In the plane, a circle $(O)$ and two distinct fixed points $B$ and $C$ on it are given, with $BC$ not a diameter. $A$ varies on $(O)$, with $A$ different from $B$ and $C$. Let $D, K$ and $J$ be the midpoints of $BC, CA$ and $AB$, respectively. Let $E, M$ and $N$ be the perpendicular projections of $A, B$ and $C$ onto the lines $BC, DJ$ and $DK$, respectively.
Prove that the tangents at $M$ and $N$ to the circumcircle of triangle $EMN$ meet at a fixed point $T$ as $A$ varies.
A rectangular field consists of $m$ rows and $n$ columns of unit squares. Pumps are placed in some squares. Each pump waters its own square, every square sharing an edge with it, and the squares in its column separated from it by exactly one intervening square. Find the smallest number of pumps needed to water the whole field when (1) $m=4$; (2) $m=3$.
Let $p\ge17$ be a prime. Prove that $t=3$ is the largest positive integer with the following property: for arbitrary integers $a,b,c,d$ such that $p\nmid abc$ and $p\mid a+b+c$, there exist integers $x,y,z\in\{0,1,\ldots,\lfloor p/t\rfloor-1\}$ for which $p\mid ax+by+cz$.
Editorial notice: The printed statement omits the term $+d$. As written it is false, since $x=y=z=0$ also works for $t=4$. The intended corrected requirement is $p\mid ax+by+cz+d$; the model solution explicitly proves that corrected version.
The positive-integer sequence $(x_n)$ is defined by $x_1=1$, $x_2=2011$, and $x_{n+2}=4022x_{n+1}-x_n$ for every positive integer $n$. Prove that $(x_{2012}+1)/2012$ is a perfect square.
Prove that $C=10\sqrt{24}$ is the largest constant with the following property: if 17 positive real numbers $a_1,\ldots,a_{17}$ satisfy
$$\sum_{i=1}^{17}a_i^2=24,\qquad \sum_{i=1}^{17}a_i^3+\sum_{i=1}^{17}a_i\lt C,$$
then $a_i,a_j,a_k$ are the side lengths of a triangle for every $1\le i\lt j\lt k\le17$.
There are 42 students at an International Mathematical Olympiad team selection examination. Each student knows exactly 20 other students. Prove that the students can be divided into either 2 groups or 21 groups of equal size so that any two students in the same group know each other. Acquaintance is understood to be mutual.