them is always equal to their difference. What is the minimum sum of these four numbers?	Not supported with pagination yet
1248	2006	autumn games	no	L2	Geometry	Angle greater than 123°	The measures of the six interior angles of a non-regular hexagon are given as integers (in degrees), all distinct from one another. What is the minimum possible measure (in degrees) for the largest angle of the hexagon?	Not supported with pagination yet
1250	2006	team games	no	C2 L1 L2	Algebra	The rest is 1	N is a 3-digit number. If the two rightmost digits are swapped, N increases by 36; if the two leftmost digits are swapped, N increases by 270. Now take the 3 digits of N, sum them together, and divide the resulting sum by 3. What is the remainder?	Not supported with pagination yet
1252	2006	team games	no	C2 L1 L2	Algebra	N=145	N is a three-digit number ( N = a b c ) that can also be written as a!+b!+c!. Find its value (or values).	Not supported with pagination yet
1254	2006	team games	no	C2 L1 L2	Geometry	The perimeter is 120 m	Piazza dell’Unità d’Italia has the shape of a regular hexagon. It occupies an area of 1020 m². If Ingrid completes a full circuit of the square, following all its sides, how many meters does she walk? (Substitute √2 with 1.4; √3 with 1.7; √5 with 2.2 should these numbers appear in the calculations).	Not supported with pagination yet
1255	2006	team games	no	C2 L1 L2	Arithmetic	The sum is worth 8000	Multiply 2002 by 111 … 111 (in this second factor, the digit '1' is repeated 2000 times!). What is the sum of the digits of the product?	Not supported with pagination yet
1256	2006	team games	no	C2 L1 L2	Algebra	y-2x=0	If you know that y equals 4x - 6, and also 5x - x², and furthermore, that it is equal to the product xy decreased by x² and then also by 3, can you determine the value of y - 2x?	Not supported with pagination yet
1257	2006	team games	no	C2 L1 L2	Combinatorics	20 cows	The grass in a meadow grows and reproduces at the same rate (and with the same density). We know that 70 cows take 24 full days to "eat" all the grass in the meadow, while 30 cows need 60 days. How many cows are "needed" to eat the grass in the meadow in 96 days?	Not supported with pagination yet
1260	2006	team games	no	C2 L1 L2	Algebra	D	The equation x^6 – 3x^5 – 6x^3 – x + 8 = 0 a) has no real roots; b) has at least two negative real roots; c) has only one negative real root; d) has no negative real roots but has at least one positive real root. N.B. This question carries a potential penalty: half of the points awarded for the question will be deducted for an incorrect answer.	Not supported with pagination yet
1261	2006	team games	no	C2 L1 L2	Logic	2	Jacob recounts last year's vacation as follows: "I had 7 half-days with rain; when it rained in the morning, the afternoon was then sunny. Without rain, I had 5 mornings and 6 afternoons." How many (completely) rain-free days did Jacob have?	Not supported with pagination yet
1262	2006	team games	no	C2 L1 L2	Geometry	There are three solutions: 15 cm, 17 cm, 25 cm	The side lengths of a right triangle are integers in cm. Inside the triangle, the inscribed circle is drawn (tangent to each of the three sides of the triangle). Its radius is 3 cm. What is the length of the hypotenuse of the triangle?	Not supported with pagination yet
1263	2006	team games	no	C2 L1 L2	Algebra	n=1	Find all positive integers n for which (n^4+4) is a prime number.	Not supported with pagination yet
1265	2006	team games	no	C2 L1 L2	Algebra	a=12, b=18, c=2	Angelo needs to verify that (a+b)/c equals 15. But he forgets the parentheses and finds 21. Since he made a mistake, he swaps a with b, calculates (b+a)/c but, again forgetting the parentheses, finds 24. What are the values of a, b, c?	Not supported with pagination yet
1266	2006	team games	no	C2 L1 L2	Algebra	The uncelebrated birthday is that of the 26 years	Renato has kept all the candles from his birthday cakes, starting from his first birthday (as many candles as his age in years), except for those from one year when he was sick. Currently, he has 1990 candles. How old was he when he couldn't celebrate his birthday?	Not supported with pagination yet
1267	2006	team games	no	C2 L1 L2	Combinatorics	The tired tourists are 3	One hundred Japanese tourists have arrived in Rome. Presented with several interesting tourist proposals, 49 of them decide to go to Venice; 42 to Florence; 35 to Naples; and 30 to Maremma. Evidently, some tourists go on more than one excursion. In particular, 24 tourists go on 2 excursions; 10 go on 3; and some, even, 4. There are also some – few and can be counted on the fingers of one hand – who, tired, prefer to remain in Rome. How many tourists are tired?	Not supported with pagination yet
1268	2006	team games	no	C2 L1 L2	Algebra	The weight of gold is 11 g	Desiderio's balance has 17 weights: 1 g, 2 g, 3 g, ..., 17 g. Ten of these weights are black; six are silver and only one is gold (... perhaps gold-plated). The difference between the sum of the silver 'weights' and that of the black 'weights' is 32 g. What is the 'value' of the gold weight?	Not supported with pagination yet
1269	2006	team games	no	C2 L1 L2	Algebra	Carla had 9 Italian stamps	Carla's stamp collection consists of 45 stamps, partly Italian and partly from Vatican City. To have stamps of a single origin, Carla goes to her friend Milena, who collects stamps from all over the world, and exchanges 3 Italian stamps for 5 Vatican City stamps with her. In the end, Carla ends up with 51 stamps, all from Vatican City. How many Italian stamps did Carla have at the beginning?	Not supported with pagination yet
1270	2006	Rosi's games	no	C2 L1 L2	Algebra	There are 8 white roses	For the upcoming Mother's Day, Anna prepared a bouquet of 24 roses of three different colors. Specifically, for every red rose, she included two white ones and three yellow ones. How many white roses are there in Anna's bouquet?	Not supported with pagination yet
1271	2006	Rosi's games	no	C2 L1 L2	Algebra	On the first day, Desiderio had made 5 training jumps	Desiderio is training for next Sunday's high jump competition. During the first day of training, he performed an odd number of jumps. The next day, he performed double the number of training jumps, and continued this pattern each day, doing double the number of jumps compared to the previous day. During the last training session, the day before the competition, Desiderio performed 80 jumps. How many jumps did he perform on the first day?	Not supported with pagination yet
1276	2006	Rosi's games	no	C2 L1 L2	Algebra	Luca's bucket contains 23 liters	Luca's bucket is larger than Sara's. Specifically, with the water contained in his bucket, Luca can fill Matilde's bucket 3 times and still has 2 liters remaining. The two buckets together contain 30 liters. How many liters does Luca's bucket contain?	Not supported with pagination yet
1277	2006	Rosi's games	no	C2 L1 L2	Algebra	The largest number is 402	The sum of 5 consecutive numbers is 2000. What is the largest of these numbers?	Not supported with pagination yet
1278	2006	Rosi's games	no	C2 L1 L2	Arithmetic	The product is worth 110	Opening his book, Marco notices that the sum of the numbers indicating the two pages is 21. What is their product?	Not supported with pagination yet
1279	2006	Rosi's games	no	C2 L1 L2	Combinatorics	The possible triangles, in this case, are 2	How many possible triangles (with integer side lengths in cm) have a perimeter of 10 cm?	Not supported with pagination yet
1280	2006	Rosi's games	no	C2 L1 L2	Arithmetic	We need 25 people.	If 10 people need 5 days to clean a beach, how many people are needed to clean the same beach in 2 days? How many people are needed to clean the beach in 2 days?	Not supported with pagination yet
1281	2006	Rosi's games	no	C2 L1 L2	Combinatorics	The possible even numbers (of 3 digits) are 450	How many 3-digit even numbers are there?	Not supported with pagination yet
1282	2006	Rosi's games	no	C2 L1 L2	Algebra	Jacob and Luca changed their schedule after 30 days	To celebrate the end of school, Jacob and his friend Luca decide to spend their holidays taking a long trip (on foot!) to go and visit Jacob's grandmother. The 800 km journey lasts 40 days. For the first days, the two friends cover 17 km per day. Then, after a certain number of days, realizing they are behind schedule, they increase their daily distance to 29 km. In this way, they arrive exactly on the 40th day at Jacob's grandmother's. After how many days from their departure did they increase the daily number of km?	Not supported with pagination yet
1283	2006	Rosi's games	no	C2 L1 L2	Combinatorics	Desire, Nando and Guido can sit in 6 different ways	Desiderio, Nando and Guido arrive at the cinema at the last moment, just before the start of the screening. Luckily, they find exactly 3 seats, one next to the other. In how many different ways can they sit?	Not supported with pagination yet
1284	2006	Rosi's games	no	C2 L1 L2	Algebra	There are 7 patients who are double	Dr. Benguarito's clinic specializes in the detoxification of alcoholics and smokers. Currently, 98 patients are admitted. Given that 10% of alcoholics also need to detox from smoking, and 20% of smokers have the same problem with alcohol, how many patients have problems with both alcohol and smoking?	Not supported with pagination yet
1285	2006	Rosi's games	no	C2 L1 L2	Algebra	At the birth of his son, Renato was 36 years old	Today, Renato is twice as old as his son, but 18 years ago, he was three times as old as his son. How old was Renato when his son was born?	Not supported with pagination yet
1287	2006	Rosi's games	no	C2 L1 L2	Algebra	Carla is 35 years old	Carla, Liliana, and Milena do not want to reveal their ages. To Marco, who is a bit curious, they only tell him that Carla is 7 years older than Milena, while Liliana is 9 years older than Milena, and, furthermore, that the sum of their ages is 100. How old is Carla?	Not supported with pagination yet
1290	2006	semifinal	no	C1	Arithmetic	3 desserts	To make 1 cake, Rosi needs 6 eggs, 500 g of flour, 300 g of sugar, and 150 g of butter. In the kitchen, she has available two packs of butter of 250 g each, 2 kg of flour, 1 kg of sugar, and two dozen eggs. How many cakes, at most, can Rosi make?	Not supported with pagination yet
1291	2006	semifinal	no	C1	Arithmetic	3.40 m	The snail Lilly lives in a beautiful garden but loves to climb a wall. In the morning, she climbs 50 cm. In the afternoon, exhausted, she falls asleep and descends 20 cm. The wall is 3.40 m high. In how many days will Lilly reach the top of the wall?	Not supported with pagination yet
1297	2006	semifinal	no	C2 L1 L2 GP	Combinatorics	19 parts	On a rectangular piece of cardboard, Desiderio has drawn three lines and observes that they divide the rectangle into 7 parts. He then draws three more lines, each parallel to one of the first three. How many parts will Desiderio obtain, at most, on his cardboard?	Not supported with pagination yet
1298	2006	semifinal	no	C2 L1 L2 GP	Combinatorics	21 digits	Milena writes a sequence of digits using only 1, 2, 3, 4 and 5 such that: - two adjacent digits are always different; - all numbers formed by two adjacent digits are different. For example: 123134251 satisfies the above conditions; 12315412 does not, because 12 appears twice. How many digits can Milena write, at most, in her sequence?	Not supported with pagination yet
1299	2006	semifinal	no	L1 L2 GP	Algebra	1806	One day, the mathematician Augustus De Morgan (who was born and died in the 19th century) replied as follows to someone who asked his age: "I celebrated turning y years old in the year whose number (a four-digit number) was equal to the square of y." In what year was Augustus De Morgan born?	Not supported with pagination yet
1301	2006	semifinal	no	L2 GP	Geometry	Nando will get 596 triangles	Nando made a cardboard hexagon with all internal angles equal and four consecutive sides that measure, in order, 9, 12, 8 and 11 centimeters. He wants to cut out of this hexagon the largest possible number of equilateral triangles with a side length of 1 cm. How many triangles will Nando get, at most?	Not supported with pagination yet
1302	2006	semifinal	no	L2 GP	Algebra	19 - 26 - 55 Euro	In Augusto's shop, three types of calculators are for sale. The most expensive (and most sophisticated) model costs more than double the mid-priced model. Three mid-priced calculators cost more than four economy-type calculators, but three economy-type calculators cost more than a single calculator of the most expensive model. The unit prices of the three models are integer numbers of Euros, and their sum is 100 Euros. What are the prices of the three models?	Not supported with pagination yet
1303	2006	semifinal	no	GP	Combinatorics	60 cm2	The pupils in Sara's class have built 120 cubes with an edge of 1 cm: 80 completely red and 40 completely white. With great patience, the teacher glues the 120 cubes together and forms a rectangular cuboid. What will be the minimum visible red area on the faces of the cuboid?	Not supported with pagination yet
1304	2006	final	no	C1	Combinatorics	Jacob's notebook has: 38 pages	To number all the pages of his notebook, starting from page 1, Jacob used the digit "3" 13 times. What is the number of the last page of Jacob's notebook?	Not supported with pagination yet
1305	2006	final	no	C1	Arithmetic	Angelo will give Desire 8 candy	Renato has 7 candies, Pietro has 3, Desiderio 2, Settimo 8 and Amerigo 9. Angelo has 21 (candies) and wants to distribute them among his five friends, so that each of them eventually has the same number of candies. How many candies will he give to Desiderio?	Not supported with pagination yet
1306	2006	final	no	C1 C2	Algebra	1984	Find a number less than 2000 which, when added to the sum of its digits, is equal to 2006.	Not supported with pagination yet
1307	2006	final	no	C1 C2	Logic	Milena: dog, Igrid: parrot, Carlo: cat	Milena Gatto, Ingrid Locane, and Carla Pappagallo are three friends. One of the three owns a cat, another a dog, and the third a parrot, but none of them owns the animal that matches her surname. When one of these friends, with her pet, visits the one who owns the parrot, the parrot says the name of the animal accompanying the visitor, unless this name matches that of the parrot's owner. Today the parrot screamed: "The cat! The cat!" What animal does each of the three friends own?	Not supported with pagination yet
1308	2006	final	no	C1 C2 L1	Algebra	87	Each of the 29 families in the village owns 1 or 3 or 5 horses. The families that own only one horse are as many as those that own 5 horses. How many horses are there in total in the village?	Not supported with pagination yet
1309	2006	final	no	C1 C2 L1	Pattern Recognition	In the box there are: 5 numbers equal to 7 odd numbers	1 ; 2 ; 3 ; 4 ; 5 ; 6 ; 7 ; 8 ; 9 ; 10 In this box there are _____ even numbers In this box there are _____ odd numbers Complete the sentences in the box with numbers (written as digits) so that the sentences contained within it are true.	Not supported with pagination yet
1320	2006	international final day 1	no	C1	Pattern Recognition	number 55	The director of a 40-room hotel is superstitious: he refuses to use the unlucky digit "3". He assigned the number 1 to one room, 2 to another, and so on, but skipping all numbers that are written using the digit "3". Thus, the third room has the number 4. What number did he assign to the fortieth room?	Not supported with pagination yet
1321	2006	international final day 1	no	C1	Algebra	3 rings of 30 pearls and 8 rings of 20 pearls.	Claudia and Anna make rings with pearls. Together, they bought 250 pearls. Anna's model uses 30 pearls, and Claudia's uses 20. They used all their pearls and made 11 rings. How many rings did Anna make, and how many did Claudia make?	Not supported with pagination yet
1323	2006	international final day 1	no	C1	Arithmetic	7 minutes	My friend Anna took the train in Paris at 14:29. This train usually takes 4 hours and 3 minutes to arrive in Strasbourg. I arrive at Strasbourg station to pick her up at 18:35 and read on the arrivals board that the train is running 10 minutes late. How long do I have to wait for my friend?	Not supported with pagination yet
1324	2006	international final day 1	no	C1	Combinatorics	28 lots	Eight classes participate in a school football tournament. Each class has three teams: the experts, the strong, and the clever ones. When two classes meet, each of the teams from one class plays once against the team of the same level from the other class: the experts against the experts, the strong against the strong, and the clever ones against the clever ones. How many matches will be played if each class meets every other class?	Not supported with pagination yet
1325	2006	international final day 1	no	C1 C2	Arithmetic	16 h 26 min.	One hundred forty cyclists take part in the Giro d'Italia prologue: it is an individual time trial. Riders depart every 2 minutes. The first one departs at 13:10. Matteo Rossi is the 99th to depart. At what time will he depart?	Not supported with pagination yet
1328	2006	international final day 1	no	C2 L1	Logic	2 solutions: BCAD or BADC.	Four cyclists, Alessandro, Bruno, Carlo, and Davide, take part in a time trial. The first one starts at 8:00 AM, then subsequent starts follow every 5 minutes. The arrival times are 8:59 AM, 9:02 AM, 9:04 AM, and 9:08 AM. The one who started immediately before Bruno and immediately after Davide took two minutes more than the winner. Alessandro arrived at the finish line before Carlo. Determine the race ranking, from fastest to slowest, knowing that there were no ties.	Not supported with pagination yet
1332	2006	international final day 1	no	L1 L2	Algebra	5 solutions: 2008, 1006, 136, 94 and 78	On each vertex of the octahedron, a non-negative integer must be written. On each face, the product of the three numbers corresponding to its three vertices is written. The sum of the eight numbers written on the faces must be equal to 2006. What is the sum of the six numbers written on the vertices of the octahedron?	Not supported with pagination yet
1334	2006	international final day 2	no	C1	Arithmetic	42 cm.	Eleonora places seven cubes of increasingly smaller size one on top of the other, so as to form a stepped pyramid. The side of the second cube is one centimeter shorter than that of the first; that of the third is one centimeter shorter than that of the second; and so on. The side of the smallest cube is 3 centimeters long. How high is Eleonora's pyramid?	Not supported with pagination yet
1336	2006	international final day 2	no	C1	Combinatorics	10 bags.	Giuliano is preparing his birthday party. He wants to offer small sweets to each guest. His mother bought him 5 different types: strawberry candies, liquorice strings, lollipops, lemon candies, and mint candies. Giuliano decides to prepare bags, each containing 30 sweets of three different types: ten of each type. He would like each guest to have a bag containing a different combination of sweets. How many bags containing a different combination of sweets can he prepare?	Not supported with pagination yet
1337	2006	international final day 2	no	C1	Algebra	6 times	The Rome-Pisa train departing at noon takes 3h 58min to reach its destination. The evening train, which makes triple the number of stops, takes 4h 26min. All stops have the same duration, between 5 and 10 minutes. When the two trains are not stopped, they travel at the same speed. How many stops does the evening train make?	Not supported with pagination yet
1339	2006	international final day 2	no	C1 C2	Logic	Noemi.	Four friends play 'five-seven'. They arrange themselves in a circle in the following order, clockwise: Noemi, Maura, Luigi, Romano. The game consists of counting in turns, each pronouncing a number: Noemi starts with "one", then Maura with "two", and so on. However, immediately after one of the four friends pronounces a multiple of 5 or 7, the direction in which the four friends take turns counting is reversed. Who will say "twenty-three"?	Not supported with pagination yet
1340	2006	international final day 2	no	C1 C2	Combinatorics	11 times.	Gianni has been punished: he must copy the phrase "I must not shout in class" 2006 times in his notebook. He has written this phrase once on his computer. What is the minimum number of times he must use the copy-paste command to obtain the 2006 required phrases? Note that "copy-paste command" refers to the double operation of copying and pasting.	Not supported with pagination yet
1342	2006	international final day 2	no	C2 L1	Combinatorics	26	Eighteen people want to divide 10 equal loaves of bread among themselves. They can cut each loaf as they wish, and it is not required that all be cut in the same way. What is the minimum number of pieces of bread that must be obtained, by adequately cutting the given loaves, so that each person has the same amount of bread?	Not supported with pagination yet
1343	2006	international final day 2	no	C2 L1	Arithmetic	214	I am a six-digit number with identical digits, and I am divisible by 49. Sum my integer divisors between 1 and 49 (inclusive). What number will you get?	Not supported with pagination yet
1344	2006	international final day 2	no	L1 L2	Pattern Recognition	515542	We write the integers in their natural order starting from 1. Then we cross out sequences of numbers, always leaving one uncrossed number between two sequences of crossed-out numbers. The sequences of crossed-out numbers consist respectively of: 10 numbers, 1 number, 10 numbers, 2 numbers, 10 numbers, 3 numbers, 10 numbers, 4 numbers, 10 numbers, 5 numbers, etc. Which will be the 2006th uncrossed number?	Not supported with pagination yet
1347	2006	international final day 2	no	L2	Combinatorics	200600	Matilde observes to Mattia that there are 76 different ways to place three non-aligned pawns on a 3x3 chessboard, such that each pawn occupies a distinct square. "So what?" asks Mattia. "If you multiply this number by 5, you get the product of our ages, that is, the product of 19 and 20." Mattia reflects and says that if you multiply by 5 the number of different ways to place three non-aligned pawns on an 8x8 chessboard, you get a remarkable number. What is this number?	Not supported with pagination yet
1348	2006	international final day 2	no	L2	Geometry	16 √3	Five molehills, treated as points, are located inside (including the borders) of a square lawn with a side length of 12 meters. No three of these molehills are collinear. What is the maximum possible value of the minimum area among all the triangles that can be formed by taking three of the molehills as vertices? Give the exact answer in m², even if it is not a rational number.	Not supported with pagination yet
1130	2007	autumn games CE	no	CE	Combinatorics	1	Pietro and Renato challenge each other to the game of "ten". Pietro starts and says a number that can only be 1 or 2. Then, Renato adds 1 or 2 to the number said by Pietro. Now, it's Pietro's turn again to add 1 or 2 to the previous sum. Pietro and Renato continue in this way, until one of them says 10 and wins the game. What number must Pietro say at the beginning to be sure of winning?	Not supported with pagination yet

1407

2005

semifinal

no

C1

Arithmetic

Liliana will pay the coat Euro 48,48

Even during sales, Renato never gives a discount of more than 20 Euros. Incidentally, he has a strange way of labeling discounted merchandise: he always ensures that, in the price, the number of Euros is equal to the number of cents. For example, the t-shirt I like and would like to buy was discounted to 29.29 Euros. My friend Liliana wants to buy a coat which, before the sales, cost 67.99 Euros. How much will she pay for it, if Renato gives her the maximum discount?

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1408

2005

semifinal

no

C1

Logic

We will finish cooking king at 4:20 p.m.

It's 2:40 PM and my friends have finished preparing some magnificent desserts; now, they just need to be cooked. In my oven, however, only one "piece" can be baked at a time and – as all good cooks know – baking cannot be interrupted. Anna has prepared a cake that bakes in half an hour; Chiara, a dessert that bakes in 20 minutes, and Ingrid a tart that now needs to rest for exactly 35 minutes before being baked for three quarters of an hour. At what time – as soon as possible! – will we have finished baking everything?

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1409

2005

semifinal

no

C1 C2

Logic

The total points were: 19

During a handball match between the Algebrists' team and the Geometers' team, which ended with a score of 23 to 19 in favor of the Algebrists, there was a moment when the Algebrists had as many points as the Geometers subsequently scored until the end of the match. How many points did the two teams have, together, at that precise moment?

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1411

2005

semifinal

no

C1 C2 L1

Logic

The number you searched for is: 163

Find a three-digit number, with all digits distinct, such that: - the sum of its digits is equal to 10; - the product of the first two digits is equal to 6; - the tens digit is the largest of the three digits.

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1412

2005

semifinal

no

C1 C2 L1 L2 GP

Combinatorics

At most the members of the group are: 8

In a group of friends, each has 10 tokens available (numbered from 1 to 10) and must choose three of them whose sum is 13. By doing so, everyone manages to form their own "group" of three tokens such that it is different from those of the others (meaning that it has at least one different "number"). How many, at most, are the members of the group?

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1413

2005

semifinal

no

C1 C2 L1 L2 GP

Geometry

The dimensions of the fourth rectangle are: 5 x 6

A cardboard square has been divided into four rectangles. Three of these rectangles have dimensions 4x6, 5x9, and 2x11. What are the dimensions of the fourth rectangle?

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1414

2005

semifinal

no

C1 C2 L1 L2 GP

Arithmetic

There are 18 pieces of chocolate

Seven students received twelve identical chocolate bars, each weighing 91 grams. They then decided to divide them equally among themselves, making the minimum number of pieces. How many pieces of chocolate are there (including whole bars) at the moment of its equal division among the seven students?

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1415

2005

semifinal

no

C2 L1 L2 GP

Algebra

The captain is 70 years old.

For his birthday, the captain – not yet a centenarian – invited his 3 daughters, his 5 grandchildren, and his 7 great-grandchildren: - the 3 daughters have consecutive ages; - the 5 grandchildren have consecutive ages; - the 7 great-grandchildren have consecutive ages; - the sum of the daughters' ages is equal to the sum of the grandchildren's ages and equal to the sum of the great-grandchildren's ages; - the captain's age is equal to two-thirds of the sum of the daughters' ages. How old is the captain?

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1417

2005

semifinal

no

L1 L2 GP

Combinatorics

The smallest value of N is 57

Among N arbitrarily chosen positive integers, we want to be sure to find two whose sum or difference is divisible by 111. What is the smallest possible value of N?

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1420

2005

final

no

C1

Pattern Recognition

Michel had, this month, at most, 9 votes of Mathematics

In Michel's school, as in all French schools, grades range from 0 to 20 (inclusive). In his gradebook, Michel has written down his Math grades for this month. They are all integers. The first grade is an 8; then, each subsequent grade is triple or half of the previous one. What is the maximum number of Math grades Michel had this month?

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1421

2005

final

no

C1

Arithmetic

Milena, Rosi, Carla, Desire

At the end of the school's cross-country race, the teacher asks his students to check their heart rates. Carla counts 25 beats in 15 seconds; Milena counts 24 beats in 20 seconds; Rosi counts 45 beats in 30 seconds; Desiderio counts 110 beats in one minute. Order the students (indicated by the initial of their name) from the one with the slowest pulse to the one with the fastest pulse.

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1422

2005

final

no

C1 C2

Logic

Luca is 1 year old

Today, Luca, Chiara, and Anna's combined age is 60 years. When Luca was born – after 1995 – Chiara and Anna were both older than 10. Anna is one year younger than Chiara. Chiara's age is a multiple of 6. What is Luca's age?

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1423

2005

final

no

C1 C2

Algebra

60 shells

Jacob's shell collection consists of 287 shells, divided into 5 boxes: a blue box, a green box, a yellow box, a red box, and a white box. The blue box and the green box contain the same number of shells. Similarly, the yellow box and the red box have the same number of shells. The box with the fewest shells contains 53; another box contains 57. What is the minimum number of shells in the box that contains the most (or one of the boxes with the most, if there are two)?

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1427

2005

final

no

C1 C2 L1 L2 GP

Algebra

13

How many three-digit integers are equal to 46 times the sum of their digits?

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1429

2005

final

no

C2 L1 L2 GP

Algebra

There are 2 solutions: 2012 2026

Marco's birth year is special in that the product of its digits is the square of a strictly positive integer. Today, in 2005, Marco is waiting for the year in which he will have an age equal to the square root of the product of the digits of his birth year. In what year will this occur?

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1430

2005

final

no

L1 L2 GP

Combinatorics

3 Solutions

We want to write 13 distinct positive integers in ascending order such that their sum is 94. How many different solutions are there?

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1431

2005

final

no

L1 L2 GP

Algebra

The speed of Angelo is 10 m/sec

Angelo and Renato are training for the upcoming Cycling World Championship, using an oval track of 360 meters. Angelo overtakes Renato every three minutes. If, however, one of them were to ride in the opposite direction, they would meet every twenty seconds. What is Angelo's speed? (Give the answer in meters per second)

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1436

2005

international final day 1

no

C1

Logic

Nicola, Maximilian, Camilla, Roman, Thomas

The time allowed for the test in category C1 is two hours. It started at 2:35 PM and must end at 4:35 PM. Romano submits three-quarters of an hour before the end. Tommaso submits one and a half hours after the start of the test. Massimo submits halfway through the allotted time. Camilla submits at 3:45 PM. Nicola submits 50 minutes after the start of the test. Rank the five participants according to their submission order (write the initials of their names).

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1439

2005

international final day 1

no

C1 C2

Logic

mark No 15

Teo and Tommaso play on an escalator that consists of 29 visible steps. Teo starts from the top and, in the time the escalator moves up one step, he descends 6 steps. At the same time, Tommaso starts from the bottom and climbs 4 steps. If they continue to descend and ascend at the same speed, at which step do they meet, counting the visible steps, at the precise moment they meet, starting from the bottom?

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1444

2005

international final day 1

no

C2 L1 L2

Combinatorics

2 solutions : 4 and 21

In the convent of the City of Mathematics, if you meet two nuns, chosen randomly from the set of nuns residing there, you have exactly a one in two chance that they are both brown. How many nuns reside in the convent?

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1445

2005

international final day 1

no

L1 L2

Combinatorics

3 solutions : 24, 40, 72

Ludovico takes a portion of the 104 cards from a canasta deck, forming a new deck which we will call "M". He shows us the seventeenth card from the top of this new deck, then reseals it without altering the cards' respective positions, and places it on the table. Finally, he asks Mina to perform the following operations: take the first card of deck M and place it at the bottom; take the next card and discard it; continue this process until only one card remains. Mina does as Ludovico asked and, to her surprise: the last card remaining is exactly the one Ludovico had shown. How many cards were in deck M?

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1448

2005

international final day 1

no

L2

Geometry

42 carrés.

Four distinct points have been chosen on a plane. Squares are to be drawn, each such that at least one of its sides (possibly extended) passes through each of these points. It is known that the four chosen points are such that it is possible to draw a finite number of squares satisfying this condition. What is the maximum number of squares satisfying this condition that can be drawn?

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1451

2005

international final day 2

no

C1 C2

Arithmetic

9 h 10 min

In the last four days, Maria has played a lot of tennis. On Saturday, she played from 18:00 to 19:30; on Sunday, from 12:00 to 13:43, then in the evening for 1 hour and 55 minutes; on Monday, her match lasted 2 hours and 5 minutes; and on Tuesday, 117 minutes. Maria, very tired, wonders how much time she has spent on the tennis courts in the last four days. Help her find the answer in hours and minutes.

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1455

2005

international final day 2

no

C2 L1 L2

Combinatorics

216 marquis

Geraldina enjoys doing favors for her neighbors who live in the same building. Here's what she needs to do this afternoon: visit Francesco, Tiberio, and Margherita to say hello; return Isa's DVD to Stan; pick up the keys from Sergio to collect his mail on the first floor. There are twelve steps between each floor, and the elevator is out of order. What is the minimum number of steps she must climb and descend to do all this (starting from her apartment and returning to it)? Note: Sergio and Stan are not in a hurry, so Geraldina does not have to bring them the mail and the DVD immediately after picking them up. 8th floor Margherita 7th floor Isa 6th floor Francesco 5th floor Sergio 4th floor Tiberio 3rd floor Géraldina 2nd floor Stan 1st floor

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1458

2005

international final day 2

no

L1 L2

Combinatorics

3 croisements

Randomly place six points on a sheet of paper. Using a pencil, connect every pair of them with lines. Suppose that any two lines - either have no point in common; - or intersect exactly once and thus have one and only one point in common. What is the minimum number of intersections you will have?

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1227

2006

autumn games CE

no

CE

Pattern Recognition

The next issue is: 18063 which corresponds to 2007x9

Desiderio begins performing the multiplications 2007 x 1; 2007 x 2; 2007 x 3; 2007 x 4; etc. Desiderio then realizes that, unlike the first two results, the result of 2007 x 3 is written with all different digits. What is the next result that possesses the same property (of being written with all different digits)?

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1229

2006

autumn games CE

no

CE

Arithmetic

The height of the Luke pyramid is 95 cm = (14 + 13 + 12 +.... + 7 + 6 + 5).

Today, at Luca's kindergarten, the 'little ones' are playing with cube-shaped boxes which they place one on top of the other, such that the side of one box always measures one centimeter less than that of the box it rests on. Luca has ten boxes. The side of the largest one measures 14 cm. His pyramid is built using all the boxes, stacked from largest to smallest. What is the height of Luca's pyramid?

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1230

2006

autumn games CE

no

CE

Logic

Anna will greet her friends in the following order: S-I-G-C Silvia, Ingrid, Giovanna and Claudia.

Before going abroad for six months for her studies, Anna wants to say goodbye to her friends who live in her apartment building. Ingrid is home between 11 AM and 11:25 AM; Giovanna returns from the supermarket at 11:30 AM; Claudia will return home at a quarter to noon; Silvia (who has to go play tennis) told her to visit before noon. Anna wants to stay 20 minutes with each of her friends and leave immediately afterward. In what order should she say goodbye to her friends to be able to leave as soon as possible? Answer by providing (in order) the initials of their names.

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1233

2006

autumn games

no

C1

Arithmetic

95 cm

Today, at Luca's kindergarten, the "little ones" are playing with cube-shaped boxes, which they place one on top of the other, such that the side of one box always measures one centimeter less than that of the box it rests on. Luca has ten boxes. The side of the largest one measures 14 cm. His pyramid is built using all the boxes, stacked from largest to smallest. What is the height of Luca's pyramid?

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1234

2006

autumn games

no

C1

Logic

S-I-G-C (Silvia, Ingrid, Giovanna, Claudia)

Before going abroad for six months for study reasons, Anna wants to say goodbye to her friends who live in her apartment building. Ingrid is home between 11:00 AM and 11:25 AM; Giovanna returns from the supermarket at 11:30 AM; Claudia will be home at a quarter to noon; Silvia (who has to go play golf) told her to stop by before noon. Anna wants to stay 20 minutes with each of her friends and leave immediately afterwards. In what order should she greet her friends to be able to leave as soon as possible? Answer by providing (in order) the initials of their names.

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1236

2006

autumn games

no

C1 C2

Combinatorics

78 €

For the end-of-year show, the physical education teacher needs to buy 49 ribbons. Here are the supplier's offers: One ribbon: 3 Euros. A pack of 2 ribbons: 5 Euros. A pack of 5 ribbons: 10 Euros. Additionally, if you buy three identical packages, the fourth one is free. If she calculates carefully, what is the minimum amount the teacher will have to spend?

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1237

2006

autumn games

no

C1 C2

Algebra

1917 or 1932

Our grandfather was born before the Second World War, but after the start of the First. When he celebrates his next birthday, the year – 2007 – will be equal to his birth year increased by five times the sum of the digits of his birth year. In what year was the grandfather born? (Provide all possible solutions)

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1238

2006

autumn games

no

C1 C2

Logic

The number of true sentences is 3

How many simultaneously true sentences can the box below contain? 1. In this box, sentence 2 is false. 2. In this box, sentence 3 is false. 3. In this box, sentence 4 is false. 4. In this box, sentence 5 is false. 5. In this box, sentence 6 is false. 6. In this box, sentence 1 is false.

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1239

2006

autumn games

no

C1 C2

Algebra

Old price 45 €, new price 54 €

Sara has finally decided to buy a video game, which she has actually been dreaming of for a long time. When the store cashier informs her of the price to pay, Sara is however surprised: "It's not possible! You've swapped the units digit with the tens digit!" "I'm sorry," the cashier replies, "but since yesterday all video game prices have increased by 20%." The new price is expressed as an integer number of Euros, less than 100. What is this price?

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1241

2006

autumn games

no

C2 L1 L2

Geometry

Angle greater than 92°

The four interior angles of a quadrilateral have measures that are distinct integers (in degrees). What is the minimum possible measure (in degrees) of the largest angle of the quadrilateral?

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1242

2006

autumn games

no

C2 L1 L2

Logic

Local time 16:40

Nando's watch gains 3 minutes every hour. His partner Giorgio's watch, however, loses 4 minutes in an hour. Although they were set correctly at the same time this morning, this afternoon, one now shows 3:55 PM while the other indicates 5:07 PM. What was the time this morning (according to an "official" clock) when the two watches were set?

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1244

2006

autumn games

no

L1 L2

Pattern Recognition

Total 765 numbers

Carla lists all numbers whose digits contain at least one repeated digit (up to and including 2007): 11, 22, 33, ..., 99, 100, 101, ..., 2006, 2007. How many numbers does Carla write in total?

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1246

2006

autumn games

no

L1 L2

Geometry

Angle greater than 90°

In a triangle with sides measuring respectively 6 cm, 8 cm, and 2√7 cm, the midpoints of its three sides are marked; then, the triangle whose vertices are these three points is drawn. What is the measure (in degrees) of the largest angle of this second triangle?

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1247

2006

autumn games

no

L1 L2

Arithmetic

Sum of numbers 35

Four positive integers are such that the GCD (greatest common divisor) of any two of them is always equal to their difference. What is the minimum sum of these four numbers?

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1248

2006

autumn games

no

L2

Geometry

Angle greater than 123°

The measures of the six interior angles of a non-regular hexagon are given as integers (in degrees), all distinct from one another. What is the minimum possible measure (in degrees) for the largest angle of the hexagon?

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1250

2006

team games

no

C2 L1 L2

Algebra

The rest is 1

N is a 3-digit number. If the two rightmost digits are swapped, N increases by 36; if the two leftmost digits are swapped, N increases by 270. Now take the 3 digits of N, sum them together, and divide the resulting sum by 3. What is the remainder?

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1252

2006

team games

no

C2 L1 L2

Algebra

N=145

N is a three-digit number ( N = a b c ) that can also be written as a!+b!+c!. Find its value (or values).

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1254

2006

team games

no

C2 L1 L2

Geometry

The perimeter is 120 m

Piazza dell’Unità d’Italia has the shape of a regular hexagon. It occupies an area of 1020 m². If Ingrid completes a full circuit of the square, following all its sides, how many meters does she walk? (Substitute √2 with 1.4; √3 with 1.7; √5 with 2.2 should these numbers appear in the calculations).

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1255

2006

team games

no

C2 L1 L2

Arithmetic

The sum is worth 8000

Multiply 2002 by 111 … 111 (in this second factor, the digit '1' is repeated 2000 times!). What is the sum of the digits of the product?

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1256

2006

team games

no

C2 L1 L2

Algebra

y-2x=0

If you know that y equals 4x - 6, and also 5x - x², and furthermore, that it is equal to the product xy decreased by x² and then also by 3, can you determine the value of y - 2x?

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1257

2006

team games

no

C2 L1 L2

Combinatorics

20 cows

The grass in a meadow grows and reproduces at the same rate (and with the same density). We know that 70 cows take 24 full days to "eat" all the grass in the meadow, while 30 cows need 60 days. How many cows are "needed" to eat the grass in the meadow in 96 days?

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1260

2006

team games

no

C2 L1 L2

Algebra

D

The equation x^6 – 3x^5 – 6x^3 – x + 8 = 0 a) has no real roots; b) has at least two negative real roots; c) has only one negative real root; d) has no negative real roots but has at least one positive real root. N.B. This question carries a potential penalty: half of the points awarded for the question will be deducted for an incorrect answer.

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1261

2006

team games

no

C2 L1 L2

Logic

2

Jacob recounts last year's vacation as follows: "I had 7 half-days with rain; when it rained in the morning, the afternoon was then sunny. Without rain, I had 5 mornings and 6 afternoons." How many (completely) rain-free days did Jacob have?

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1262

2006

team games

no

C2 L1 L2

Geometry

There are three solutions: 15 cm, 17 cm, 25 cm

The side lengths of a right triangle are integers in cm. Inside the triangle, the inscribed circle is drawn (tangent to each of the three sides of the triangle). Its radius is 3 cm. What is the length of the hypotenuse of the triangle?

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1263

2006

team games

no

C2 L1 L2

Algebra

n=1

Find all positive integers n for which (n^4+4) is a prime number.

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1265

2006

team games

no

C2 L1 L2

Algebra

a=12, b=18, c=2

Angelo needs to verify that (a+b)/c equals 15. But he forgets the parentheses and finds 21. Since he made a mistake, he swaps a with b, calculates (b+a)/c but, again forgetting the parentheses, finds 24. What are the values of a, b, c?

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1266

2006

team games

no

C2 L1 L2

Algebra

The uncelebrated birthday is that of the 26 years

Renato has kept all the candles from his birthday cakes, starting from his first birthday (as many candles as his age in years), except for those from one year when he was sick. Currently, he has 1990 candles. How old was he when he couldn't celebrate his birthday?

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1267

2006

team games

no

C2 L1 L2

Combinatorics

The tired tourists are 3

One hundred Japanese tourists have arrived in Rome. Presented with several interesting tourist proposals, 49 of them decide to go to Venice; 42 to Florence; 35 to Naples; and 30 to Maremma. Evidently, some tourists go on more than one excursion. In particular, 24 tourists go on 2 excursions; 10 go on 3; and some, even, 4. There are also some – few and can be counted on the fingers of one hand – who, tired, prefer to remain in Rome. How many tourists are tired?

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1268

2006

team games

no

C2 L1 L2

Algebra

The weight of gold is 11 g

Desiderio's balance has 17 weights: 1 g, 2 g, 3 g, ..., 17 g. Ten of these weights are black; six are silver and only one is gold (... perhaps gold-plated). The difference between the sum of the silver 'weights' and that of the black 'weights' is 32 g. What is the 'value' of the gold weight?

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1269

2006

team games

no

C2 L1 L2

Algebra

Carla had 9 Italian stamps

Carla's stamp collection consists of 45 stamps, partly Italian and partly from Vatican City. To have stamps of a single origin, Carla goes to her friend Milena, who collects stamps from all over the world, and exchanges 3 Italian stamps for 5 Vatican City stamps with her. In the end, Carla ends up with 51 stamps, all from Vatican City. How many Italian stamps did Carla have at the beginning?

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1270

2006

Rosi's games

no

C2 L1 L2

Algebra

There are 8 white roses

For the upcoming Mother's Day, Anna prepared a bouquet of 24 roses of three different colors. Specifically, for every red rose, she included two white ones and three yellow ones. How many white roses are there in Anna's bouquet?

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1271

2006

Rosi's games

no

C2 L1 L2

Algebra

On the first day, Desiderio had made 5 training jumps

Desiderio is training for next Sunday's high jump competition. During the first day of training, he performed an odd number of jumps. The next day, he performed double the number of training jumps, and continued this pattern each day, doing double the number of jumps compared to the previous day. During the last training session, the day before the competition, Desiderio performed 80 jumps. How many jumps did he perform on the first day?

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1276

2006

Rosi's games

no

C2 L1 L2

Algebra

Luca's bucket contains 23 liters

Luca's bucket is larger than Sara's. Specifically, with the water contained in his bucket, Luca can fill Matilde's bucket 3 times and still has 2 liters remaining. The two buckets together contain 30 liters. How many liters does Luca's bucket contain?

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1277

2006

Rosi's games

no

C2 L1 L2

Algebra

The largest number is 402

The sum of 5 consecutive numbers is 2000. What is the largest of these numbers?

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1278

2006

Rosi's games

no

C2 L1 L2

Arithmetic

The product is worth 110

Opening his book, Marco notices that the sum of the numbers indicating the two pages is 21. What is their product?

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1279

2006

Rosi's games

no

C2 L1 L2

Combinatorics

The possible triangles, in this case, are 2

How many possible triangles (with integer side lengths in cm) have a perimeter of 10 cm?

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1280

2006

Rosi's games

no

C2 L1 L2

Arithmetic

We need 25 people.

If 10 people need 5 days to clean a beach, how many people are needed to clean the same beach in 2 days? How many people are needed to clean the beach in 2 days?

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1281

2006

Rosi's games

no

C2 L1 L2

Combinatorics

The possible even numbers (of 3 digits) are 450

How many 3-digit even numbers are there?

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1282

2006

Rosi's games

no

C2 L1 L2

Algebra

Jacob and Luca changed their schedule after 30 days

To celebrate the end of school, Jacob and his friend Luca decide to spend their holidays taking a long trip (on foot!) to go and visit Jacob's grandmother. The 800 km journey lasts 40 days. For the first days, the two friends cover 17 km per day. Then, after a certain number of days, realizing they are behind schedule, they increase their daily distance to 29 km. In this way, they arrive exactly on the 40th day at Jacob's grandmother's. After how many days from their departure did they increase the daily number of km?

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1283

2006

Rosi's games

no

C2 L1 L2

Combinatorics

Desire, Nando and Guido can sit in 6 different ways

Desiderio, Nando and Guido arrive at the cinema at the last moment, just before the start of the screening. Luckily, they find exactly 3 seats, one next to the other. In how many different ways can they sit?

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1284

2006

Rosi's games

no

C2 L1 L2

Algebra

There are 7 patients who are double

Dr. Benguarito's clinic specializes in the detoxification of alcoholics and smokers. Currently, 98 patients are admitted. Given that 10% of alcoholics also need to detox from smoking, and 20% of smokers have the same problem with alcohol, how many patients have problems with both alcohol and smoking?

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1285

2006

Rosi's games

no

C2 L1 L2

Algebra

At the birth of his son, Renato was 36 years old

Today, Renato is twice as old as his son, but 18 years ago, he was three times as old as his son. How old was Renato when his son was born?

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1287

2006

Rosi's games

no

C2 L1 L2

Algebra

Carla is 35 years old

Carla, Liliana, and Milena do not want to reveal their ages. To Marco, who is a bit curious, they only tell him that Carla is 7 years older than Milena, while Liliana is 9 years older than Milena, and, furthermore, that the sum of their ages is 100. How old is Carla?

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1290

2006

semifinal

no

C1

Arithmetic

3 desserts

To make 1 cake, Rosi needs 6 eggs, 500 g of flour, 300 g of sugar, and 150 g of butter. In the kitchen, she has available two packs of butter of 250 g each, 2 kg of flour, 1 kg of sugar, and two dozen eggs. How many cakes, at most, can Rosi make?

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1291

2006

semifinal

no

C1

Arithmetic

3.40 m

The snail Lilly lives in a beautiful garden but loves to climb a wall. In the morning, she climbs 50 cm. In the afternoon, exhausted, she falls asleep and descends 20 cm. The wall is 3.40 m high. In how many days will Lilly reach the top of the wall?

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1297

2006

semifinal

no

C2 L1 L2 GP

Combinatorics

19 parts

On a rectangular piece of cardboard, Desiderio has drawn three lines and observes that they divide the rectangle into 7 parts. He then draws three more lines, each parallel to one of the first three. How many parts will Desiderio obtain, at most, on his cardboard?

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1298

2006

semifinal

no

C2 L1 L2 GP

Combinatorics

21 digits

Milena writes a sequence of digits using only 1, 2, 3, 4 and 5 such that: - two adjacent digits are always different; - all numbers formed by two adjacent digits are different. For example: 123134251 satisfies the above conditions; 12315412 does not, because 12 appears twice. How many digits can Milena write, at most, in her sequence?

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1299

2006

semifinal

no

L1 L2 GP

Algebra

1806

One day, the mathematician Augustus De Morgan (who was born and died in the 19th century) replied as follows to someone who asked his age: "I celebrated turning y years old in the year whose number (a four-digit number) was equal to the square of y." In what year was Augustus De Morgan born?

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1301

2006

semifinal

no

L2 GP

Geometry

Nando will get 596 triangles

Nando made a cardboard hexagon with all internal angles equal and four consecutive sides that measure, in order, 9, 12, 8 and 11 centimeters. He wants to cut out of this hexagon the largest possible number of equilateral triangles with a side length of 1 cm. How many triangles will Nando get, at most?

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1302

2006

semifinal

no

L2 GP

Algebra

19 - 26 - 55 Euro

In Augusto's shop, three types of calculators are for sale. The most expensive (and most sophisticated) model costs more than double the mid-priced model. Three mid-priced calculators cost more than four economy-type calculators, but three economy-type calculators cost more than a single calculator of the most expensive model. The unit prices of the three models are integer numbers of Euros, and their sum is 100 Euros. What are the prices of the three models?

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1303

2006

semifinal

no

GP

Combinatorics

60 cm2

The pupils in Sara's class have built 120 cubes with an edge of 1 cm: 80 completely red and 40 completely white. With great patience, the teacher glues the 120 cubes together and forms a rectangular cuboid. What will be the minimum visible red area on the faces of the cuboid?

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1304

2006

final

no

C1

Combinatorics

Jacob's notebook has: 38 pages

To number all the pages of his notebook, starting from page 1, Jacob used the digit "3" 13 times. What is the number of the last page of Jacob's notebook?

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1305

2006

final

no

C1

Arithmetic

Angelo will give Desire 8 candy

Renato has 7 candies, Pietro has 3, Desiderio 2, Settimo 8 and Amerigo 9. Angelo has 21 (candies) and wants to distribute them among his five friends, so that each of them eventually has the same number of candies. How many candies will he give to Desiderio?

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1306

2006

final

no

C1 C2

Algebra

1984

Find a number less than 2000 which, when added to the sum of its digits, is equal to 2006.

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1307

2006

final

no

C1 C2

Logic

Milena: dog, Igrid: parrot, Carlo: cat

Milena Gatto, Ingrid Locane, and Carla Pappagallo are three friends. One of the three owns a cat, another a dog, and the third a parrot, but none of them owns the animal that matches her surname. When one of these friends, with her pet, visits the one who owns the parrot, the parrot says the name of the animal accompanying the visitor, unless this name matches that of the parrot's owner. Today the parrot screamed: "The cat! The cat!" What animal does each of the three friends own?

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1308

2006

final

no

C1 C2 L1

Algebra

87

Each of the 29 families in the village owns 1 or 3 or 5 horses. The families that own only one horse are as many as those that own 5 horses. How many horses are there in total in the village?

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1309

2006

final

no

C1 C2 L1

Pattern Recognition

In the box there are: 5 numbers equal to 7 odd numbers

1 ; 2 ; 3 ; 4 ; 5 ; 6 ; 7 ; 8 ; 9 ; 10 In this box there are _____ even numbers In this box there are _____ odd numbers Complete the sentences in the box with numbers (written as digits) so that the sentences contained within it are true.

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1320

2006

international final day 1

no

C1

Pattern Recognition

number 55

The director of a 40-room hotel is superstitious: he refuses to use the unlucky digit "3". He assigned the number 1 to one room, 2 to another, and so on, but skipping all numbers that are written using the digit "3". Thus, the third room has the number 4. What number did he assign to the fortieth room?

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1321

2006

international final day 1

no

C1

Algebra

3 rings of 30 pearls and 8 rings of 20 pearls.

Claudia and Anna make rings with pearls. Together, they bought 250 pearls. Anna's model uses 30 pearls, and Claudia's uses 20. They used all their pearls and made 11 rings. How many rings did Anna make, and how many did Claudia make?

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1323

2006

international final day 1

no

C1

Arithmetic

7 minutes

My friend Anna took the train in Paris at 14:29. This train usually takes 4 hours and 3 minutes to arrive in Strasbourg. I arrive at Strasbourg station to pick her up at 18:35 and read on the arrivals board that the train is running 10 minutes late. How long do I have to wait for my friend?

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1324

2006

international final day 1

no

C1

Combinatorics

28 lots

Eight classes participate in a school football tournament. Each class has three teams: the experts, the strong, and the clever ones. When two classes meet, each of the teams from one class plays once against the team of the same level from the other class: the experts against the experts, the strong against the strong, and the clever ones against the clever ones. How many matches will be played if each class meets every other class?

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1325

2006

international final day 1

no

C1 C2

Arithmetic

16 h 26 min.

One hundred forty cyclists take part in the Giro d'Italia prologue: it is an individual time trial. Riders depart every 2 minutes. The first one departs at 13:10. Matteo Rossi is the 99th to depart. At what time will he depart?

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1328

2006

international final day 1

no

C2 L1

Logic

2 solutions: BCAD or BADC.

Four cyclists, Alessandro, Bruno, Carlo, and Davide, take part in a time trial. The first one starts at 8:00 AM, then subsequent starts follow every 5 minutes. The arrival times are 8:59 AM, 9:02 AM, 9:04 AM, and 9:08 AM. The one who started immediately before Bruno and immediately after Davide took two minutes more than the winner. Alessandro arrived at the finish line before Carlo. Determine the race ranking, from fastest to slowest, knowing that there were no ties.

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1332

2006

international final day 1

no

L1 L2

Algebra

5 solutions: 2008, 1006, 136, 94 and 78

On each vertex of the octahedron, a non-negative integer must be written. On each face, the product of the three numbers corresponding to its three vertices is written. The sum of the eight numbers written on the faces must be equal to 2006. What is the sum of the six numbers written on the vertices of the octahedron?

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1334

2006

international final day 2

no

C1

Arithmetic

42 cm.

Eleonora places seven cubes of increasingly smaller size one on top of the other, so as to form a stepped pyramid. The side of the second cube is one centimeter shorter than that of the first; that of the third is one centimeter shorter than that of the second; and so on. The side of the smallest cube is 3 centimeters long. How high is Eleonora's pyramid?

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1336

2006

international final day 2

no

C1

Combinatorics

10 bags.

Giuliano is preparing his birthday party. He wants to offer small sweets to each guest. His mother bought him 5 different types: strawberry candies, liquorice strings, lollipops, lemon candies, and mint candies. Giuliano decides to prepare bags, each containing 30 sweets of three different types: ten of each type. He would like each guest to have a bag containing a different combination of sweets. How many bags containing a different combination of sweets can he prepare?

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1337

2006

international final day 2

no

C1

Algebra

6 times

The Rome-Pisa train departing at noon takes 3h 58min to reach its destination. The evening train, which makes triple the number of stops, takes 4h 26min. All stops have the same duration, between 5 and 10 minutes. When the two trains are not stopped, they travel at the same speed. How many stops does the evening train make?

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1339

2006

international final day 2

no

C1 C2

Logic

Noemi.

Four friends play 'five-seven'. They arrange themselves in a circle in the following order, clockwise: Noemi, Maura, Luigi, Romano. The game consists of counting in turns, each pronouncing a number: Noemi starts with "one", then Maura with "two", and so on. However, immediately after one of the four friends pronounces a multiple of 5 or 7, the direction in which the four friends take turns counting is reversed. Who will say "twenty-three"?

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1340

2006

international final day 2

no

C1 C2

Combinatorics

11 times.

Gianni has been punished: he must copy the phrase "I must not shout in class" 2006 times in his notebook. He has written this phrase once on his computer. What is the minimum number of times he must use the copy-paste command to obtain the 2006 required phrases? Note that "copy-paste command" refers to the double operation of copying and pasting.

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1342

2006

international final day 2

no

C2 L1

Combinatorics

26

Eighteen people want to divide 10 equal loaves of bread among themselves. They can cut each loaf as they wish, and it is not required that all be cut in the same way. What is the minimum number of pieces of bread that must be obtained, by adequately cutting the given loaves, so that each person has the same amount of bread?

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1343

2006

international final day 2

no

C2 L1

Arithmetic

214

I am a six-digit number with identical digits, and I am divisible by 49. Sum my integer divisors between 1 and 49 (inclusive). What number will you get?

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1344

2006

international final day 2

no

L1 L2

Pattern Recognition

515542

We write the integers in their natural order starting from 1. Then we cross out sequences of numbers, always leaving one uncrossed number between two sequences of crossed-out numbers. The sequences of crossed-out numbers consist respectively of: 10 numbers, 1 number, 10 numbers, 2 numbers, 10 numbers, 3 numbers, 10 numbers, 4 numbers, 10 numbers, 5 numbers, etc. Which will be the 2006th uncrossed number?

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1347

2006

international final day 2

no

L2

Combinatorics

200600

Matilde observes to Mattia that there are 76 different ways to place three non-aligned pawns on a 3x3 chessboard, such that each pawn occupies a distinct square. "So what?" asks Mattia. "If you multiply this number by 5, you get the product of our ages, that is, the product of 19 and 20." Mattia reflects and says that if you multiply by 5 the number of different ways to place three non-aligned pawns on an 8x8 chessboard, you get a remarkable number. What is this number?

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1348

2006

international final day 2

no

L2

Geometry

16 √3

Five molehills, treated as points, are located inside (including the borders) of a square lawn with a side length of 12 meters. No three of these molehills are collinear. What is the maximum possible value of the minimum area among all the triangles that can be formed by taking three of the molehills as vertices? Give the exact answer in m², even if it is not a rational number.

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1130

2007

autumn games CE

no

CE

Combinatorics

1

Pietro and Renato challenge each other to the game of "ten". Pietro starts and says a number that can only be 1 or 2. Then, Renato adds 1 or 2 to the number said by Pietro. Now, it's Pietro's turn again to add 1 or 2 to the previous sum. Pietro and Renato continue in this way, until one of them says 10 and wins the game. What number must Pietro say at the beginning to be sure of winning?

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