Given a sequence $s$ consisting of digits $0$ and $1$. For any positive integer $k$, define $v_k$ the maximum number of ways in any sequence of length $k$ that several consecutive digits can be identified, forming the sequence $s$. (For example, if $s=0110$, then $v_7=v_8=2$, because in sequences $0110110$ and $01101100$ one can find consecutive digits $0110$ in two places, and three pairs of $0110$ cannot meet in a sequence of length $7$ or $8$.) It is known that $v_n<v_{n+1}<v_{n+2}$ for some positive integer $n$. Prove that in the sequence $s$, all the numbers are the same. A. Golovanov

For every positive integer $m$ prove the inquality $|\{\sqrt{m}\} - \frac{1}{2}| \geq \frac{1}{8(\sqrt m+1)} $ (The integer part $[x]$ of the number $x$ is the largest integer not exceeding $x$. The fractional part of the number $x$ is a number $\{x\}$ such that $[x]+\{x\}=x$.) A. Golovanov

In a triangle $ABC$, $M$ is the midpoint of the $AB$. A point $B_1$ is marked on $AC$ such that $CB=CB_1$. Circle $\omega$ and $\omega_1$, the circumcircles of triangles $ABC$ and $BMB_1$, respectively, intersect again at $K$. Let $Q$ be the midpoint of the arc $ACB$ on $\omega$. Let $B_1Q$ and $BC$ intersect at $E$. Prove that $KC$ bisects $B_1E$. M. Kungozhin

Integers $x,y,z,t$ satisfy $x^2+y^2=z^2+t^2$and$xy=2zt$ prove that $xyzt=0$ Proposed by $M. Abduvaliev$