Let $AXYZB$ be a convex pentagon inscribed in a semicircle of diameter $AB$.
Denote by $P$, $Q$, $R$, $S$ the feet of the perpendiculars
from $Y$ onto lines $AX$, $BX$, $AZ$, $BZ$, respectively.
Prove that the acute angle formed by lines $PQ$ and $RS$ is half the size of $\angle XOZ$,
where $O$ is the midpoint of segment $AB$.
Proposed by Titu Andreescu (credit in the supplied third-party notes).
There are $n$ students standing in a circle, one behind the other.
The students have heights $h_1\lt h_2\lt \ldots \lt h_n$.
If a student with height $h_k$ is standing directly
behind a student with height $h_{k-2}$ or less,
the two students are permitted to switch places.
Prove that it is not possible to make more than
$\binom{n}{3}$ such switches
before reaching a position in which no
further switches are possible.
Proposed by David Speyer (credit in the supplied third-party notes).
The $2010$ positive real numbers $a_1,a_2,\ldots,a_{2010}$
satisfy the inequality $a_i a_j \le i+j$
for all $1 \le i \lt j \le 2010$.
Determine, with proof, the largest possible value
of the product $a_1a_2\ldots a_{2010}$.
Proposed by Gabriel Carroll (credit in the supplied third-party notes).
Let $ABC$ be a triangle with $\angle A = 90^\circ$.
Points $D$ and $E$ lie on sides $AC$ and $AB$, respectively,
such that $\angle ABD = \angle DBC$ and $\angle ACE = \angle ECB$.
Segments $BD$ and $CE$ meet at $I$.
Determine whether or not it is possible for segments $AB$, $AC$, $BI$, $ID$, $CI$, $IE$
to all have integer lengths.
Proposed by Zuming Feng (credit in the supplied third-party notes).
Let $q = \frac{3p-5}{2}$ where $p$ is an odd prime, and let
$$ S_q = \frac{1}{2\cdot 3 \cdot 4} + \frac{1}{5\cdot 6 \cdot 7}
+ \ldots + \frac{1}{q(q+1)(q+2)}. $$
Prove that if $\frac{1}{p}-2S_q = \frac{m}{n}$ for integers $m$ and $n$,
then $m - n$ is divisible by $p$.
Proposed by Titu Andreescu (credit in the supplied third-party notes).
There are $68$ ordered pairs (not necessarily distinct)
of nonzero integers on a blackboard.
It's known that for no integer $k$ does both $(k,k)$ and $(-k,-k)$ appear.
A student erases some of the $136$ integers such that
no two erased integers have sum zero, and scores one point
for each ordered pair with at least one erased integer.
What is the maximum possible score the student can guarantee?
Proposed by Gerhard Woeginger (credit in the supplied third-party notes).