International Mathematics Tournament of the Towns

Practice Problems

Arithmetics

1. A jar completely filled with water weighs 5000 grams. The same jar, half-filled with water, weighs 3250 grams. How much does the empty jar weigh?
2. Each postcard costs the same integral number of cents. If nine postcards cost less than $10, but ten cost more than $11, what is the cost of one postcard?
3. Peter puts $100 in the bank. The bank offers him either an annual interest of 7% or a monthly interest of 7/12 %. Which plan should he choose in order to have more money in the end of 5 years?
4. Which takes more time: sailing 10 kilometres upstream against the current and then 10 kilometres downstream with the current, or sailing 20 kilometres in still water?
5. A eats a cake in 1/2 hour, B in 1 hour and C in 5 minutes. At these rates, how long would it take for three of them to eat the cake?

Divisibility and Modular Arithmetic

1. A grasshopper jumps along a straight line in either direction. Each jump is of length 6 centimeters or 8 centimeters. Can the grasshopper get to a point which is
   • 3 centimeters
   • 4 centimeters
   from its starting position?
2. Basil tears a sheet of newspaper into 8 pieces, then he tears one of the pieces into 8 pieces, and so on. Can he end up with 2003 pieces?
3. A certain number leaves a remainder of 1 when divided by 2, and a remainder of 2 when divided by 3. What is the remainder when it is divided by 6?
4. Prove that k³-k is divisible by 6 for every integer k.
5. What is the last digit of 3²⁰⁰³?

Combinatorics

1. A fence consists of 20 boards. Each of them is to be painted in blue, green or yellow, and any two adjacent boards must have different colours. In how many ways can the fence be painted? In how many ways can the fence be painted if there must be at least one blue board?
2. There are 25 students in a class. In how many ways can they elect
   • one helper for the mathematics teacher and one helper for the language arts teacher;
   • two helpers for the mathematics teacher;
   • three helpers for the mathematics teacher?
3. Peter has 5 mathematical books and Basil has 7 history books. Peter should trade a pair of his books for a pair of Basil's books. In how many ways can they make such trade?
4. In how many ways can a room with five independent light bulbs be illuminated?
5. The menu in a school cafeteria never changes. It consists of 10 different dishes. Peter decides to make his school lunch different everyday. For each lunch, he may eat any number of dishes, but no two are identical.
   • What is the maximum number of days Peter can do so?
   • What is the total number of dishes Peter has consumed during this period?

Pigeonhole Principle

1. There are 25 students in a class.
   • Prove that there are 2 students who are born in the same month.
   • Are there necessarily three students who are born in the same month?
2. 15 boys altogether collected 100 nuts. Prove that there are two the boys who collected the same number of nuts.
3. Prove that out of 10 positive integers, none of which is divisible by 10, one can find
   • two numbers whose difference is divisible by 10;
   • several numbers whose sum is divisible by 10.
4. If we choose 26 numbers out of 1, 2, .... , 49, 50, must we have chosen two consecutive numbers?
5. Can one cover an equilateral triangle by two smaller equilateral triangles?

Logic

1. We have two unmarked jars of capacities 2 and 5 litres respectively. We have an unlimited supply of water from a tap. By filing the jugs, emptying them and pouring from one into the other, can we leave exactly 4 litres of water in the larger jug?
2. In the number 3141592653589793 cross out seven digits so as to leave behind as large a number as possible.
3. A said, "Dimitry has more than a thousand books." B said, "Wrong, he has less than a thousand books." C said, "He has at least one book." If exactly one of these statements is true, how many books does Dimitry have?
4. Serge had 7 potatoes, Paul had 5 and Nick none. They boiled all 12 and shared equally. Nick gave 12 cents to Serge and Paul as his share. How should Serge and Paul divide this amount?
5. An archery tournament was held in two days. the first day each participant got as many points as all the other participants got on the second day. Prove that each participant got the same number of points in this tournament.

   Geometry

6. Draw on the plane
   • four;
   • five;
   • six
   points such that every three of them formed an isosceles triangle.
7. Into how many parts can a plane be divided by
   • three;
   • four
   different straight lines? Give an example for each possible case.
8. In the right triangle ABC, AB=BC=1. D is an arbitrary point on AC. Is it possible to determine the sum of the distances from D to AB and BC?
9. Is it possible to cut a triangle into
   • two acute triangles?
   • two obtuse triangles?
10. On an infinite grid, draw a square whose area is 5 times that of a square on the grid. Use a straight-edge but not a compass.
11. The midpoints of two sides of a triangle are given. Using only a straight-edge but not a compass, construct the midpoint of the third side.
12. In quadrilateral ABCD, AD is parallel to and longer than BC. Which is larger: the sum of angles A and D or the sum of angles B and C?
13. In a triangle, two of its altitudes are not shorter than the sides onto which they are dropped. Determine the angles of this triangle.
14. K is a point outside the square ABCD of side 1 such that AKB is an equilateral triangle. Determine the radii of the circle passing through C, K and D.

Miscellaneous

1. A king with his entourage moves from his Palace to the village, with speed 5 kilometres per hour. Every hour, he sends a messenger to the village. The messengers move with speed 20 kilometres per hour. At what time interval will the messengers be arriving at the village?
2. A logging company decided to cut a forest but Green-Peace activists protested. The president of the company stated that 99% of the trees were pines, which were the only kind of tree being cut. After logging, he reassured the activities, the pines would still constitute 98% of the trees left. How many percent of the trees was the company going to cut?
3. Is it possible for each of the 25 students in a class to shake hands with exactly 7 classmates?
4. The sum of the squares of two integers is divisible by 3. Prove that each of these numbers is divisible by 3.
5. In a country there are 15 cities. Each of them is directly connected by a road to at least 7 other cities. The roads do not meet except at the cities. Prove that one can reach any city from any other city along the roads.
6. In a class with 28 students, each girl shook hands with 4 boys and each boy shook hands with 3 girls. How many boys and girls are there in this class?
7. In a class with at least two students, prove that there are two who shook hands with the same number of classmates.
8. Consider the numbers from 1 to 1000 inclusive.
   • How many of them contain the digit 3?
   • How many of them contain both the digit 1 and the digit 2?
9. How many different necklaces are there if each consists of 5 beads which may be green or blue?
10. Is it possible to fill some of the squares of 5 x 5 table with numbers in such way that the sum of the numbers in each column is 8 while the sum of the number in each row is 9?
11. 8 x 8 square is covered by 32 of 1 x 2 dominos. Prove that two of the dominoes form a 2 x 2 square .
12. It is known that the sum of any 23 of 2002 given numbers is positive. Prove that the sum of all of them is also positive.
13. Can one cut out of 10 x 10 square several circles such that the sum of their diameters exceeds 500?
14. There are two staircases. Each of them has the height 1 meter and the base 2 meters. The first staircase has 7 steps, and the second one has 9 steps. Is a carpet covering the first staircase long enough to cover the second one?
15. One can easily cut 3 x 3 x 3 cube into 27 unit cubes with six cuts. Is it possible to do so with a smaller number of cuts if pieces may be moved and rearranged after each cut?