SILK-ROAD 2018

In an acute-angled triangle $ABC$ on the sides $AB$ , $BC$ , $AC$ the points $H$ , $L$ , $K$ so that $CH \perp AB$ , $HL \parallel AC$ , $HK \parallel BC$ . Let $P$ and $Q$ feet of altitudes of a triangle $HBL$ , drawn from the vertices $H$ and $B$ respectively. Prove that the feet of the altitudes of the triangle $AKH$ , drawn from the vertices $A$ and $H$ lie on the line $PQ$ .

Find all functions $f:\ \mathbb{R}\rightarrow\mathbb{R}$ such that for any real number $x$ the equalities are true: $f\left(x+1\right)=1+f(x)$ and $f\left(x^4-x^2\right)=f^4(x)-f^2(x).$ source

Given the natural $n$ . We shall call word sequence from $n$ letters of the alphabet, and distance $\rho(A, B)$ between words $A=a_1a_2\dots a_n$ and $B=b_1b_2\dots b_n$ , the number of digits in which they differ (that is, the number of such $i$ , for which $a_i\ne b_i$ ). We will say that the word $C$ lies between words $A$ and $B$ , if $\rho (A,B)=\rho(A,C)+\rho(C,B)$ . What is the largest number of words you can choose so that among any three, there is a word lying between the other two?

Does there exist a sequence of positive integers $a_1,a_2,...$ such that every positive integer occurs exactly once and that the number $\tau (na_{n+1}^n+(n+1)a_n^{n+1})$ is divisible by $n$ for all positive integer.

Here $\tau (n)$ denotes the number of positive divisor of $n$ .