Let $P(n)$ be the number of permutations $(a_1,\ldots,a_n)$ of $1,2,\ldots,n$ such that $ka_k$ is a perfect square for every $1\le k\le n$. Find, with proof, the smallest $n$ for which $2010\mid P(n)$.
Proposed by Andy Niedermier (credit in the supplied notes).
Let $n\gt 1$ be an integer. Find all positive integer sequences $x_1,\ldots,x_{n-1}$ such that (a) $x_1\lt \cdots\lt x_{n-1}$; (b) $x_i+x_{n-i}=2n$ for $1\le i\le n-1$; (c) whenever $x_i+x_j\lt 2n$ (indices may coincide), the sum is one of the terms of the sequence.
Proposed by Răzvan Gelca (credit in the supplied notes).
Let $AXYZB$ be a convex pentagon inscribed in a semicircle of diameter $AB$. Let $P,Q,R,S$ be the feet from $Y$ to the lines $AX,BX,AZ,BZ$, respectively. Prove that the acute angle between $PQ$ and $RS$ is half of $\angle XOZ$, where $O$ is the midpoint of $AB$.
Proposed by Titu Andreescu (credit in the supplied notes).
A triangle is called parabolic if its vertices lie on $y=x^2$. Prove that for every nonnegative integer $n$ there exist an odd integer $m$ and a parabolic triangle with three distinct integer-coordinate vertices whose area is $(2^n m)^2$.
Proposed by Zuming Feng (credit in the supplied notes).
Two permutations $(a_1,\ldots,a_{2010})$ and $(b_1,\ldots,b_{2010})$ of $1,\ldots,2010$ intersect if $a_k=b_k$ for some $k$. Prove that there exist 1006 permutations such that every other permutation intersects at least one of them.
Proposed by Gregory Galperin (credit in the supplied notes).
In triangle $ABC$, $\angle A=90^\circ$. Points $D\in AC$ and $E\in AB$ satisfy $\angle ABD=\angle DBC$ and $\angle ACE=\angle ECB$. Let $I=BD\cap CE$. Can all six lengths $AB,AC,BI,ID,CI,IE$ be integers? Prove your answer.
Proposed by Zuming Feng (credit in the supplied notes).