Let $A, B, C, A', B', C'$ be six pairs of different points. Prove that the Circles $BCA'$, $CAB'$ and $ABC'$ have a common point, then the Circles $B'C'A, C'A'B$ and $A'B'C$ also share a common point. Note: For three pairs of different points $X, Y$ and $Z$ we define the [i]Circle [/i] $XYZ$ as the circumcircle of the triangle $XYZ$, or - in the case when the points $X, Y$ and $Z$ lie on a straight line - this straight line.

We are given a right triangle $ABC$ and the median $BD$ drawn from the vertex $B$ of the right angle. Let the circle inscribed in $\vartriangle ABD$ be tangent to side $AD$ at $K$. Find the angles of $\vartriangle ABC$ if $K$ divides $AD$ in halves.

Six tennis players gather to play in a tournament where each pair of persons play one game, with one person declared the winner and the other person the loser. A triplet of three players $\{\mathit{A}, \mathit{B}, \mathit{C}\}$ is said to be [i]cyclic[/i] if $\mathit{A}$ wins against $\mathit{B}$, $\mathit{B}$ wins against $\mathit{C}$ and $\mathit{C}$ wins against $\mathit{A}$. [list] [*] (a) After the tournament, the six people are to be separated in two rooms such that none of the two rooms contains a cyclic triplet. Prove that this is always possible. [*] (b) Suppose there are instead seven people in the tournament. Is it always possible that the seven people can be separated in two rooms such that none of the two rooms contains a cyclic triplet?[/list]

Let $n$ be the answer to this problem. An urn contains white and black balls. There are $n$ white balls and at least two balls of each color in the urn. Two balls are randomly drawn from the urn without replacement. Find the probability, in percent, that the rst ball drawn is white and the second is black.

Suppose that a function $f:\mathbb R\to\mathbb R$ is such that for any real $h$ there exist at most $19990509$ different values of $x$ for which $f(x)\ne f(x+h)$. Prove that there is a set of at most $9995256$ real numbers such that $f$ is constant outside of this set.

Let $ABC$ be a triangle with orthocenter $H$ and circumcenter $O$. Let $K$ be the midpoint of $AH$. point $P$ lies on $AC$ such that $\angle BKP=90^{\circ}$. Prove that $OP\parallel BC$.

There are $n$ red sticks and $n$ blue sticks. The sticks of each colour have the same total length, and can be used to construct an $n$-gon. We wish to repaint one stick of each colour in the other colour so that the sticks of each colour can still be used to construct an $n$-gon. Is this always possible if (a) $n = 3$, (b) $n > 3$ ?

Let $q > 1$ be a power of $2$. Let $f: \mathbb{F}_{q^2} \to \mathbb{F}_{q^2}$ be an affine map over $\mathbb{F}_2$. Prove that the equation \[ f(x) = x^{q+1} \] has at most $2q - 1$ solutions.

Let $N=6+66+666+....+666..66$, where there are hundred $6's$ in the last term in the sum. How many times does the digit $7$ occur in the number $N$

In convex quadrilateral $KALE$, angles $\angle KAL$, $\angle AKL$, and $\angle ELK$ measure $110^\circ$, $50^\circ$, and $10^\circ$, respectively. Given that $KA = LE$ and that $\overline{KL}$ and $\overline{AE}$ intersect at point $X$, compute the value of $\tfrac{KX^2}{AL\cdot EX}$.

The faces of a cube contain the number 1, 2, 3, 4, 5, 6 such that the sum of the numbers on each pair of opposite faces is 7. For each of the cube’s eight corners, we multiply the three numbers on the faces incident to that corner, and write down its value. (In the diagram, the value of the indicated corner is 1 x 2 x 3 = 6.) What is the sum of the eight values assigned to the cube’s corners?

Triangle $ ABC$ is isosceles with $ AC \equal{} BC$ and $ \angle{C} \equal{} 120^o$. Points $ D$ and $ E$ are chosen on segment $ AB$ so that $ |AD| \equal{} |DE| \equal{} |EB|$. Find the sizes of the angles of triangle $ CDE$.

Let $O$ be a point equally distanced from the vertices of the tetrahedron $ABCD$. If the distances from $O$ to the planes $(BCD)$, $(ACD)$, $(ABD)$ and $(ABC)$ are equal, prove that the sum of the distances from a point $M \in \textrm{int}[ABCD]$, to the four planes, is constant.

How many planes of symmetry can a triangular pyramid have?

Suppose there are $n$ points on the plane, no three of which are collinear. Draw $n-1$ non-intersecting segments (except possibly at endpoints) between pairs of points, such that it is possible to travel between any two points by travelling along the segments. Such a configuration of points and segments is called a [i]network[/i]. Given a network, we may assign labels from $1$ to $n-1$ to each segment such that each segment gets a different label. Define a [i]spin[/i] as the following operation: $\bullet$ Choose a point $v$ and rotate the labels of its adjacent segments clockwise. Formally, let $e_1,e_2,\cdots,e_k$ be the segments which contain $v$ as an endpoint, sorted in clockwise order (it does not matter which segment we choose as $e_1$). Then, the label of $e_{i+1}$ is replaced with the label of $e_{i}$ simultaneously for all $1 \le i \le k$. (where $e_{k+1}=e_{1}$) A network is [i]nontrivial[/i] if there exists at least $2$ points with at least $2$ adjacent segments each. A network is [i]versatile[/i] if any labeling of its segments can be obtained from any initial labeling using a finite amount of spins. Find all integers $n \ge 5$ such that any nontrivial network with $n$ points is versatile. [i]Proposed by Yeoh Zi Song[/i]

Show that there is no function $f : \mathbb N \to \mathbb N$ satisfying $f(f(n)) = n + 1$ for each positive integer $n.$

If the expression $ \begin{pmatrix}a & c \\ d & b \end{pmatrix}$ has the value $ ab\minus{}cd$ for all values of $a, b, c$ and $d$, then the equation $ \begin{pmatrix}2x & 1 \\ x & x \end{pmatrix} = 3$: $\textbf{(A)}\ \text{Is satisfied for only 1 value of }x \qquad\\ \textbf{(B)}\ \text{Is satisified for only 2 values of }x \qquad\\ \textbf{(C)}\ \text{Is satisified for no values of }x \qquad\\ \textbf{(D)}\ \text{Is satisfied for an infinite number of values of }x \qquad\\ \textbf{(E)}\ \text{None of these.}$

SQUARING OFF: Master Chief and Samus Aran take turns firing rockets at one another from across the Cartesian plane. Master Chief’s movement is restricted to lattice points within the $10\times 10$ square with vertices $(0,0)$, $(0,10)$, $(10,0)$, and $(10,10)$, while Samus Aran’s movement is restricted to lattice points inside the $10\times 10$ square with vertices $(0,0)$, $(-10,0)$, $(0,-10)$, and $(-10,-10)$. Neither player can be located on or beyond the border of his or her square. Both players randomly choose a lattice point at which they begin the game, and do not move the rest of the game (until either they are killed or kill the other player). Each player’s turn consists of firing a rocket, targeted at a specific undestroyed lattice point inside the border of the opponent’s movement square, which hits immediately. When a rocket hits its intended lattice point, it explodes, destroying the surrounding $3\times 3$ square ($8$ additional adjacent lattice points). The game ends when one player is hit by a rocket (when the player is located within the $3\times 3$ grid hit by a rocket). If the highest possible probability that Samus Aran wins the game in three turns or less, assuming Master Chief goes first, is expressed as $a/b$, where $a$ and $b$ are relatively prime integers, find $a+b$.

Suppose $A\subseteq \mathbb R^m$ is closed and non-empty. Let $f:A\to A$ is a lipchitz function with constant less than 1. (ie there exist $c<1$ that $|f(x)-f(y)|<c|x-y|,\ \forall x,y \in A)$. Prove that there exists a unique point $x\in A$ such that $f(x)=x$.

A polygon has twice as many diagonals as it has sides. How many sides does it have?

Given that $\frac{1+3+5+\cdots+(2n-1)}{2+4+6+\cdots+(2n)}=\frac{2011}{2012}$, determine n.

Find the minimum value of $\int_{-1}^1 \sqrt{|t-x|}\ dt$

Quadrilateral $ABCD$ is inscribed in circle $\Gamma$. Segments $AC$ and $BD$ intersect at $E$. Circle $\gamma$ passes through $E$ and is tangent to $\Gamma$ at $A$. Suppose the circumcircle of triangle $BCE$ is tangent to $\gamma$ at $E$ and is tangent to line $CD$ at $C$. Suppose that $\Gamma$ has radius $3$ and $\gamma$ has radius $2$. Compute $BD$.

Let $\bigoplus$ and $\bigotimes$ be two binary boolean operators, i.e. functions that send $\{\text{True}, \text{False}\}\times \{\text{True}, \text{False}\}$ to $\{\text{True}, \text{False}\}$. Find the number of such pairs $(\bigoplus, \bigotimes)$ such that $\bigoplus$ and $\bigotimes$ distribute over each other, that is, for any three boolean values $a, b, c$, the following four equations hold: 1) $c \bigotimes (a \bigoplus b) = (c \bigotimes a) \bigoplus (c \bigotimes b);$ 2) $(a \bigoplus b) \bigotimes c = (a \bigotimes c) \bigoplus (b \bigotimes c);$ 3) $c \bigoplus (a \bigotimes b) = (c \bigoplus a) \bigotimes (c \bigoplus b);$ 4) $(a \bigotimes b) \bigoplus c = (a \bigoplus c) \bigotimes (b \bigoplus c).$ [i]Proposed by Yannick Yao

Given $2025$ pairwise distinct positive integer numbers \(a_1, a_2, \ldots, a_{2025}\), find the maximum possible number of equal numbers among the fractions of the form \[ \frac{a_i^2 + a_j^2}{a_i + a_j} \] [i]Proposed by Mykhailo Shtandenko[/i]