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.
Let the real sequence $(x_n)$ be defined by $x_1=3$ and $x_n=\dfrac{n+2}{3n}(x_{n-1}+2)$ for every $n\ge2$. Prove that the sequence has a finite limit as $n\to+\infty$, and find that limit.
Let $(a_n)$ and $(b_n)$ be arithmetic progressions, and let $m\gt 2$ be an integer. Consider $P_k(x)=x^2+a_kx+b_k$, $k=1,2,\ldots,m$. Prove that if $P_1$ and $P_m$ have no real roots, then none of the remaining polynomials has a real root.
In the plane, let $ABCD$ be a convex quadrilateral inscribed in a circle with center $O$, with neither pair of opposite sides parallel. Let $M=AB\cap CD$ and $N=AD\cap BC$, where the intersections are of the supporting lines. Let $P,Q,S,T$ respectively be the intersections of the internal angle bisectors of the pairs $\angle MAN$ and $\angle MBN$; $\angle MBN$ and $\angle MCN$; $\angle MCN$ and $\angle MDN$; $\angle MDN$ and $\angle MAN$. Assume that $P,Q,S,T$ are pairwise distinct.
(1) Prove that $P,Q,S,T$ lie on one circle, and let $I$ be its center.
(2) Let $E=AC\cap BD$. Prove that $E,O,I$ are collinear.
Let $n$ be a positive integer. There are $n$ male students and $n$ female students standing in a row in an arbitrary order. Each student $X$ receives as many candies as there are ways to choose two students of the opposite sex to $X$, one on each side of $X$. Prove that the total number of candies received by all $2n$ students does not exceed $\dfrac13n(n^2-1)$.
A group consists of five girls, denoted by $G_1,G_2,G_3,G_4,G_5$, and twelve boys. Seventeen chairs are arranged in a row. Seat the group so that all the following conditions hold:
1. Every chair has exactly one occupant.
2. From left to right, the girls occur in the order $G_1,G_2,G_3,G_4,G_5$.
3. At least three boys sit between $G_1$ and $G_2$.
4. At least one and at most four boys sit between $G_4$ and $G_5$.
How many such arrangements are there? Two arrangements are different if some chair has different occupants in them.
Let $a,b$ be odd natural numbers such that $a$ divides $b^2+2$ and $b$ divides $a^2+2$. Prove that $a$ and $b$ are terms of the natural-number sequence $(v_n)$ defined by $v_1=v_2=1$ and $v_n=4v_{n-1}-v_{n-2}$ for every $n\ge3$.
Find all functions $f:\mathbb R\to\mathbb R$ satisfying all three conditions: (1) $f$ is surjective; (2) $f$ is increasing on $\mathbb R$; (3) $f(f(x))=f(x)+12x$ for every real $x$.