Problems

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

is true.

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.

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.

Prove that \(\sqrt{\frac{a^2 + b^2}{2}} \geq \frac{a+b}{2}\).

We are given rational positive numbers \(p, q\) where \(1/p + 1/q = 1\). Prove that for positive \(a\) and \(b\), the following inequality holds: \(ab \leq \frac{a^p}{p} + \frac{b^q}{q}\).

Let \(p\) and \(q\) be positive numbers where \(1 / p + 1 / q = 1\). Prove that \[a_1b_1 + a_2b_2 + \dots + a_nb_n \leq (a_1^p + \dots a_n^p)^{1/p}(b_1^q +\dots + b_n^q)^{1/q}\] The values of the variables are considered positive.

Find the largest value of the expression \(a + b + c + d - ab - bc - cd - da\), if each of the numbers \(a\), \(b\), \(c\) and \(d\) belongs to the interval \([0, 1]\).