Let $ABCD$ be a quadrilateral inscribed in a circle $\omega$. The lines $AB$ and $CD$ meet at $P$, the lines $AD$ and $BC$ meet at $Q$, and the diagonals $AC$ and $BD$ meet at $R$. Let $M$ be the midpoint of the segment $PQ$, and let $K$ be the common point of the segment $MR$ and the circle $\omega$. Prove that the circumcircle of the triangle $KPQ$ and $\omega$ are tangent to one another.

Triangle $ABC$ is inscribed in a circle. Chords $AM$ and $AN$ intersect side $BC$ at points $K$ and $L$ respectively. Prove that if a circle passes through all of the points $K, L, M$ and $N$, then $ABC$ is an isosceles triangle. (V Zhgun)

Given an odd integer $n \geq 3$. Let $V$ be the set of vertices of a regular $n$-gon, and $P$ be the set of all regular polygons formed by points in $V$. For instance, when $n=15$, $P$ consists of $1$ regular $15$-gon, $3$ regular pentagons, and $5$ regular triangles. Initially, all points in $V$ are uncolored. Two players, $A$ and $B$, play a game where they take turns coloring an uncolored point, with player $A$ starting and coloring points red, and player $B$ coloring points blue. The game ends when all points are colored. A regular polygon in $P$ is called $\textit{good}$ if it has more red points than blue points. Find the largest positive integer $k$ such that no matter how player $B$ plays, player $A$ can ensure that there are at least $k$ $\textit{good}$ polygons.

Let $\triangle{ABC}$ be an acute angled triangle. The circle with diameter AB intersects the sides AC and BC at points E and F respectively. The tangents drawn to the circle through E and F intersect at P. Show that P lies on the altitude through the vertex C.

Triangle $ABC$ is inscribed in circle $C.$ The bisectors of the angles $A,B$ and $C$ meet the circle $C$ again at the points $A', B', C'$. Let $I$ be the incenter of $ABC,$ prove that \[\frac{IA'}{IA} + \frac{IB'}{IB}+\frac{IC'}{IC} \geq 3\]\[, IA'+IB'+IC' \geq IA+IB+IC\]

A circle of radius $2$ is centered at $A$. An equilateral triangle with side $4$ has a vertex at $A$. What is the difference between the area of the region that lies inside the circle but outside the triangle and the area of the region that lies inside the triangle but outside the circle? $ \textbf {(A) } 8-\pi \qquad \textbf {(B) } \pi + 2 \qquad \textbf {(C) } 2\pi - \frac {\sqrt{2}}{2} \qquad \textbf {(D) } 4(\pi - \sqrt{3}) \qquad \textbf {(E) } 2\pi + \frac {\sqrt{3}}{2} $

For any positive reals $a,b,c,d$ that satisfy $a^2 + b^2 + c^2 + d^2 = 4,$ show that $$\frac{a^3}{a+b} + \frac{b^3}{b+c} + \frac{c^3}{c+d} + \frac{d^3}{d+a} + 4abcd \leq 6.$$

Let $(a_n), (b_n)$, $n = 1,2,...$ be two sequences of integers defined by $a_1 = 1, b_1 = 0$ and for $n \geq 1$ $a_{n+1} = 7a_n + 12b_n + 6$ $b_{n+1} = 4a_n + 7b_n + 3$ Prove that $a_n^2$ is the difference of two consecutive cubes.

Given a convex polygon with 20 vertexes, there are many ways of traingulation it (as 18 triangles). We call the diagram of triangulation, meaning the 20 vertexes, with 37 edges(17 triangluation edges and the original 20 edges), a T-diagram. And the subset of this T-diagram with 10 edges which covers all 20 vertexes(meaning any two edges in the subset doesn't cover the same vertex) calls a "perfect matching" of this T-diagram. Among all the T-diagrams, find the maximum number of "perfect matching" of a T-diagram.

Is there a power of $2$ such that it is possible to rearrange the digits giving another power of $2$?

Partition $\{1,2,3, ... ,2024\}$ into $506$ sets $\{a_i, b_i, c_i, d_i\}$ such that $a_i<b_i<c_i<d_i$. Find the maximum of \[\sum_{i=1}^{506} (a_i-b_i-c_i+d_i)\] over all partitions. [i]Tiebreaker #2[/i]

Suppose that $\omega$ is a primitive $2007^{\text{th}}$ root of unity. Find $\left(2^{2007}-1\right)\displaystyle\sum_{j=1}^{2006}\dfrac{1}{2-\omega^j}$.

We call a table of size $n \times n$ self-describing if each cell of the table contains the total number of even numbers in its row and column other than itself. How many self-describing tables of size a) $3 \times 3$ exist? b) $4 \times 4$ exist? c) $5 \times 5$ exist? Two tables are different if they differ in at least one cell.

Let $0 < x_i < \pi$ for $i=1,2,\ldots, n$ and set $$x= \frac{ x_1 +x_2 + \ldots+ x_n }{n}.$$ Prove that $$ \prod_{i=1}^{n} \frac{ \sin x_i }{x_i } \leq \left( \frac{ \sin x}{x}\right)^{n}.$$

Larry always orders pizza with exactly two of his three favorite toppings: pepperoni, bacon, and sausage. If he has ordered a total of $600$ pizzas and has had each topping equally often, how many pizzas has he ordered with pepperoni? $\text{(A) }200\qquad\text{(B) }300\qquad\text{(C) }400\qquad\text{(D) }500\qquad\text{(E) }600$

Let $ABCD$ be any convex quadrilateral and let $P, Q, R, S$ be points on the segments $AB, BC, CD$, and $DA$, respectively. It is given that the segments $PR$ and $QS$ dissect $ABCD$ into four quadrilaterals, each of which has perpendicular diagonals. Show that the points $P, Q, R, S$ are concyclic.

Suppose that $x$, $y$, and $z$ are real numbers such that $x + y + z = 3$ and $x^2 + y^2 + z^2 = 6$. What is the largest possible value of $z$?

What is the largest $n$ such that $n! + 1$ is a square?

Let $n$, $k$ be fixed integers. On a $n \times n$ board, label each square $0$ or $1$ such that in each $2k \times 2k$ sub-square of the board, the number of $0$'s and $1$'s written are the same. What is the largest possible sum of numbers written on the $n\times n$ board?

In triangle $ABC$ we have $\angle C = 90^o$ and $AC = BC$. Furthermore $M$ is an interior pont in the triangle so that $MC = 1 , MA = 2$ and $MB =\sqrt2$. Determine $AB$

What is the value of $$2^{\left(0^{\left(1^9\right)}\right)}+\left(\left(2^0\right)^1\right)^9\,?$$ $\textbf{(A) } 0 \qquad\textbf{(B) } 1 \qquad\textbf{(C) } 2 \qquad\textbf{(D) } 3 \qquad\textbf{(E) } 4$

Players $A$ and $B$ play a game on a blackboard that initially contains 2020 copies of the number 1 . In every round, player $A$ erases two numbers $x$ and $y$ from the blackboard, and then player $B$ writes one of the numbers $x+y$ and $|x-y|$ on the blackboard. The game terminates as soon as, at the end of some round, one of the following holds: $(1)$ one of the numbers on the blackboard is larger than the sum of all other numbers; $(2)$ there are only zeros on the blackboard. Player $B$ has to pay to player $A$ an amount in reais equivalent to the quantity of numbers left on the blackboard after the game ends. Show that player $A$ can earn at least 8 reais regardless of the moves taken by $B$ Ps.: Easier version of [url = https://artofproblemsolving.com/community/c6h2625868p22698110]ISL 2020 C8[/url]

Given two primes $p$ and $q$, is $v_p(q^n+n^q)$ unbounded as $n$ varies? [i](Proposed by Ivan Chan Kai Chin)[/i]

The points $X, Y,Z$ are marked on the sides $AB, BC,AC$ of the triangle $ABC$, respectively. Points $A',B', C'$ are on the $XZ, XY, YZ$ sides of the triangle $XYZ$, respectively, so that $\frac{AB}{A'B'} = \frac{AB}{A'B'} =\frac{BC}{B'C'}= 2$ and $ABB'A',BCC'B',ACC'A'$ are trapezoids in which the sides of the triangle $ABC$ are bases. a) Determine the ratio between the area of the trapezium $ABB'A'$ and the area of the triangle $A'B'X$. b) Determine the ratio between the area of the triangle $XYZ$ and the area of the triangle $ABC$.

The sum of the following seven numbers is exactly 19: \[a_1=2.56,\qquad a_2=2.61,\qquad a_3=2.65,\qquad a_4=2.71,\] \[a_5=2.79,\qquad a_6=2.82,\qquad a_7=2.86.\] It is desired to replace each $a_i$ by an integer approximation $A_i$, $1 \le i \le 7$, so that the sum of the $A_i$'s is also 19 and so that $M$, the maximum of the "errors" $|A_i - a_i|$, is as small as possible. For this minimum $M$, what is $100M$?