PL-OM 2011

Find all integers $n\geq 1$ such that there exists a permutation $(a_1,a_2,...,a_n)$ of $(1,2,...,n)$ such that $a_1+a_2+...+a_k$ is divisible by $k$ for $k=1,2,...,n$

The incircle of triangle $ABC$ is tangent to $BC,CA,AB$ at $D,E,F$ respectively. Consider the triangle formed by the line joining the midpoints of $AE,AF$ , the line joining the midpoints of $BF,BD$ , and the line joining the midpoints of $CD,CE$ . Prove that the circumcenter of this triangle coincides with the circumcenter of triangle $ABC$ .

Let $n\geq 3$ be an odd integer. Determine how many real solutions there are to the set of $n$ equations $$\left\{\begin{array}{cc}x_1(x_1+1)=x_2(x_2-1)x_2(x_2+1)=x_3(x_3-1) \vdots x_n(x_n+1) = x_1(x_1-1)\end{array}\right.$$

Determine all pairs of functions $f,g:\mathbb{R}\rightarrow\mathbb{R}$ such that for any $x,y\in \mathbb{R}$ , $$f(x)f(y)=g(x)g(y)+g(x)+g(y).$$

In a tetrahedron $ABCD$ , the four altitudes are concurrent at $H$ . The line $DH$ intersects the plane $ABC$ at $P$ and the circumsphere of $ABCD$ at $Q\neq D$ . Prove that $PQ=2HP$ .

Prove that it is impossible for polynomials $f_1(x),f_2(x),f_3(x),f_4(x)\in \mathbb{Q}[x]$ to satisfy $$f_1^2(x)+f_2^2(x)+f_3^2(x)+f_4^2(x) = x^2+7.$$