Problems

Prove that for all \(x\), \(0 < x < \pi /3\), we have the inequality \(\sin 2x + \cos x > 1\).

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

is true.

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

Prove that for all \(x \in (0;\pi /2)\) for \(n > m\), where \(n, m\) are natural, we have the inequality \(2 | \sin^n x-\cos^n x | \leq 3 | \sin^m x-\cos^m x |\);

The polynomial \(P (x)\) of degree \(n\) has \(n\) distinct real roots.

What is the largest number of its coefficients that can be equal to zero?

Members of the State parliament formed factions in such a way that for any two factions \(A\) and \(B\) (not necessarily different)

– also a faction (through

the set of all parliament members not included in \(C\) is denoted). Prove that for any two factions \(A\) and \(B\), \(A \cup % \includegraphics{https://problems-static.s3.eu-west-2.amazonaws.com/static/test/task_images/82/109909-3.png} B\) is also a faction.

We call a number \(x\) rational if it can be represented as \(x=\frac{p}{q}\) for coprime integers \(p\) and \(q\). Otherwise we call the number irrational.
Non-zero numbers \(a\) and \(b\) satisfy the equality \(a^2b^2 (a^2b^2 + 4) = 2(a^6 + b^6)\). Prove that at least one of them is irrational.

The sum of the positive numbers \(a, b, c\) is \(\pi / 2\). Prove that \(\cos a + \cos b + \cos c > \sin a + \sin b + \sin c\).

Prove that for each \(x\) such that \(\sin x \neq 0\), there is a positive integer \(n\) such that \(|\sin nx| \geq \sqrt{3}/2\).

For what natural numbers \(n\) are there positive rational but not whole numbers \(a\) and \(b\), such that both \(a + b\) and \(a^n + b^n\) are integers?