Let $ABC$ be a triangle with orthocenter $H$ and $\ell$ a line that passes through $H$, and is parallel to $BC$. Let $m$ and $n$ be the reflections of $\ell$ on the sides of $AB$ and $AC$, respectively, $m$ and $n$ are intersect at $P$. If $HP$ and $BC$ intersect at $Q$, prove that the parallel to $AH$ through $Q$ and $AP$ intersect at the circumcenter of the triangle $ABC$.

Mari and Juri ordered a round pizza. Juri cut the pizza into four pieces by two straight cuts, none of which passed through the centre point of the pizza. Mari can choose two pieces not aside of these four, and Juri gets the rest two pieces. Prove that if Mari chooses the piece that covers the centre point of the pizza, she will get more pizza than Juri.

Find all natural numbers $n$ such that the equation $x^2 + y^2 + z^2 = nxyz$ has solutions in positive integers

A room has $2017$ boxes in a circle. A set of boxes is [i]friendly[/i] if there are at least two boxes in the set, and for each boxes in the set, if we go clockwise starting from the box, we would pass either $0$ or odd number of boxes before encountering a new box in the set. $30$ students enter the room and picks a set of boxes so that the set is friendly, and each students puts a letter inside all of the boxes that he/she chose. If the set of the boxes which have $30$ letters inside is not friendly, show that there exists two students $A, B$ and boxes $a, b$ satisfying the following condition. (i). $A$ chose $a$ but not $b$, and $B$ chose $b$ but not $a$. (ii). Starting from $a$ and going clockwise to $b$, the number of boxes that we pass through, not including $a$ and $b$, is not an odd number, and none of $A$ or $B$ chose such boxes that we passed.

Soda is sold in packs of 6, 12 and 24 cans. What is the minimum number of packs needed to buy exactly 90 cans of soda? $ \textbf{(A)}\ 4\qquad\textbf{(B)}\ 5\qquad\textbf{(C)}\ 6\qquad\textbf{(D)}\ 8\qquad\textbf{(E)}\ 15 $

Each of the digits $1$, $3$, $7$ and $9$ occurs at least once in the decimal representation of some positive integers. Prove that one can permute the digits of this integer such that the resulting integer is divisible by $7$.

Let $B$ be a set of $k$ sequences each having $n$ terms equal to $1$ or $-1$. The product of two such sequences $(a_1, a_2, \ldots , a_n)$ and $(b_1, b_2, \ldots , b_n)$ is defined as $(a_1b_1, a_2b_2, \ldots , a_nb_n)$. Prove that there exists a sequence $(c_1, c_2, \ldots , c_n)$ such that the intersection of $B$ and the set containing all sequences from $B$ multiplied by $(c_1, c_2, \ldots , c_n)$ contains at most $\frac{k^2}{2^n}$ sequences.

In the quadrilateral $MARE$ inscribed in a unit circle $\omega,$ $AM$ is a diameter of $\omega,$ and $E$ lies on the angle bisector of $\angle RAM.$ Given that triangles $RAM$ and $REM$ have the same area, find the area of quadrilateral $MARE.$

The sides of a convex $200$-gon $A_1 A_2 \dots A_{200}$ are colored red and blue in an alternating fashion. Suppose the extensions of the red sides determine a regular $100$-gon, as do the extensions of the blue sides. Prove that the $50$ diagonals $\overline{A_1A_{101}},\ \overline{A_3A_{103}},\ \dots, \ \overline{A_{99}A_{199}}$ are concurrent. [i]Proposed by: [b]Ankan Bhattacharya[/b][/i]

Toward the end of a game of Fish, the $2$ through $7$ of spades, inclusive, remain in the hands of three distinguishable players: DBR, RB, and DB, such that each player has at least one card. If it is known that DBR either has more than one card or has an even-numbered spade, or both, in how many ways can the players’ hands be distributed?

The area of the triangle bounded by the lines $y = x, y = -x$ and $y = 6$ is $ \mathbf{(A)}\; 12\qquad \mathbf{(B)}\; 12\sqrt2\qquad \mathbf{(C)}\; 24\qquad \mathbf{(D)}\; 24\sqrt2\qquad \mathbf{(E)}\; 36$

Two concentric circles have radii $R$ and $r$ respectively. Determine the greatest possible number of circles that are tangent to both these circles and mutually nonintersecting. Prove that this number lies between $\frac 32 \cdot \frac{\sqrt R +\sqrt r }{\sqrt R -\sqrt r } -1$ and $\frac{63}{20} \cdot \frac{R+r}{R-r}.$

Let $m$ and $n$ be positive integers greater than $1$. In each unit square of an $m\times n$ grid lies a coin with its tail side up. A [i]move[/i] consists of the following steps. [list=1] [*]select a $2\times 2$ square in the grid; [*]flip the coins in the top-left and bottom-right unit squares; [*]flip the coin in either the top-right or bottom-left unit square. [/list] Determine all pairs $(m,n)$ for which it is possible that every coin shows head-side up after a finite number of moves. [i]Thanasin Nampaisarn, Thailand[/i]

What is the largest number of cells that can be colored in a $7\times7$ table in such a way that any $2\times2$ subtable has at most 2 colored cells?

Find all prime numbers $p, q$ and $r$ such that $p>q>r$ and the numbers $p-q, p-r$ and $q-r$ are also prime.

A game has two players. In the game there is a rectangular chocolate bar, with $60$ pieces, arranged in a $6 \times 1 0$ formation , which can be broken only along the lines dividing the pieces. The first player breaks the bar along one line, discarding one section . The second player then breaks the remaining section, discarding one section. The first player repeats this process with the remaining section , and so on. The game is won by the player who leaves a single piece. In a perfect game which player wins? {S. Fomin , Leningrad)

A woman, her brother, her son and her daughter are chess players (all relations by birth). The worst player's twin (who is one of the four players) and the best player are of opposite sex. The worst player and the best player are the same age. Who is the worst player? $ \textbf{(A)}\ \text{the woman} \qquad\textbf{(B)}\ \text{her son} \qquad\textbf{(C)}\ \text{her brother} \qquad\textbf{(D)}\ \text{her daughter} \\ \qquad\textbf{(E)}\ \text{No solution is consistent with the given information} $

If $n \ge 2$ is an integer, prove the equality $$\prod_{k=1}^n \tan \frac{\pi}{3}\left(1+\frac{3^k}{3^n-1}\right)=\prod_{k=1}^n \cot \frac{\pi}{3}\left(1-\frac{3^k}{3^n-1}\right)$$

Let $ABCD$ be a cyclic quadrilateral such that $AD=BD=AC$. A point $P$ moves along the circumcircle $\omega$ of triangle $ABCD$. The lined $AP$ and $DP$ meet the lines $CD$ and $AB$ at points $E$ and $F$ respectively. The lines $BE$ and $CF$ meet point $Q$. Find the locus of $Q$.

For every positive integer $n$ write all its divisors in increasing order: $1=d_1<d_2<\ldots<d_k=n$. Find all $n$ such that $2025 \cdot n=d_{20} \cdot d_{25}$. [i]I. Voronovich[/i]

A wooden cube with edge length $ n$ units (where $ n$ is an integer $ >2$) is painted black all over. By slices parallel to its faces, the cube is cut into $ n^3$ smaller cubes each of unit length. If the number of smaller cubes with just one face painted black is equal to the number of smaller cubes completely free of paint, what is $ n$? $ \textbf{(A)}\ 5 \qquad \textbf{(B)}\ 6 \qquad \textbf{(C)}\ 7 \qquad \textbf{(D)}\ 8 \qquad \textbf{(E)}\ \text{none of these}$

Find all functions $f:\mathbb{Q}\rightarrow\mathbb{Q}$ and $g:\mathbb{Q}\rightarrow\mathbb{Q}$ such that $$f(f(x)+yg(x))=(x+1)g(y)+f(y)$$ for any $x;y\in\mathbb{Q}$

Find the smallest positive integer $k$ which is representable in the form $k=19^n-5^m$ for some positive integers $m$ and $n$.

Find the largest natural number $n$ for which $4^{995} +4^{1500} +4^n$ is a square.

Solve the equation: $$ \frac{xy}{z}+\frac{yz}{x}+\frac{zx}{y}=3$$ where $x$, $y$ and $z$ are integers