We solve problems

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

2022 points are selected from a cube, whose edge is equal to 13 units. Is it possible to place a cube with edge of 1 unit in this cube so that there is not one selected point inside it?

Problem #PRU-78175

Problems Algebra and arithmetic Number systems Decimal number system

12–15 2.0

Note that if you turn over a sheet on which numbers are written, then the digits 0, 1, 8 will not change and the digits 6 and 9 will switch places, whilst the others will lose their meaning. How many nine-digit numbers exist that do not change when a sheet is turned over?

Problem #PRU-78206

Problems Methods Algebraic methods Counting in two ways Pigeonhole principle Pigeonhole principle (other)

13–14 3.0

30 pupils in years 7 to 11 took part in the creation of 40 maths problems. Every possible pair of pupils in the same year created the same number of problems. Every possible pair of pupils in different years created a different number of problems. How many pupils created exactly one problem?

Problem #PRU-78234

Problems Methods Pigeonhole principle Algebraic methods Processes and operations

13–15 4.0

The number $A$ is divisible by $1, 2, 3, \dots, 9$ . Prove that if $2 A$ is presented in the form of a sum of some natural numbers smaller than 10, $2 A = a_{1} + a_{2} + \dots + a_{k}$ , then we can always choose some of the numbers $a_{1}, a_{2}, \dots, a_{k}$ so that the sum of the chosen numbers is equal to $A$ .

Problem #PRU-78259

Problems Methods Examples and counterexamples. Constructive proofs Pigeonhole principle Pigeonhole principle (other)

14–16 4.0

A table of $4 \times 4$ cells is given, in some cells of which a star is placed. Show that you can arrange seven stars so that when you remove any two rows and any two columns of this table, there will always be at least one star in the remaining cells. Prove that if there are fewer than seven stars, you can always remove two rows and two columns so that all the remaining cells are empty.

Problem #PRU-78270

Problems Geometry Plane geometry Area Area of curvilinear shapes Methods Pigeonhole principle Pigeonhole principle (area and volume)

13–15 4.0

120 unit squares are placed inside a $20 \times 25$ rectangle. Prove that it will always be possible to place a circle with diameter 1 inside the rectangle, without it overlapping with any of the unit squares.

Problem #PRU-78475

Problems Geometry Plane geometry Lines, segments, rays, and angles

13–14 3.0

You are given $7$ straight lines on a plane, no two of which are parallel. Prove that there will be two lines such that the angle between them is less than $26^{\circ}$ .

Problem #PRU-78494

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

12–15 3.0

All of the integers from 1 to 64 are written in an $8 \times 8$ table. Prove that in this case there are two adjacent numbers, the difference between which is not less than 5. (Numbers that are in cells which share a common side are called adjacent).

Problem #PRU-78496

Problems Methods Examples and counterexamples. Constructive proofs Pigeonhole principle Pigeonhole principle (other)

13–14 4.0

What is the largest amount of numbers that can be selected from the set 1, 2, ..., 1963 so that the sum of any two numbers is not divisible by their difference?

Problem #PRU-78501

Problems Methods Pigeonhole principle Pigeonhole principle (other)

14–15 3.0

All of the integers from 1 to 81 are written in a $9 \times 9$ table. Prove that in this case there are two adjacent numbers, the difference between which is not less than 6. (Numbers that are in cells which share a common side are called adjacent.)