Let $\triangle ABC$ be an acute triangle with $AB =AC$ and let $D$ be a point on the side $BC$. The circle with centre $D$ passing through $C$ intersects $\odot(ABD)$ at points $P$ and $Q$, where $Q$ is the point closer to $B$. The line $BQ$ intersects $AD$ and $AC$ at points $X$ and $Y$ respectively. Prove that quadrilateral $PDXY$ is cyclic.

Let $\mathcal K$ be a field with $2^{n}$ elements, $n \in \mathbb N^\ast$, and $f$ be the polynomial $X^{4}+X+1$. Prove that: (a) if $n$ is even, then $f$ is reducible in $\mathcal K[X]$; (b) if $n$ is odd, then $f$ is irreducible in $\mathcal K[X]$. [hide="Remark."]I saw the official solution and it wasn't that difficult, but I just couldn't solve this bloody problem.[/hide]

Given an arbitrary prime $p>2011$. Prove that there exist positive integers $a, b, c$ not all divisible by $p$ such that for all positive integers $n$ that $p\mid n^4- 2n^2+ 9$, we have $p\mid 24an^2 + 5bn + 2011c$.

Let $k,l$ be two given integers. Prove that there exist infinite many integers $m\ge k$ such that $\gcd\left(\binom{m}{k},l\right)=1$.

Let $\Omega$ be the $A$-excircle of triangle $ABC$, and suppose that $\Omega$ is tangent to lines $BC$, $CA$, and $AB$ at points $D$, $E$, and $F$, respectively. Let $M$ be the midpoint of segment $EF$. Two more points $P$ and $Q$ are on $\Omega$ such that $EP$ and $FQ$ are both parallel to $DM$. Let $BP$ meet $CQ$ at point $X$. Prove that the line $AM$ is the angle bisector of $\angle XAD$. [i]Proposed by Shuang-Yen Lee[/i]

Let $a$ and $m$ be positive integers and $p$ be an odd prime number such that $p^m\mid a-1$ and $p^{m+1}\nmid a-1$. Prove that (a) $p^{m+n}\mid a^{p^n}-1$ for all $n\in\mathbb N$, and (a) $p^{m+n+1}\nmid a^{p^n}-1$ for all $n\in\mathbb N$.

In the cells of an $8\times 8$ board, marbles are placed one by one. Initially there are no marbles on the board. A marble could be placed in a free cell neighboring (by side) with at least three cells which are still free. Find the greatest possible number of marbles that could be placed on the board according to these rules.

Estimate the value of $e^f$ , where $f = e^e$ .

Let $ W$ be a dense, open subset of the real line $ \mathbb{R}$. Show that the following two statements are equivalent: (1) Every function $ f : \mathbb{R} \rightarrow \mathbb{R}$ continuous at all points of $ \mathbb{R} \setminus W$ and nondecreasing on every open interval contained in $ W$ is nondecreasing on the whole $ \mathbb{R}$. (2) $ \mathbb{R} \setminus W$ is countable. [i]E. Gesztelyi[/i]

Prove that there exists only one pair $ (p,q) $ of odd primes satisfying the properties that $ p^2\equiv 4\pmod q $ and $ q^2\equiv 1\pmod p. $ [i]Ana Maria Acu[/i]

Let $P(x)$ be a polynomial with integer coefficients. Show that if $Q(x) = P(x) +12$ has at least six distinct integer roots, then $P(x)$ has no integer roots.

On an infinite square grid, Gru and his $2022$ minions play a game, making moves in a cyclic order with Gru first. On any move, the current player selects $2$ adjacent cells of their choice, and paints their shared border. A border cannot be painted over more than once. Gru wins if after any move there is a $2 \times 1$ or $1 \times 2$ subgrid with its border (comprising of $6$ segments) completely colored, but the $1$ segment inside it uncolored. Can he guarantee a win? [i]Evan Chang[/i] [size=50](oops)[/size]

Point $P$ is taken inside the square $ABCD$ such that $BP + DP=25$, $CP - AP = 15$ and $\angle$[b]ABP =[/b] $\angle$[b]ADP[/b]. What is the radius of the circumcircle of $ABCD$?

The grid below contains six rows with six points in each row. Points that are adjacent either horizontally or vertically are a distance two apart. Find the area of the irregularly shaped ten sided figure shown. [asy] import graph; size(5cm); pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); pen dotstyle = black; draw((-2,5)--(-3,4), linewidth(1.6)); draw((-3,4)--(-2,1), linewidth(1.6)); draw((-2,1)--(1,0), linewidth(1.6)); draw((1,0)--(2,1), linewidth(1.6)); draw((2,1)--(1,3), linewidth(1.6)); draw((1,3)--(1,4), linewidth(1.6)); draw((1,4)--(2,5), linewidth(1.6)); draw((2,5)--(0,5), linewidth(1.6)); draw((-2,5)--(-1,4), linewidth(1.6)); draw((-1,4)--(0,5), linewidth(1.6)); dot((-3,5),linewidth(6pt) + dotstyle); dot((-2,5),linewidth(6pt) + dotstyle); dot((-1,5),linewidth(6pt) + dotstyle); dot((0,5),linewidth(6pt) + dotstyle); dot((1,5),linewidth(6pt) + dotstyle); dot((2,5),linewidth(6pt) + dotstyle); dot((2,4),linewidth(6pt) + dotstyle); dot((2,3),linewidth(6pt) + dotstyle); dot((2,2),linewidth(6pt) + dotstyle); dot((2,1),linewidth(6pt) + dotstyle); dot((2,0),linewidth(6pt) + dotstyle); dot((-3,4),linewidth(6pt) + dotstyle); dot((-3,3),linewidth(6pt) + dotstyle); dot((-3,2),linewidth(6pt) + dotstyle); dot((-3,1),linewidth(6pt) + dotstyle); dot((-3,0),linewidth(6pt) + dotstyle); dot((-2,0),linewidth(6pt) + dotstyle); dot((-2,1),linewidth(6pt) + dotstyle); dot((-2,2),linewidth(6pt) + dotstyle); dot((-2,3),linewidth(6pt) + dotstyle); dot((-2,4),linewidth(6pt) + dotstyle); dot((-1,4),linewidth(6pt) + dotstyle); dot((0,4),linewidth(6pt) + dotstyle); dot((1,4),linewidth(6pt) + dotstyle); dot((1,3),linewidth(6pt) + dotstyle); dot((0,3),linewidth(6pt) + dotstyle); dot((-1,3),linewidth(6pt) + dotstyle); dot((-1,2),linewidth(6pt) + dotstyle); dot((-1,1),linewidth(6pt) + dotstyle); dot((-1,0),linewidth(6pt) + dotstyle); dot((0,0),linewidth(6pt) + dotstyle); dot((1,0),linewidth(6pt) + dotstyle); dot((1,1),linewidth(6pt) + dotstyle); dot((1,2),linewidth(6pt) + dotstyle); dot((0,2),linewidth(6pt) + dotstyle); dot((0,1),linewidth(6pt) + dotstyle); [/asy]

Suppose $ABCD$ is a cyclic quadrilateral with $BC = CD$. Let $\omega$ be the circle with center $C$ tangential to the side $BD$. Let $I$ be the centre of the incircle of triangle $ABD$. Prove that the straight line passing through $I$, which is parallel to $AB$, touches the circle $\omega$.

Let $n$ be an integer, and let $X$ be a set of $n+2$ integers each of absolute value at most $n$. Show that there exist three distinct numbers $a, b, c \in X$ such that $c=a+b$.

Find all triples $(a, b, c)$ of positive integers with $a\le b\le c$ such that\[\left(1+\dfrac1{a}\right)\left(1+\dfrac1{b}\right)\left(1+\dfrac1{c}\right)=3.\]

Let $G$ be a simple graph on the infinite vertex set $V=\{v_1, v_2, v_3,\ldots\}$. Suppose every subgraph of $G$ on a finite vertex subset is $10$-colorable, Prove that $G$ itself is $10$-colorable.

Twenty-one girls and twenty-one boys took part in a mathematical competition. It turned out that each contestant solved at most six problems, and for each pair of a girl and a boy, there was at least one problem that was solved by both the girl and the boy. Show that there is a problem that was solved by at least three girls and at least three boys.

Let $ABCD$ be a square with side length $6$. Let $E$ be a point on the perimeter of $ABCD$ such that the area of $\triangle{AEB}$ is $\frac{1}{6}$ the area of $ABCD$. Find the maximum possible value of $CE^2$. [i]Proposed by Anthony Yang[/i]

It is not possible to construct a segment of length $\pi$ using a straightedge, compass, and a given segment of length $1$. The following construction, given in $1685$ by Adam Kochansky, yields a segment whose length agrees with $\pi$ to five decimal places: Construct a circle of radius $1$ and call its center $O$. Construct a diameter $AB$ of this circle and a line $\ell$ tangent to the circle at $A$. Next, draw a circle with radius $1$ centered at $A$, and call one of the intersections with the original circle $C$. Now from C draw an arc of radius $1$ intersecting the circle around $A$ at $D$, where $D$ lies outside of the circle centered at $O$. Draw $OD$ and let $E$ be its point of intersection with $\ell$ . Construct $H$ on $AE$ such that $A$ is between $H$ and $E$, and $HE=3$. The distance between $B$ and $H$ is then close to $\pi$; calculate its exact value.

Let $ABC$ be an acute-angled triangle with $AB<AC<BC$ and $c(O,R)$ the circumscribed circle. Let $D, E$ be points in the small arcs $AC, AB$ respectively. Let $K$ be the intersection point of $BD,CE$ and $N$ the second common point of the circumscribed circles of the triangles $BKE$ and $CKD$. Prove that $A, K, N$ are collinear if and only if $K$ belongs to the symmedian of $ABC$ passing from $A$.

Let $(K,+,\cdot)$ be a field with the property $-x=x^{-1},\forall x\in K,x\ne0$. Prove that: $$(K,+,\cdot)\simeq(\mathbb Z_2,+,\cdot)$$ [i]Proposed by Ovidiu Pop[/i]

Let $n\ge 2$ be an integer. $a_1,a_2,\dotsc,a_n$ are arbitrarily chosen positive integers with $(a_1,a_2,\dotsc,a_n)=1$. Let $A=a_1+a_2+\dotsb+a_n$ and $(A,a_i)=d_i$. Let $(a_2,a_3,\dotsc,a_n)=D_1, (a_1,a_3,\dotsc,a_n)=D_2,\dotsc, (a_1,a_2,\dotsc,a_{n-1})=D_n$. Find the minimum of $\prod\limits_{i=1}^n\dfrac{A-a_i}{d_iD_i}$

Prove that the vertices of any planar graph can be colored with $3$ colors such that there is no monochromatic cycle.