XK-TST 2022

Find all functions $f:\mathbb{R}\rightarrow\mathbb{R}$ such that for all real numbers $x$ and $y$ , $$f(x^2)+2f(xy)=xf(x+y)+yf(x).$$ Proposed by Dorlir Ahmeti, Kosovo

Let $ABC$ be a triangle and $D$ point on side $BC$ such that $AD$ is angle bisector of angle $\angle BAC$ . Let $E$ be the intersection of the side $AB$ with circle $\omega_1$ which has diameter $CD$ and let $F$ be the intersection of the side $AC$ with circle $\omega_2$ which has diameter $BD$ . Suppose that there exist points $P\in\omega_1$ and $Q\in\omega_2$ such that $E, P, Q$ and $F$ are collinear and on this order. Prove that $AD, BQ$ and $CP$ are concurrent.

Proposed by Dorlir Ahmeti, Kosovo and Noah Walsh, U.S.A.

On a board, Ana writes $a$ different integers, while Ben writes $b$ different integers. Then, Ana adds each of her numbers with with each of Ben's numbers and she obtains $c$ different integers. On the other hand, Ben substracts each of his numbers from each of Ana's numbers and he gets $d$ different integers.

For each integer $n$ , let $f(n)$ be the number of ways that $n$ may be written as sum of one number of Ana and one number of Ben.

a) Show that there exist an integer $n$ such that, $$f(n)\geq\frac{ab}{c}.$$ b) Does there exist an integer $n$ such that, $$f(n)\geq\frac{ab}{d}?$$ Proposed by Besfort Shala, Kosovo