In quadrilateral $ABCD$, a circle $\omega$ is inscribed. A point $K$ is chosen on diagonal $AC$. Segment $BK$ intersects $\omega$ at a unique point $X$, and segment $DK$ intersects $\omega$ at a unique point $Y$. It turns out that $XY$ is the diameter of $\omega$. Prove that it is perpendicular to $AC$. [i]Proposed by Tseren Frantsuzov[/i]

Let $S$ be a finite set of points on a plane, where no three points are collinear, and the convex hull of $S$, $\Omega$, is a $2016-$gon $A_1A_2\ldots A_{2016}$. Every point on $S$ is labelled one of the four numbers $\pm 1,\pm 2$, such that for $i=1,2,\ldots , 1008,$ the numbers labelled on points $A_i$ and $A_{i+1008}$ are the negative of each other. Draw triangles whose vertices are in $S$, such that any two triangles do not have any common interior points, and the union of these triangles is $\Omega$. Prove that there must exist a triangle, where the numbers labelled on some two of its vertices are the negative of each other.

Let $ABCD$ be a quadrilateral with sides $AB$, $BC$, $CD$, $DA$ and diagonals $AC$, $BD$. Suppose that all sides of the quadrilateral have length greater than $ 1$, and that the difference between any side and diagonal is less than 1. Prove that the following inequality holds $$(AB + BC + CD + DA + AC + BD)^2 > 2|AC^3 - BC^3| + 2|BD^3 - AD^3| - (AB + CD)^3$$

We define the following operation which will be applied to a row of bars being situated side-by-side on positions $1, 2, \ldots ,N$. Each bar situated at an odd numbered position is left as is, while each bar at an even numbered position is replaced by two bars. After that, all bars will be put side-by- side in such a way that all bars form a new row and are situated on positions $1, \ldots,M$. From an initial number $a_0 > 0$ of bars there originates a sequence $(a_n)_{n \geq 0}$, where an is the number of bars after having applied the operation $n$ times. [b](a)[/b] Prove that for no $n > 0$ can we have $a_n = 1997$. [b](b)[/b] Determine all natural numbers that can only occur as $a_0$ or $a_1$.

Given a non-degenerate triangle $ABC$, let $A_{1}, B_{1}, C_{1}$ be the point of tangency of the incircle with the sides $BC, AC, AB$. Let $Q$ and $L$ be the intersection of the segment $AA_{1}$ with the incircle and the segment $B_{1}C_{1}$ respectively. Let $M$ be the midpoint of $B_{1}C_{1}$. Let $T$ be the point of intersection of $BC$ and $B_{1}C_{1}$. Let $P$ be the foot of the perpendicular from the point $L$ on the line $AT$. Prove that the points $A_{1}, M, Q, P$ lie on a circle.

Prove that there exists a polynomial $P(x)$ with real coefficients and degree greater than 1 such that both of the following conditions are true $\cdot$ $P(a)+P(b)+P(c)\ge 2021$ holds for all nonnegative real numbers $a,b,c$ such that $a+b+c=3$ $\cdot$ There are infinitely many triples $(a,b,c)$ of nonnegative real numbers such that $a+b+c=3$ and $P(a)+P(b)+P(c)= 2021$

The perimeter of a square is equal in value to its area. Determine the length of one of its sides.

$k > 1$ is fixed. Show that for $n$ sufficiently large for every partition of $\{1,2,\dots,n\}$ into $k$ disjoint subsets we can find $a \neq b$ such that $a$ and $b$ are in the same subset and $a+1$ and $b+1$ are in the same subset. What is the smallest $n$ for which this is true?

Find all positive integers $n>1$ such that \[\tau(n)+\phi(n)=n+1\] Which in this case, $\tau(n)$ represents the amount of positive divisors of $n$, and $\phi(n)$ represents the amount of positive integers which are less than $n$ and relatively prime with $n$. [i]Raja Oktovin, Pekanbaru[/i]

Consider a function $f: \mathbb Z \to \mathbb Z$ such that for every integer $n \ge 0$, there are at most $0.001n^2$ pairs of integers $(x,y)$ for which $f(x+y) \neq f(x)+f(y)$ and $\max\{ \lvert x \rvert, \lvert y \rvert \} \le n$. Is it possible that for some integer $n \ge 0$, there are more than $n$ integers $a$ such that $f(a) \neq a \cdot f(1)$ and $\lvert a \rvert \le n$? [i]Proposed by David Yang[/i]

Prove that there exists an infinite set of points \[ \dots, \; P_{-3}, \; P_{-2},\; P_{-1},\; P_0,\; P_1,\; P_2,\; P_3,\; \dots \] in the plane with the following property: For any three distinct integers $a,b,$ and $c$, points $P_a$, $P_b$, and $P_c$ are collinear if and only if $a+b+c=2014$.

Let $f(x)$ be a function such that $f(x)+2f\left(\frac{x+2010}{x-1}\right)=4020 - x$ for all $x \ne 1$. Then the value of $f(2012)$ is (A) $2010$, (B) $2011$, (C) $2012$, (D) $2014$, (E) None of the above.

Prove that one can cut $a \times b$ rectangle, $\frac{b}{2} < a < b$, into three pieces and rearrange them into a square (without overlaps and holes).

Find the point $ p $ in the first quadrant on the line $ y = 2x $ such that the distance between $ p $ and $ p' $, the point reflected across the line $ y = x $, is equal to $ \sqrt{32} $.

Evaluate the following integrals. (1) $\int_0^{\pi} \cos mx\cos nx\ dx\ (m,\ n=1,\ 2,\ \cdots).$ (2) $\int_1^3 \left(x-\frac{1}{x}\right)(\ln x)^2dx.$

Prove the following inequality $$\dfrac{6}{2024^3} < \left(1-\dfrac{3}{4}\right)\left(1-\dfrac{3}{5}\right)\left(1-\dfrac{3}{6}\right)\left(1-\dfrac{3}{7}\right)\ldots\left(1-\dfrac{3}{2025}\right).$$

$d$ is a positive integer and $f : [0,d] \rightarrow \mathbb{R}$ is a continuous function with $f(0) = f(d)$. Show that there exists $x \in [0,d-1]$ such that $f(x) = f(x+1)$.

For natural numbers, when $P$ is divided by $D$, the quotient is $Q$ and the remainder is $R$. When $Q$ is divided by $D'$, the quotient is $Q'$ and the remainder is $R'$. Then, when $P$ is divided by $DD'$, the remainder is: $\textbf{(A)}\ R+R'D\qquad \textbf{(B)}\ R'+RD\qquad \textbf{(C)}\ RR'\qquad \textbf{(D)}\ R\qquad \textbf{(E)}\ R'$

If $ P(x),Q(x),R(x)$, and $ S(x)$ are all polynomials such that \[ P(x^5)\plus{}xQ(x^5)\plus{}x^2R(x^5)\equal{}(x^4\plus{}x^3\plus{}x^2\plus{}x\plus{}1)S(x),\] prove that $ x\minus{}1$ is a factor of $ P(x)$.

There are 10 numbers on a board. The product of any four of them is divisible by 30. Prove that at least one of the numbers on the board is divisible by 30.

Prove that the sum of two nagelians is greater than the semiperimeter of a triangle. (The nagelian is the segment between the vertex of a triangle and the tangency point of the opposite side with the correspondent excircle.)

In The ring $\mathbf R$, we have $\forall x \in \mathbf R : x^2=x$. Prove that in this ring [b]i)[/b] Every element is equals to its additive inverse. [b]ii)[/b] This ring has commutative property.

Given $\cos (\alpha + \beta) + sin (\alpha - \beta) = 0$, $\tan \beta =\frac{1}{2000}$, find $\tan \alpha$.

Let $ABCDE$ be a regular pentagon. We drew two circles around $A$ and $B$ with radius $AB$. Let $F$ mark the intersection of the two circles that is inside the pentagon. Let $G$ mark the intersection of lines $EF$ and $AD$. What is the degree measure of angle $AGE$?

Two consecutive positive integers $n$ and $n+1$ have the property that they both have $6$ divisors but a different number of distinct prime factors. Find the sum of the possible values of $n$. [i]Proposed by Harry Kim[/i]