A triangular pyramid $ABCD$ is given. A sphere $\omega_A$ is tangent to the face $BCD$ and to the planes of other faces in points don't lying on faces. Similarly, sphere $\omega_B$ is tangent to the face $ACD$ and to the planes of other faces in points don't lying on faces. Let $K$ be the point where $\omega_A$ is tangent to $ACD$, and let $L$ be the point where $\omega_B$ is tangent to $BCD$. The points $X$ and $Y$ are chosen on the prolongations of $AK$ and $BL$ over $K$ and $L$ such that $\angle CKD = \angle CXD + \angle CBD$ and $\angle CLD = \angle CYD +\angle CAD$. Prove that the distances from the points $X$, $Y$ to the midpoint of $CD$ are the same. [hide=thanks ]Thanks to the user Vlados021 for translating the problem.[/hide]

Consider a triangular grid of equilateral triangles with unit sides. Assume that $\mathcal{P}$ is a non-self-intersecting polygon with perimeter 1399 and sides from the grid. Prove that $\mathcal{P}$ has either an internal or an external 120-degree angle. [i]Proposed by Seyed Hessam Firouzi[/i]

Does there exist a function $f\colon \mathbb{R}\to\mathbb{R}$ such that for any $x>y,$ it satisfies $f(x)-f(y)>\sqrt{x-y}.$

Let $ABC$ be a triangle with all angles $\leq 120^{\circ}$. Let $F$ be the Fermat point of triangle $ABC$, that is, the interior point of $ABC$ such that $\angle AFB = \angle BFC = \angle CFA = 120^\circ$. For each one of the three triangles $BFC$, $CFA$ and $AFB$, draw its Euler line - that is, the line connecting its circumcenter and its centroid. Prove that these three Euler lines pass through one common point. [i]Remark.[/i] The Fermat point $F$ is also known as the [b]first Fermat point[/b] or the [b]first Toricelli point[/b] of triangle $ABC$. [i]Floor van Lamoen[/i]

The equation $$(x-1)(x-2)\cdots(x-2016)=(x-1)(x-2)\cdots (x-2016)$$ is written on the board, with $2016$ linear factors on each side. What is the least possible value of $k$ for which it is possible to erase exactly $k$ of these $4032$ linear factors so that at least one factor remains on each side and the resulting equation has no real solutions?

Let $M$ be the point of intersection of the diagonals $AC$ and $BD$ of the convex quadrilateral $ABCD$. Let$ K$ be the intersection point of the extension of side $AB$ (from $A$) with the bisector of the $\angle ACD$. If $MA \cdot MC + MA \cdot CD = MB\cdot MD$ , prove that $\angle BKC = \angle CDB$.

A semicircle $(O)$ is drawn with the center $O$, where $O$ lies on a line $\ell$. $C$ and $D$ lie on the circle $(O)$, and the tangent lines of $(O)$ at points $C$ and $D$ intersects the line $\ell$ at points $B$ and $A$, respectively, such that $O$ lies between points $B$ and $A$. Let $E$ be the intersection point between $AC$ and $BD$, and $F$ the point on $\ell$ so that $EF $ is perpendicular to line $\ell$. Prove that $EF$ bisects the angle $\angle CFD$.

Find the number of ordered pairs $(x,y)$, where $x$ and $y$ are both integers between $1$ and $9$, inclusive, such that the product $x\times y$ ends in the digit $5$. [i]Proposed by Andrew Wen[/i]

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.