Problems

To each pair of numbers \(x\) and \(y\) some number \(x * y\) is placed in correspondence. Find \(1993 * 1935\) if it is known that for any three numbers \(x, y, z\), the following identities hold: \(x * x = 0\) and \(x * (y * z) = (x * y) + z\).

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.

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

is true.

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

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.

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 \(| f (x + y) + \sin x + \sin y | < 2\)?

The real numbers \(x\) and \(y\) are such that for any distinct prime odd \(p\) and \(q\) the number \(x^p + y^q\) is rational. Prove that \(x\) and \(y\) are rational numbers.

The functions \(f (x) - x\) and \(f (x^2) - x^6\) are defined for all positive \(x\) and increase. Prove that the function

also increases for all positive \(x\).

Prove that if the numbers \(x, y, z\) satisfy the following system of equations for some values of \(p\) and \(q\): \[\begin{aligned} y &= x^2 + px + q,\\ z &= y^2 + py + q,\\ x &= z^2 + pz + q, \end{aligned}\] then the inequality \(x^2y + y^2z + z^2x \geq x^2z + y^2x + z^2y\) is satisfied.

Solve the equation \(xy = x + y\) in integers.