We solve problems

Problems Algebra and arithmetic Mean values Methods Proof by contradiction

Is it possible for the mean of some 35 whole numbers to equal \(6.35\)?

Problems Combinatorics Packing problems Methods Pigeonhole principle Pigeonhole principle (finite number of poits, lines etc.)

11–13 2.0

Is it possible to place 12 identical coins along the edges of a square box so that touching each edge there were exactly: a) 2 coins, b) 3 coins, c) 4 coins, d) 5 coins, e) 6 coins, f) 7 coins.

You are allowed to place coins on top of one another. In the cases where it is possible, draw how this could be done. In the other cases, prove that doing so is impossible.

Problem #PRU-107704

Problems Algebra and arithmetic Arithmetics. Mental maths Set theory and logic Set theory Inclusion-exclusion principle

11–13 3.0

In a class there are 50 children. Some of the children know all the letters except “h” and they miss this letter out when writing. The rest know all the letters except “c” which they also miss out. One day the teacher asked 10 of the pupils to write the word “cat”, 18 other pupils to write “hat” and the rest to write the word “chat”. The words “cat” and “hat” each ended up being written 15 times. How many of the pupils wrote their word correctly?

Problem #PRU-107802

Problems Methods Examples and counterexamples. Constructive proofs Algebra and arithmetic Word problems Tables and tournaments Number tables and its properties Pigeonhole principle Pigeonhole principle (other)

12–14 4.0

Eight schoolchildren solved \(8\) tasks. It turned out that \(5\) schoolchildren solved each problem. Prove that there are two schoolchildren, who solved every problem at least once.
If each problem is solved by \(4\) pupils, prove that it is not necessary to have two schoolchildren who would solve each problem.

Problem #PRU-107821

Problems Algebra and arithmetic Word problems Tables and tournaments Chessboards and chess pieces Methods Pigeonhole principle Pigeonhole principle (finite number of poits, lines etc.)

12–14 3.0

Some squares on a chess board contain a chess piece. It is known that each row contains at least one chess piece, but that different rows all have different numbers of pieces. Prove that it is always possible to mark 8 pieces so that each row and each column of the board contains exactly one marked piece.

Problem #PRU-107845

Problems Algebra and arithmetic Number theory. Divisibility Equations in integer numbers

11–14 3.0

Is it possible to find natural numbers \(x\), \(y\) and \(z\) which satisfy the equation \(28x+30y+31z=365\)?

Problem #PRU-107986

Graph Theory Problems Methods Pigeonhole principle Set theory and logic

12–15 4.0

Peter has 28 classmates. Each 2 out of these 28 have a different number of friends in the class. How many friends does Peter have?

Problem #PRU-108181

Problems Methods Pigeonhole principle Pigeonhole principle (angles and lengths)

12–14 3.0

Prove that any convex polygon contains not more than \(35\) vertices with an angle of less than \(170^\circ\).

Problem #PRU-108731

Problems Algebra and arithmetic Word problems Percentages and ratios

12–14 2.0

A field that will be used to grow wheat has a rectangular shape. This year, the farmer responsible for this field decided to increase the length of one of the sides by \(20\%\) and decrease the length of another side by \(20\%\). The field remains rectangular. Will the harvest of wheat change this year and, if so, then by how much?

Problem #PRU-108737

Problems Set theory and logic Algorithm Theory Algorithm Theory

10–12 2.0

Solve problem number 108736 for the inscription \(A\), \(BC\), \(DEF\), \(CGH\), \(CBE\), \(EKG\).