We solve problems

Problems Set theory and logic Joke problem

The angle at the top of a crane is $20^{\circ}$ . How will the magnitude of this angle change when looking at the crane with binoculars which triple the size of everything?

Problem #PRU-97979

Problems Algebra and arithmetic Number theory. Divisibility Division with remainders. Arithmetic of remainders Arithmetic of remainders Methods Pigeonhole principle Pigeonhole principle (other)

12–14 2.0

Prove that in any group of 7 natural numbers – not necessarily consecutive – it is possible to choose three numbers such that their sum is divisible by 3.

Problem #PRU-97991

Problems Algebra and arithmetic Word problems Tables and tournaments Chessboards and chess pieces

10–15 3.0

What is the minimum number of squares that need to be marked on a chessboard, so that:

1) There are no horizontally, vertically, or diagonally adjacent marked squares.

2) Adding any single new marked square breaks rule 1.

Problem #PRU-98016

Problems Combinatorics Integer lattices Integer lattices (other) Set theory and logic Order relations Methods Pigeonhole principle Pigeonhole principle (other)

10–15 4.0

We are given 101 rectangles with integer-length sides that do not exceed 100.

Prove that amongst them there will be three rectangles $A, B, C$ , which will fit completely inside one another so that $A \subset B \subset C$ .

Problem #PRU-98017

Problems Methods Pigeonhole principle Pigeonhole principle (other)

12–14 3.0

A staircase has 100 steps. Vivian wants to go down the stairs, starting from the top, and she can only do so by jumping down and then up, down and then up, and so on. The jumps can be of three types – six steps (jumping over five to land on the sixth), seven steps or eight steps. Note that Vivian does not jump onto the same step twice. Will she be able to go down the stairs?

Problem #PRU-98031

Problems Methods Pigeonhole principle Pigeonhole principle (other) Proof by contradiction

12–14 2.0

10 friends sent one another greetings cards; each sent 5 cards. Prove that there will be two friends who sent cards to one another.

Problem #PRU-98143

Problems Methods Pigeonhole principle Pigeonhole principle (other)

12–14 3.0

A gang contains 101 gangsters. The whole gang has never taken part in a raid together, but every possible pair of gangsters have taken part in a raid together exactly once. Prove that one of the gangsters has taken part in no less than 11 different raids.

Problem #PRU-98170

Problems Methods Extremal principle Extremal principle (other) Algebraic methods Partitions into pairs and groups; bijections Pigeonhole principle Pigeonhole principle (other)

10–14 4.0

Ben noticed that all 25 of his classmates have a different number of friends in this class. How many friends does Ben have?

Problem #PRU-98230

Problems Methods Pigeonhole principle Pigeonhole principle (other)

12–14 4.0

A village infant school has 20 pupils. If we pick any two pupils they will have a shared granddad.

Prove that one of the granddads has no fewer than 14 grandchildren who are pupils at this school.

Problem #PRU-98363

Problems Algebra and arithmetic Word problems Tables and tournaments Chessboards and chess pieces Set theory and logic Theory of algotithms Theory of algorithms (other)

10–13 2.0

Initially, on each cell of a $1 \times n$ board a checker is placed. The first move allows you to move any checker onto an adjacent cell (one of the two, if the checker is not on the edge), so that a column of two pieces is formed. Then one can move each column in any direction by as many cells as there are checkers in it (within the board); if the column is on a non-empty cell, it is placed on a column standing there and unites with it. Prove that in $n - 1$ moves you can collect all of the checkers on one square.