We solve problems

Problem #PRU-79515

Problems Algebra and arithmetic Number systems Decimal number system Methods Pigeonhole principle Pigeonhole principle (other)

13–15 4.0

7 different digits are given. Prove that for any natural number $n$ there is a pair of these digits, the sum of which ends in the same digit as the number.

Problem #PRU-86112

Problems Geometry Plane geometry Quadrilateral Arbitrary quadrilaterals Algebra and arithmetic Trigonometry Trigonometry (other)

14–15 2.0

Does there exist a flat quadrilateral in which the tangents of all interior angles are equal?

Problem #PRU-86120

Problems Combinatorics Geometry on grid paper Methods Pigeonhole principle Pigeonhole principle (other)

13–15 4.0

A board of size $2005 \times 2005$ is divided into square cells with a side length of 1 unit. Some board cells are numbered in some order by numbers 1, 2, ... so that from any non-numbered cell there is a numbered cell within a distance of less than 10. Prove that there can be found two cells with a distance between them of less than 150, which are numbered by numbers that differ by more than 23. (The distance between the cells is the distance between their centres.)

Problem #PRU-87046

Problems Geometry Geometry (other)

15–16 2.0

Let $M$ be the point of intersection of the medians of the triangle $A B C$ , and $O$ an arbitrary point on a plane. Prove that $O M^{2} = 1 / 3 (O A^{2} + O B^{2} + O C^{2}) - 1 / 9 (A B^{2} + B C^{2} + A C^{2}) .$

Problem #PRU-87048

Problems Geometry Geometry (other)

15–16 2.0

Three non-coplanar vectors are given. Is it possible to find a fourth vector perpendicular to the three vectors given?

Problem #PRU-87266

Problems Geometry Geometry (other)

15–16 2.0

Find the volume of an inclined triangular prism whose base is an equilateral triangle with sides equal to a if the side edge of the prism is equal to the side of the base and is inclined to the plane of the base at an angle of $60^{\circ}$ .

Problem #PRU-88105

Problems Algebra and arithmetic Sequences Progressions Arithmetic progression

12–15 2.0

Is the sum of the numbers $1 + 2 + 3 + \dots + 1999$ divisible by 1999?

Problem #PRU-97823

Problems Algebra and arithmetic Word problems Tables and tournaments Chessboards and chess pieces Methods Examples and counterexamples. Constructive proofs Combinatorics Painting problems

14–16 4.0

Given an endless piece of chequered paper with a cell side equal to one. The distance between two cells is the length of the shortest path parallel to cell lines from one cell to the other (it is considered the path of the center of a rook). What is the smallest number of colors to paint the board (each cell is painted with one color), so that two cells, located at a distance of 6, are always painted with different colors?

Problem #PRU-97855

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

13–16 4.0

A group of numbers $A_{1}, A_{2}, \dots, A_{100}$ is created by somehow re-arranging the numbers $1, 2, \dots, 100$ .

100 numbers are created as follows: $B_{1} = A_{1}, B_{2} = A_{1} + A_{2}, B_{3} = A_{1} + A_{2} + A_{3}, \dots, B_{100} = A_{1} + A_{2} + A_{3} \dots + A_{100} .$

Prove that there will always be at least 11 different remainders when dividing the numbers $B_{1}, B_{2}, \dots, B_{100}$ by 100.

Problem #PRU-97984

Problems Algebra and arithmetic Number theory. Divisibility Division with remainders. Arithmetic of remainders Arithmetic of remainders Methods Examples and counterexamples. Constructive proofs

13–16 3.0

a) We are given two cogs, each with 14 teeth. They are placed on top of one another, so that their teeth are in line with one another and their projection looks like a single cog. After this 4 teeth are removed from each cog, the same 4 teeth on each one. Is it always then possible to rotate one of the cogs with respect to the other so that the projection of the two partially toothless cogs appears as a single complete cog? The cogs can be rotated in the same plane, but cannot be flipped over.

b) The same question, but this time two cogs of 13 teeth each from which 4 are again removed?