We solve problems

Problems Geometry Solid geometry Visual geometry in space

Find the first 99 decimal places in the number expansion of \((\sqrt{26} + 5)^{99}\).

Problems Algebra and arithmetic Algebraic equations and systems of equations Higher order equations. Palindromic polynomial equations Equations of higher order (other) Calculus Real numbers Integer and fractional parts. Archimedean property

12–14 2.0

Let \(M\) be a finite set of numbers. It is known that among any three of its elements there are two, the sum of which belongs to \(M\).

What is the largest number of elements in \(M\)?

Problem #PRU-35575

Problems Methods Methods of calculus Continuity considerations Geometry Plane geometry Convex and non-convex figures Convex polygons Calculus Functions of one variable. Continuity Intermediate value theorem. Connectedness.

14–16 5.0

A convex figure and point \(A\) inside it are given. Prove that there is a chord (that is, a segment joining two boundary points of a convex figure) passing through point \(A\) and dividing it in half at point \(A\).

Problem #PRU-35598

Problems Geometry Solid geometry Visual geometry in space

11–13 2.0

In a communication system consisting of 2001 subscribers, each subscriber is connected with exactly \(n\) others. Determine all the possible values of \(n\).

Problem #PRU-35632

Problems Geometry Solid geometry Visual geometry in space

11–13 2.0

A raisin bag contains 2001 raisins with a total weight of 1001 g, and no raisin weighs more than 1.002 g.

Prove that all the raisins can be divided onto two scales so that they show a difference in weight not exceeding 1 g.

Problem #PRU-35635

Problems Calculus Real numbers Integer and fractional parts. Archimedean property Methods Pigeonhole principle Pigeonhole principle (angles and lengths)

13–15 3.0

Two players play the following game. They take turns. One names two numbers that are at the ends of a line segment. The next then names two other numbers, which are at the ends of a segment nested in the previous one. The game goes on indefinitely. The first aims to have at least one rational number within the intersection of all of these segments, and the second aims to prevent such occurring. Who wins in this game?

Problem #PRU-35708

Problems Calculus Real numbers Integer and fractional parts. Archimedean property Methods Pigeonhole principle Pigeonhole principle (angles and lengths)

13–15 3.0

The positive irrational numbers \(a\) and \(b\) are such that \(1/a + 1/b = 1\). Prove that among the numbers \(\lfloor ma\rfloor , \lfloor nb\rfloor\) each natural number occurs exactly once.

Problem #PRU-35714

Problems Methods Mathematical induction Mathematical induction (other) Set theory and logic Theory of algotithms Theory of algorithms (other)

12–14 3.0

There are 8 glasses of water on the table. You are allowed to take any two of the glasses and make them have equal volumes of water (by pouring some water from one glass into the other). Prove that, by using such operations, you can eventually get all the glasses to contain equal volumes of water.

Problem #PRU-35733

Problems Algebra and arithmetic Rational functions Identical transformations Set theory and logic Theory of algotithms Theory of algorithms (other)

12–14 3.0

A broken calculator carries out only one operation “asterisk”: \(a*b = 1 - a/b\). Prove that using this calculator it is possible to carry out all four arithmetic operations (addition, subtraction, multiplication, division).

Problem #PRU-35734

Problems Geometry Solid geometry Visual geometry in space

11–13 2.0

There are 20 students in a class, and each one is friends with at least 14 others. Can you prove that there are four students in this class who are all friends?