Problems

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.

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.

Find the largest natural number \(n\) which satisfies \(n^{200} <5^{300}\).

Of the four inequalities \(2x > 70\), \(x < 100\), \(4x > 25\) and \(x > 5\), two are true and two are false. Find the value of \(x\) if it is known that it is an integer.

Solve the inequality: \(\lfloor x\rfloor \times \{x\} < x - 1\).

Prove that \(\frac {1}{2} (x^2 + y^2) \geq xy\) for any \(x\) and \(y\).

Prove that for \(x \geq 0\) the inequality is valid: \(2x + \frac {3}{8} \ge \sqrt[4]{x}\).

On a plane, there are 1983 points and a circle of unit radius. Prove that there is a point on the circle, from which the sum of the distances to these points is no less than 1983.