20th Tournament of Towns

Spring 1999, O-level.

Your total score is based on the three problems for which you earn the most points; the scores for the individual parts of a single problem are summed. Points for each problem are shown in brackets [ ].

Juniors

1. [3] A father and son skate around a circular skating rink. From time to time the father overtakes his son. After the son starts to skate in the opposite direction, they began to meet five times more often. How much faster does the father skate than his son?

Tairova

2. [4] A square ABDE is constructed to the outward of the hypothenuse AB of the right triangle ABC. We know that AC=1cm and BC=3cm. What is the ratio into which the side DE is divided by the bissector of angle C?

A.Blinkov

3. [4] Several positive numbers a0, a1, a2, ... , an are written on a board. On a second board, we write the amount b0 of numbers written on the first board, the amount b1 of numbers on the first board exceeding 1, the amount b2 of numbers greater than 2, and so on while the b's are still positive; then we stop (we don't write any zeros). On a third board we write the numbers c0, c1, c2, ... by using the same rules as before, but applied to the numbers b0, b1, b2, ... of the second board. Prove that the numbers written on the first and on the third board are the same.

A.Lebesque, A.Kanel-Belov

4. [5] A black regular triangle is drawn on the plane. There are nine tiles of the same size and form. We must place them on the plane so that they don't overlap and each tile covers at least one inner point of the black triangle. How can this be done?

folklore

5. [5] A square was cut into 100 rectangles by means of 9 straight lines parallel to one of the sides and 9 lines parallel to another. It turned out that exactly 9 of the rectangles were actually squares. Prove that we can find two equal squares among the 9.

V.Proizvolov

Seniors

1. [3] 1999 numbers are written one after the other. The first number is 1. It is known that each number (except the first and last) is equal to the sum of its neighbors. Find the last number.

V.Senderov

2. [3] A square ABDE is constructed to the outward of the hypothenuse AB of the right triangle ABC. We know that AC=1cm and BC=3cm. What is the ratio int which the side DE is divided by the bissector of angle C?

A.Blinkov

3. [3] Several positive numbers a0, a1, a2, ... , an are written on a board. On a second board, we write the amount b0 of numbers written on the first board, the amount b1 of numbers on the first board exceeding 1, the amount b2 of numbers greater than 2, and so on while the b's are still positive; then we stop (we don't write any zeros). On a third board we write the numbers c0, c1, c2, ... by using the same rules as before, but applied to the numbers b0, b1, b2, ... of the second board. Prove that the numbers written on the first and on the third board are the same.

A.Lebesque, A.Kanel-Belov

4. [5] A black square is drawn on the plane. There are seven square tiles of the same size. We must place them on the plane so that they don't overlap and each tile covers at least one inner point of the black square. How can this be done?

folklore

5. [5] Two perons play a game on a 9 by 9 square-lined sheet of paper. They move alternatively. At each move the first player draws a cross inside some empty little square, the second one draws a little circle. When all the 81 squares are filled, the number K of rows and columns in which there are more crosses than circles and the number H of rows and columns in which there are more circles (the total number of rows and columns is 18) is counted. The difference B = K - H is the value of the win of the first player. Find a value of B such that (a) the first player can ensure that his win will be no less than B, no matter how the second one plays while (b) the second player can ensure that the first player's win will be no greater than B, no matter how the first player plays.

A.Kanel-Belov