BxMO 2009

Find all functions $f:\mathbb{Z}_{>0}\rightarrow\mathbb{Z}_{>0}$ that satisfy the following two conditions: [list] $\bullet\ f(n)$ is a perfect square for all $n\in\mathbb{Z}_{>0}$ $\bullet\ f(m+n)=f(m)+f(n)+2mn$ for all $m,n\in\mathbb{Z}_{>0}$ .[/list]

Let $n$ be a positive integer and let $k$ be an odd positive integer. Moreover, let $a,b$ and $c$ be integers (not necessarily positive) satisfying the equations $$a^n+kb=b^n+kc=c^n+ka$$ Prove that $a=b=c$ .

Let $n\ge 1$ be an integer. In town $X$ there are $n$ girls and $n$ boys, and each girl knows each boy. In town $Y$ there are $n$ girls, $g_1,g_2,\ldots ,g_n$ , and $2n-1$ boys, $b_1,b_2,\ldots ,b_{2n-1}$ . For $i=1,2,\ldots ,n$ , girl $g_i$ knows boys $b_1,b_2,\ldots ,b_{2i-1}$ and no other boys. Let $r$ be an integer with $1\le r\le n$ . In each of the towns a party will be held where $r$ girls from that town and $r$ boys from the same town are supposed to dance with each other in $r$ dancing pairs. However, every girl only wants to dance with a boy she knows. Denote by $X(r)$ the number of ways in which we can choose $r$ dancing pairs from town $X$ , and by $Y(r)$ the number of ways in which we can choose $r$ dancing pairs from town $Y$ . Prove that $X(r)=Y(r)$ for $r=1,2,\ldots ,n$ .

Given trapezoid $ABCD$ with parallel sides $AB$ and $CD$ , let $E$ be a point on line $BC$ outside segment $BC$ , such that segment $AE$ intersects segment $CD$ . Assume that there exists a point $F$ inside segment $AD$ such that $\angle EAD=\angle CBF$ . Denote by $I$ the point of intersection of $CD$ and $EF$ , and by $J$ the point of intersection of $AB$ and $EF$ . Let $K$ be the midpoint of segment $EF$ , and assume that $K$ is different from $I$ and $J$ .

Prove that $K$ belongs to the circumcircle of $\triangle ABI$ if and only if $K$ belongs to the circumcircle of $\triangle CDJ$ .