We solve problems

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-86492

Problems Set theory and logic Mathematical logic Mathematical logic (other) Algebra and arithmetic Number theory. Divisibility Odd and even numbers

12–14 2.0

On an island there are 1,234 residents, each of whom is either a knight (who always tells the truth) or a liar (who always lies). One day, all of the inhabitants of the island were broken up into pairs, and each one said: “He is a knight!" or “He is a liar!" about his partner. Could it eventually turn out to be that the number of “He is a knight!" and “He is a liar!" phrases is the same?

Problem #PRU-86496

Problems Algebra and arithmetic Absolute value Inequalities containing absolute values

13–14 2.0

Solve the inequality: $| x + 2000 | < | x - 2001 |$ .

Problem #PRU-86508

Problems Set theory and logic Mathematical logic Mathematical logic (other) Algebra and arithmetic Polynomials Quadratic polynomials Quadratic equations and systems of equations

13–14 2.0

Solving the problem: “What is the solution of the expression $x^{2000} + x^{1999} + x^{1998} + 1000 x^{1000} + 1000 x^{999} + 1000 x^{998} + 2000 x^{3} + 2000 x^{2} + 2000 x + 3000$ ( $x$ is a real number) if $x^{2} + x + 1 = 0$ ?”, Vasya got the answer of 3000. Is Vasya right?

Problem #PRU-86522

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

13–14 3.0

Prove that amongst the numbers of the form $19991999 \dots 19990 \dots 0$ – that is 1999 a number of times, followed by a number of 0s – there will be at least one divisible by 2001.

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-87939

Problems Combinatorics Dissections, partitions, covers and tilings Dissections with certain properties Set theory and logic Set theory Unusual constructions

10–14 2.0

Is it possible to cut out such a hole in a sheet of paper through which a person could climb through?