RU-239-OPEN 2023

There are $n{}$ wise men in a hall and everyone sees each other. Each man will wear a black or white hat. The wise men should simultaneously write down on their piece of paper a guess about the color of their hat. If at least one does not guess, they will all be executed.

The wise men can discuss a strategy before the test and they know that the layout of the hats will be chosen randomly from the set of all $2^n$ layouts. They want to choose their strategy so that the number of layouts for which everyone guesses correctly is as high as possible. What is this number equal to?

The excircles of triangle $ABC$ touch its sides $BC$ , $CA$ , and $AB$ at points $A_1$ , $B_1$ , and $C_1$ , respectively. Let $B_2$ and $C_2$ be the midpoints of segments $BB_1$ and $CC_1$ , respectively. Line $B_2C_2$ intersects line $BC$ at point $W$ . Prove that $AW = A_1W$ .

Let $n>1$ be a natural number and $x_k{}$ be the residue of $n^2$ modulo $\lfloor n^2/k\rfloor+1$ for all natural $k{}$ . Compute the sum $$\bigg\lfloor\frac{x_2}{1}\bigg\rfloor+\bigg\lfloor\frac{x_3}{2}\bigg\rfloor+\cdots+\left\lfloor\frac{x_n}{n-1}\right\rfloor.$$