Given an acute-angled triangle $ABC$ with orthocenter $H$. Reflection of nine-point circle about $AH$ intersects circumcircle at points $X$ and $Y$. Prove that $AH$ is the external bisector of $\angle XHY$. [i]Proposed by Mohammad Javad Shabani[/i]

Show that if $a, b, c$ are the lengths of the sides of a triangle and if $2S = a + b + c$, then \[\frac{a^n}{b+c} + \frac{b^n}{c+a} +\frac{c^n}{a+b} \geq \left(\dfrac 23 \right)^{n-2}S^{n-1} \quad \forall n \in \mathbb N \] [i]Proposed by Greece.[/i]

What algebraic relationship holds between $ \alpha $, $ \beta $ and $ \gamma $ when the equality is satisfied $$ \tan \alpha + \tan \beta + \tan \gamma = \tan \alpha \tan \beta \tan \gamma?$$

Find all positive integers $n$ for which number $A = n^3-n^2+n-1$ is prime

In tetrahedron $ABCD$, $AB=4$, $CD=7$, and $AC=AD=BC=BD=5$. Let $I_A$, $I_B$, $I_C$, and $I_D$ denote the incenters of the faces opposite vertices $A$, $B$, $C$, and $D$, respectively. It is provable that $AI_A$ intersects $BI_B$ at a point $X$, and $CI_C$ intersects $DI_D$ at a point $Y$. Compute $XY$.

If $ a$, $ b$, and $ c$ are in geometric progression (G.P.) with $ 1 < a < b < c$ and $ n > 1$ is an integer, then $ \log_an$, $ \log_b n$, $ \log_c n$ form a sequence $ \textbf{(A)}\ \text{which is a G.P} \qquad$ $ \textbf{(B)}\ \text{whichi is an arithmetic progression (A.P)} \qquad$ $ \textbf{(C)}\ \text{in which the reciprocals of the terms form an A.P} \qquad$ $ \textbf{(D)}\ \text{in which the second and third terms are the }n\text{th powers of the first and second respectively} \qquad$ $ \textbf{(E)}\ \text{none of these}$

Define the numbers $a_0, a_1, \ldots, a_n$ in the following way: \[ a_0 = \frac{1}{2}, \quad a_{k+1} = a_k + \frac{a^2_k}{n} \quad (n > 1, k = 0,1, \ldots, n-1). \] Prove that \[ 1 - \frac{1}{n} < a_n < 1.\]

Compute the number of ordered pairs of integers $(a, b)$, where $0 \le a < 17$ and $0 \le b < 17$, such that $y^2 \equiv x^3 +ax +b \pmod{17}$ has an even number of solutions $(x, y)$, where $0 \le x < 17$ and $0 \le y < 17$ are integers.

Show that $19^{93} - 13^{99}$ is a positive integer divisible by $162$.

Is it possible to use $2 \times 1$ dominoes to cover a $2k \times 2k$ checkerboard which has $2$ squares, one of each colour, removed ?

Find all positive integers $m$ such that there exists integers $x$ and $y$ that satisfies $$m \mid x^2+11y^2+2022.$$

If $x,y,z$ are real numbers such that $xyz=1$, evaluate $$\frac{x+1}{xy+x+1}+\frac{y+1}{yz+y+1}+\frac{z+1}{zx+z+1}.$$

We have $2^m$ sheets of paper, with the number $1$ written on each of them. We perform the following operation. In every step we choose two distinct sheets; if the numbers on the two sheets are $a$ and $b$, then we erase these numbers and write the number $a + b$ on both sheets. Prove that after $m2^{m -1}$ steps, the sum of the numbers on all the sheets is at least $4^m$ . [i]Proposed by Abbas Mehrabian, Iran[/i]

Given a convex polyhedron and a sphere intersecting each its edge at two points so that each edge is trisected (divided into three equal parts). Is it necessarily true that all faces of the polyhedron are (a) congruent polygons? (b) regular polygons?

Let $r,s,t$ be the roots of $x^3+6x^2+7x+8$. Find $$(r^2+s+t)(s^2+t+r)(t^2+r+s).$$ [i]Proposed by Evan Chang (squareman), USA[/i]

Peter has many squares of equal side. Some of the squares are black, some are white. Peter wants to assemble a big square, with side equal to $n$ sides of the small squares, so that the big square has no rectangle formed by the small squares such that all the squares in the vertices of the rectangle are of equal colour. How big a square is Peter able to assemble?

Consider a checkered $3m\times 3m$ square, where $m$ is an integer greater than $1.$ A frog sits on the lower left corner cell $S$ and wants to get to the upper right corner cell $F.$ The frog can hop from any cell to either the next cell to the right or the next cell upwards. Some cells can be [i]sticky[/i], and the frog gets trapped once it hops on such a cell. A set $X$ of cells is called [i]blocking[/i] if the frog cannot reach $F$ from $S$ when all the cells of $X$ are sticky. A blocking set is [i] minimal[/i] if it does not contain a smaller blocking set.[list=a][*]Prove that there exists a minimal blocking set containing at least $3m^2-3m$ cells. [*]Prove that every minimal blocking set containing at most $3m^2$ cells.

Let $ABCDA'B'C'D'$ be a cube (where $ABCD$ is a square and $AA'\parallel BB'\parallel CC'\parallel DD'$). Furthermore, let $\mathcal R$ be a rotation (with respect some line) that maps vertex $A$ to $B.$ Find the set of all images $X=\mathcal R(C)$ such that $X$ lies on the surface of the cube for some rotation $\mathcal R(A)=B.$

Let $ a, b \in \mathbb{N}$ with $ 1 \leq a \leq b,$ and $ M \equal{} \left[\frac {a \plus{} b}{2} \right].$ Define a function $ f: \mathbb{Z} \mapsto \mathbb{Z}$ by \[ f(n) \equal{} \begin{cases} n \plus{} a, & \text{if } n \leq M, \\ n \minus{} b, & \text{if } n >M. \end{cases} \] Let $ f^1(n) \equal{} f(n),$ $ f_{i \plus{} 1}(n) \equal{} f(f^i(n)),$ $ i \equal{} 1, 2, \ldots$ Find the smallest natural number $ k$ such that $ f^k(0) \equal{} 0.$

Let $ [a]$ be the integer such that $ [a]\le a<[a]\plus{}1$. Find all real numbers $ (a,b,c)$ such that \[ \{a\}\plus{}[b]\plus{}\{c\}\equal{}2.9\\\{b\}\plus{}[c]\plus{}\{a\}\equal{}5.3\\\{c\}\plus{}[a]\plus{}\{b\}\equal{}4.0.\]

A positive integer $n$ is [i]stacked[/i] if $2n$ has the same number of digits as $n$ and the digits of $2n$ are multiples of the corresponding digits of $n.$ For example, $1203$ is stacked because $2 \times 1203 = 2406,$ and $2, 4, 0, 6$ are multiples of $1, 2, 0, 3,$ respectively. Compute the number of stacked integers less than $1000.$

In how many ways a cube can be painted using seven different colors in such a way that no two faces are in same color? $ \textbf{(A)}\ 154 \qquad\textbf{(B)}\ 203 \qquad\textbf{(C)}\ 210 \qquad\textbf{(D)}\ 240 \qquad\textbf{(E)}\ \text{None of the above} $

Find all positive integers $n$ such that $4^n+6^n+9^n$ is a square. [i]David Yang, Alex Zhu.[/i]

Let $S$ be a set of integers which has the following properties: 1) There exists $x,y\in S$ such that $(x,y)=(x-2,y-2)=1$; 2) For $\forall$ $x,y\in S, x^2-y\in S$. Prove that $S\equiv \mathbb{Z}$ .

$ABCDE$ is a regular pentagon (with vertices in that order) inscribed in a circle of radius $1$. Find $AB \cdot AC$.