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Problem #PRU-79285

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

12–14 4.0

The sum of 100 natural numbers, each of which is no greater than 100, is equal to 200. Prove that it is possible to pick some of these numbers so that their sum is equal to 100.

Problem #PRU-79322

Problems Geometry Solid geometry Spheres Circle on a sphere Methods Pigeonhole principle Pigeonhole principle (area and volume) Extremal principle Smallest or largest distance (length)

14–16 4.0

A spherical sun is observed to have a finite number of circular sunspots, each of which covers less than half of the sun’s surface. These sunspots are said to be enclosed, that is no two sunspots can touch, and they do not overlap with one another. Prove that the sun will have two diametrically opposite points that are not covered by sunspots.

Problem #PRU-79384

Problems Combinatorics Integer lattices Integer lattices (other) Methods Pigeonhole principle Pigeonhole principle (finite number of poits, lines etc.)

13–15 3.0

There are several squares on a rectangular sheet of chequered paper of size \(m \times n\) cells, the sides of which run along the vertical and horizontal lines of the paper. It is known that no two squares coincide and no square contains another square within itself. What is the largest number of such squares?

Problem #PRU-79399

Problems Methods Pigeonhole principle Pigeonhole principle (finite number of poits, lines etc.) Geometry Plane geometry Polygons Regular polygons Quadrilateral Trapeziums

13–15 4.0

In a regular 1981-gon 64 vertices were marked. Prove that there exists a trapezium with vertices at the marked points.

Problem #PRU-79407

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

13–13 3.0

A square \(ABCD\) contains 5 points. Prove that the distance between some pair of these points does not exceed \(\frac{1}{2} AC\).

Problem #PRU-79426

Problems Geometry Plane geometry Triangles Types of triangles Equilateral triangle Combinatorics Painting problems Methods Pigeonhole principle Pigeonhole principle (finite number of poits, lines etc.)

12–14 3.0

A white plane is arbitrarily sprinkled with black ink. Prove that for any positive \(l\) there exists a line segment of length \(l\) with both ends of the same colour.

Problem #PRU-79462

Problems Combinatorics Painting problems Methods Pigeonhole principle Pigeonhole principle (other) Proof by contradiction Algebra Word Problems Tables and tournaments Tables and tournaments (other)

12–14 3.0

The judges of an Olympiad decided to denote each participant with a natural number in such a way that it would be possible to unambiguously reconstruct the number of points received by each participant in each task, and that from each two participants the one with the greater number would be the participant which received a higher score. Help the judges solve this problem!

Problem #PRU-79473

Problems Methods Pigeonhole principle Pigeonhole principle (other) Algebra Algebraic inequalities and systems of inequalities Quadratic inequalites (several variables)

13–14 3.0

The numbers \(a_1, a_2, \dots , a_{1985}\) are the numbers \(1, 2, \dots , 1985\) rearranged in some order. Each number \(a_k\) is multiplied by its number \(k\), and then the largest number is chosen among the resulting 1985 products. Prove that it is not less than \(993^2\).

Problem #PRU-79488

Problems Methods Pigeonhole principle Pigeonhole principle (other) Algebra Number theory. Divisibility The fundamental theorm of arithmetic. Prime factorisation.

13–15 4.0

The product of 1986 natural numbers has exactly 1985 different prime factors. Prove that either one of these natural numbers, or the product of several of them, is the square of a natural number.

Problem #PRU-79498

Problems Methods Pigeonhole principle Pigeonhole principle (other) Algebra Number theory. Divisibility The fundamental theorm of arithmetic. Prime factorisation.

13–15 4.0

The product of a group of 48 natural numbers has exactly 10 prime factors. Prove that the product of some four of the numbers in the group will always give a square number.