We solve problems

Problem #PRU-109527

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

11–13 3.0

What is the largest number of counters that can be put on the cells of a chessboard so that on each horizontal, vertical and diagonal (not only on the main ones) there is an even number of counters?

Problem #PRU-109599

Problems Calculus Number sequences Boundedness, monotonicity Algebra Number theory. Divisibility Divisibility of a number. General properties Methods Examples and counterexamples. Constructive proofs Polynomials Algebraic identities for polynomials Identical transformations Sequences Sequences

14–16 5.0

Prove that for any natural number \(a_1> 1\) there exists an increasing sequence of natural numbers \(a_1, a_2, a_3, \dots\), for which \(a_1^2+ a_2^2 +\dots+ a_k^2\) is divisible by \(a_1+ a_2+\dots+ a_k\) for all \(k \geq 1\).

Problem #PRU-109603

Problems Algebra Number theory. Divisibility Divisibility of a number. General properties Methods Examples and counterexamples. Constructive proofs Calculus Number sequences Number sequences (other)

14–16 5.0

Is there a sequence of natural numbers in which every natural number occurs exactly once, and for any \(k = 1, 2, 3, \dots\) the sum of the first \(k\) terms of the sequence is divisible by \(k\)?

Problem #PRU-109685

Problems Algebra Algebraic inequalities and systems of inequalities Linear inequalites and systems of inequalities Methods Proof by contradiction Calculus Real numbers Rational and irrational numbers

14–16 4.0

At all rational points of the real line, integers are arranged. Prove that there is a segment such that the sum of the numbers at its ends does not exceed twice the number on its middle.

Problem #PRU-109704

Problems Algebra Algebraic inequalities and systems of inequalities Algebraic inequalities (other) Methods Algebraic methods Counting in two ways Calculus Real numbers Integer and fractional parts. Archimedean property

13–15 4.0

Prove that for any positive integer \(n\) the inequality

is true.

Problem #PRU-109715

Problems Algebra Sequences Progressions Geometric progression Calculus Real numbers Integer and fractional parts. Archimedean property Sums of number sequences and difference series

13–15 4.0

Find the sum \(1/3 + 2/3 + 2^2/3 + 2^3/3 + \dots + 2^{1000}/3\).

Problem #PRU-109766

Problems Algebra Word Problems Tables and tournaments Number tables and its properties Methods Pigeonhole principle Pigeonhole principle (other)

12–14 4.0

Is it possible to arrange natural numbers from 1 to \(2002^2\) in the cells of a \(2002\times2002\) table so that for each cell of this table one could choose a triplet of numbers, from a row or column, where one of the numbers is equal to the product of the other two?

Problem #PRU-109787

Problems Algebra Polynomials Algebraic identities for polynomials Factoring polynomials Calculus Real numbers Rational and irrational numbers

14–16 4.0

A numeric set \(M\) containing 2003 distinct numbers is such that for every two distinct elements \(a, b\) in \(M\), the number \(a^2+ b\sqrt 2\) is rational. Prove that for any \(a\) in \(M\) the number \(q\sqrt 2\) is rational.

Problem #PRU-109946

Problems Algebra Algebraic equations and systems of equations Algebraic equations and systems of equations Calculus Real numbers Integer and fractional parts. Archimedean property Polynomials Quadratic polynomials Investigating the quadratic polynomial

14–15 3.0

Solve the equation \((x + 1)^3 = x^3\).

Problem #PRU-110035

Problems Calculus Functions of one variable. Continuity Certain properties of a function and recurrence relations. Algebra Absolute value Inequalities containing absolute values Trigonometry Trigonometrical inequalities

15–16 4.0

Does there exist a function \(f (x)\) defined for all \(x \in \mathbb{R}\) and for all \(x, y \in \mathbb{R}\) satisfying the inequality \(| f (x + y) + \sin x + \sin y | < 2\)?