Problems

There are $n$ integers. Prove that among them either there are several numbers whose sum is divisible by $n$ or there is one number divisible by $n$ itself.

A cherry which is a ball of radius r is dropped into a round glass whose axial section is the graph of the function $y=x^4$. At what maximum r will the ball touch the most bottom point of the bottom of the glass? (In other words, what is the maximum radius r of a circle lying in the region $y\ge x^4$ and containing the origin?).

Cut the interval $[-1,1]$ into black and white segments so that the integrals of any a) linear function; b) a square trinomial in white and black segments are equal.

$x_1$ is the real root of the equation $x^2+ax+b=0$, $x_2$ is the real root of the equation $x^2-ax-b=0$.

Prove that the equation $x^2+2ax+2b=0$ has a real root, enclosed between $x_1$ and $x_2$. ($a$ and $b$ are real numbers).

Given a square trinomial $f(x)=x^2+ax+b$. It is known that for any real $x$ there exists a real number $y$ such that $f(y)=f(x)+y$. Find the greatest possible value of $a$.

We are given a polynomial $P(x)$ and numbers $a_1$, $a_2$, $a_3$, $b_1$, $b_2$, $b_3$ such that $a_1{a}_2{a}_3\ne 0$. It turned out that $P(a_{1}x+b_1)+P(a_{2}x+b_2)=P(a_{3}x+b_3)$ for any real $x$. Prove that $P(x)$ has at least one real root.

It is known that a certain polynomial at rational points takes rational values. Prove that all its coefficients are rational.

Prove that multiplying the polynomial $(x+1{)}^{n-1}$ by any polynomial different from zero, we obtain a polynomial having at least $n$ nonzero coefficients.

Prove that there are infinitely many composite numbers among the numbers $\lfloor 2^k\sqrt{2}\rfloor$ ($k=0,1,\dots$).

It is known that $\cos {\alpha }^{\circ }=1/3$. Is $\alpha$ a rational number?