Given a cyclic quadrilateral $ABCD$ with circumcenter $O$, let the circle $(AOD)$ intersect the segments $AB$, $AC$, $DB$, $DC$ at $P$, $Q$, $R$, $S$ respectively. Suppose $X$ is the reflection of $D$ about $PQ$ and $Y$ is the reflection of $A$ about $RS$. Prove that the circles $(AOD)$, $(BPX)$, $(CSY)$ meet at a common point. [i]Proposed by Leia Mayssa & Ivan Chan Kai Chin[/i]

Find all $l,m,n \in\mathbb{N}$ that satisfies the equation $5^l43^m+1=n^3$

Given four points $O,\ A,\ B,\ C$ on a plane such that $OA=4,\ OB=3,\ OC=2,\ \overrightarrow{OB}\cdot \overrightarrow{OC}=3.$ Find the maximum area of $\triangle{ABC}$.

A magician has one hundred cards numbered 1 to 100. He puts them into three boxes, a red one, a white one and a blue one, so that each box contains at least one card. A member of the audience draws two cards from two different boxes and announces the sum of numbers on those cards. Given this information, the magician locates the box from which no card has been drawn. How many ways are there to put the cards in the three boxes so that the trick works?

We consider a prism which has the upper and inferior basis the pentagons: $A_{1}A_{2}A_{3}A_{4}A_{5}$ and $B_{1}B_{2}B_{3}B_{4}B_{5}$. Each of the sides of the two pentagons and the segments $A_{i}B_{j}$ with $i,j=1,\ldots,5$ is colored in red or blue. In every triangle which has all sides colored there exists one red side and one blue side. Prove that all the 10 sides of the two basis are colored in the same color.

Prove that there exist distinct positive integers $a_1, a_2,\ldots , a_{2024}$ such that for each $i\in\{1,2,\ldots,2024\}$\[a_i\mid a_1a_2\cdots a_{i-1}a_{i+1}\cdots a_{2024}+k,\]where a) $k=1$ and b) $k=2024.$ [i]Proposed by Ishan Nath[/i]

Call an integer $k$ [i]debatable[/i] if the number of odd factors of $k$ is a power of two. What is the largest positive integer $n$ such that there exists $n$ consecutive debatable numbers? (Here, a power of two is defined to be any number of the form $2^m$, where $m$ is a nonnegative integer.) [i]Proposed by Lewis Chen[/i]

Let $M$ be a subset of the set of $2021$ integers $\{1, 2, 3, ..., 2021\}$ such that for any three elements (not necessarily distinct) $a, b, c$ of $M$ we have $|a + b - c | > 10$. Determine the largest possible number of elements of $M$.

What is $3^{2002}$ in $\bmod 11$? $ \textbf{a)}\ 1 \qquad\textbf{b)}\ 3 \qquad\textbf{c)}\ 4 \qquad\textbf{d)}\ 5 \qquad\textbf{e)}\ \text{None of above} $

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.

A positive integer $n$ is called $\textit{good}$ if $2 \mid \tau(n)$ and if its divisors are $$1=d_1<d_2<\ldots<d_{2k-1}<d_{2k}=n, $$ then $d_{k+1}-d_k=2$ and $d_{k+2}-d_{k-1}=65$. Find the smallest $\textit{good}$ number.

The numbers $1, 2, \dots , 50$ are written on a board. Letícia performs the following actions: she erases two numbers $a$ and $b$ on the board, writes the number $a+b$ on it and notes the number $ab(a+b)$ in her notebook. After performing these operations $49$ times, when there is only one number written on the board, Letícia calculates the sum $S$ of the $49$ numbers in the notebook. a) Prove that $S$ doesn't depend on the order Letícia chooses the numbers to perform the operations. b) Find the value of $S$.

Given is a parallelogram $ABCD$, with $AB <AC <BC$. Points $E$ and $F$ are selected on the circumcircle $\omega$ of $ABC$ so that the tangenst to $\omega$ at these points pass through point $D$ and the segments $AD$ and $CE$ intersect. It turned out that $\angle ABF = \angle DCE$. Find the angle $\angle{ABC}$. A. Yakubov, S. Berlov

Find all integers $n$, $n>1$, with the following property: for all $k$, $0\le k < n$, there exists a multiple of $n$ whose digits sum leaves a remainder of $k$ when divided by $n$.

Let $A$ be a non-empty subset of the set of all positive integers $N^*$. If any sufficient big positive integer can be expressed as the sum of $2$ elements in $A$(The two integers do not have to be different), then we call that $A$ is a divalent radical. For $x \geq 1$, let $A(x)$ be the set of all elements in $A$ that do not exceed $x$, prove that there exist a divalent radical $A$ and a constant number $C$ so that for every $x \geq 1$, there is always $\left| A(x) \right| \leq C \sqrt{x}$.

Let $ a,b,c$ be positive numbers, satisfying $ abc\geq 1$. Prove that \[ a^{3} \plus{} b^{3} \plus{} c^{3} \geq ab \plus{} bc \plus{} ca.\]

Suppose $\overline{AB}$ is a segment of unit length in the plane. Let $f(X)$ and $g(X)$ be functions of the plane such that $f$ corresponds to rotation about $A$ $60^\circ$ counterclockwise and $g$ corresponds to rotation about $B$ $90^\circ$ clockwise. Let $P$ be a point with $g(f(P))=P$; what is the sum of all possible distances from $P$ to line $AB$?

We are given a triangle $ABC$ and two circles ($k_1$ and $k_2$) so the diameter of $k_1$ is $AB$ and the diameter of $k_2$ is $AC$. Let the intersection of $BC$ line segment and $k_1$ (that isn’t $B$) be $P,$ and the intersection of $BC$ line segment and $k_2$ (that isn’t $B$) be $Q$. We know, that $AB = 3003$ and $AC = 4004$ and $BC = 5005$. What is the distance between $P$ and $Q$?

A triangle $ABC$ is inscribed in a circle $G$. For any point $M$ inside the triangle, $A_1$ denotes the intersection of the ray $AM$ with $G$. Find the locus of point $M$ for which $\frac{BM\cdot CM}{MA_1}$ is minimal, and find this minimum value.

The sequence $(a_n)_{n\in\mathbb{N}}$ is defined by $a_1=3$ and $$a_n=a_1a_2\cdots a_{n-1}-1$$ Show that there exist infinitely many prime number that divide at least one number in this sequences

Each of the players in a tennis tournament played one match against each of the others. If every player won at least one match, show that there are three players A,B, and C such that A beats B, B beats C, and C beats A. Such a triple of player is called a cycle. Determine the number of maximum cycles such a tournament can have.

If $a,b$ and $c$ are three (not necessarily different) numbers chosen randomly and with replacement from the set $\{1,2,3,4,5 \}$, the probability that $ab+c$ is even is $\textbf{(A)}\ \dfrac{2}{5} \qquad \textbf{(B)}\ \dfrac{59}{125} \qquad \textbf{(C)}\ \dfrac{1}{2} \qquad \textbf{(D)}\ \dfrac{64}{125} \qquad \textbf{(E)}\ \dfrac{3}{5}$

Let $B = (-1, 0)$ and $C = (1, 0)$ be fixed points on the coordinate plane. A nonempty, bounded subset $S$ of the plane is said to be [i]nice[/i] if $\text{(i)}$ there is a point $T$ in $S$ such that for every point $Q$ in $S$, the segment $TQ$ lies entirely in $S$; and $\text{(ii)}$ for any triangle $P_1P_2P_3$, there exists a unique point $A$ in $S$ and a permutation $\sigma$ of the indices $\{1, 2, 3\}$ for which triangles $ABC$ and $P_{\sigma(1)}P_{\sigma(2)}P_{\sigma(3)}$ are similar. Prove that there exist two distinct nice subsets $S$ and $S'$ of the set $\{(x, y) : x \geq 0, y \geq 0\}$ such that if $A \in S$ and $A' \in S'$ are the unique choices of points in $\text{(ii)}$, then the product $BA \cdot BA'$ is a constant independent of the triangle $P_1P_2P_3$.

A polynomial is called Fermat polynomial if it can be written as the sum of squares of two polynomials with integer coefficients. Suppose that $f(x)$ is a Fermat polynomial such that $f(0)=1000$. Prove that $f(x)+2x$ is not a fermat polynomial

Find all functions $f: \mathbb{R}^{+} \rightarrow \mathbb{R}^{+}$, such that $f(x+f(x)+f(y))=2f(x)+y$ for all positive reals $x,y$. [i]Proposed by Athanasios Kontogeorgis, Greece[/i]