We solve problems

Problem #PRU-21982

Problems Geometry Plane geometry Geometrical inequalities Inequalities in a triangle A segment inside the triangle is smaller than the largest side ,Problems Methods Pigeonhole principle Pigeonhole principle (angles and lengths)

11–13 3.0

Prove that it is not possible to completely cover an equilateral triangle with two smaller equilateral triangles.

Problem #PRU-21983

Problems Geometry Plane geometry Geometrical inequalities Inequalities in a triangle A segment inside the triangle is smaller than the largest side ,Problems Methods Pigeonhole principle Pigeonhole principle (finite number of poits, lines etc.)

12–14 3.0

51 points were thrown into a square of side 1 m. Prove that it is possible to cover some set of 3 points with a square of side 20 cm.

Problem #PRU-21984

Problems Methods Pigeonhole principle Pigeonhole principle (other)

11–13 2.0

At the end of the month 5 workers were paid a total of £1,500 between them. Each wants to buy themselves a smartphone that costs £320. Prove that one of them will have to wait another month in order to do so.

Problem #PRU-21985

Problems Methods Algebraic methods Counting in two ways Partitions into pairs and groups; bijections Pigeonhole principle Pigeonhole principle (other)

12–13 3.0

The total age of a group of 7 people is 332 years. Prove that it is possible to choose three members of this group so that the sum of their ages is no less than 142 years.

Problem #PRU-21987

Problems Methods Pigeonhole principle Pigeonhole principle (other)

11–13 2.0

Prove that within a group of \(51\) whole numbers there will be two whose difference of squares is divisible by \(100\).

Problem #PRU-21988

Problems Algebra Number systems Decimal number system Number theory. Divisibility Division with remainders. Arithmetic of remainders Division with remainder Methods Pigeonhole principle Pigeonhole principle (other)

12–14 3.0

Prove that amongst numbers written only using the number 1, i.e.: 1, 11, 111, etc, there is a number than is divisible by 1987.

Problem #PRU-21989

Problems Algebra Number theory. Divisibility Division with remainders. Arithmetic of remainders Arithmetic of remainders Number systems Decimal number system Euler's theorem Methods Pigeonhole principle Pigeonhole principle (other)

12–14 3.0

Prove that there is a power of 3 that ends in 001.

Problem #PRU-21990

Problems Algebra Word Problems Tables and tournaments Number tables and its properties Methods Pigeonhole principle Pigeonhole principle (other)

11–13 2.0

A \(3\times 3\) square is filled with the numbers \(-1, 0, +1\). Prove that two of the 8 sums in all directions – each row, column, and diagonal – will be equal.

Problem #PRU-21991

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

11–13 2.0

100 people are sitting around a round table. More than half of them are men. Prove that there are two males sitting opposite one another.

Problem #PRU-21993

Problems Methods Pigeonhole principle Pigeonhole principle (other) Algebra Number theory. Divisibility Products and factorials

12–14 3.0

The numbers \(1, 2, \dots , 9\) are divided into three groups. Prove that the product of the numbers in one of the groups will always be no less than 72.