On the side $AB$ of the cyclic quadrilateral $ABCD$ there is a point $X$ such that diagonal $AC$ bisects the segment $DX$, and the diagonal $BD$ bisects the segment $CX$. What is the smallest possible ratio $|AB | : |CD|$ in such a quadrilateral ?

The feet of the altitudes drawn from vertices $A$ and $B$ of an acute triangle $ABC$ are $K$ and $L$, respectively. Prove that if $|BK| = |KL|$ then the triangle $ABC$ is isosceles.

Square $S_{1}$ is $1\times 1.$ For $i\ge 1,$ the lengths of the sides of square $S_{i+1}$ are half the lengths of the sides of square $S_{i},$ two adjacent sides of square $S_{i}$ are perpendicular bisectors of two adjacent sides of square $S_{i+1},$ and the other two sides of square $S_{i+1},$ are the perpendicular bisectors of two adjacent sides of square $S_{i+2}.$ The total area enclosed by at least one of $S_{1}, S_{2}, S_{3}, S_{4}, S_{5}$ can be written in the form $m/n,$ where $m$ and $n$ are relatively prime positive integers. Find $m-n.$ [asy] size(250); path p=rotate(45)*polygon(4); int i; for(i=0; i<5; i=i+1) { draw(shift(2-(1/2)^(i-1),0)*scale((1/2)^i)*p); } label("$S_1$", (0,-0.75)); label("$S_2$", (1,-0.75)); label("$S_3$", (3/2,-0.75)); label("$\cdots$", (7/4, -3/4)); label("$\cdots$", (2.25, 0));[/asy]

In decimal representation, we call an integer [i]$k$-carboxylic[/i] if and only if it can be represented as a sum of $k$ distinct integers, all of them greater than $9$, whose digits are the same. For instance, $2008$ is [i]$5$-carboxylic[/i] because $2008=1111+666+99+88+44$. Find, with an example, the smallest integer $k$ such that $8002$ is [i]$k$-carboxylic[/i].

On a chessboard (with $64$ squares) there are situated $32$ white and $32$ black pools. We say that two pools form a mixed pair when they are with different colors and they lie on the same row or column. Find the maximum and the minimum of the mixed pairs for all possible situations of the pools. [i]K. Dochev[/i]

Let $ABC$ and $PQR$ be two triangles such that [list] [b](a)[/b] $P$ is the mid-point of $BC$ and $A$ is the midpoint of $QR$. [b](b)[/b] $QR$ bisects $\angle BAC$ and $BC$ bisects $\angle QPR$ [/list] Prove that $AB+AC=PQ+PR$.

A triangle $ABC$ is inscribed in circle $(O)$. $P1, P2$ are two points in the plane of the triangle. $P_1A, P_1B, P_1C$ meet $(O)$ again at $A_1,B_1,C_1$ . $P_2A, P_2B, P_2C$ meet $(O)$ again at $A_2,B_2,C_2$. a) $A_1A_2, B_1B_2, C_1C_2$ intersect $BC,CA,AB$ at $A_3,B_3,C_3$. Prove that three points $A_3,B_3,C_3$ are collinear. b) $P$ is a point on the line $P_1P_2. A_1P,B_1P,C_1P$ meet (O) again at $A_4,B_4,C_4$. Prove that three lines $A_2A_4,B_2B_4,C_2C_4$ are concurrent. Trần Quang Hùng

On a table there is a pile with $ T$ tokens which incrementally shall be converted into piles with three tokens each. Each step is constituted of selecting one pile removing one of its tokens. And then the remaining pile is separated into two piles. Is there a sequence of steps that can accomplish this process? a.) $ T \equal{} 1000$ (Cono Sur) b.) $ T \equal{} 2001$ (BWM)

Non-empty set $M$, that consists of positive integer numbers, has the following property: if for some(not necessarily distinct) positive integers $a_1,\ldots,a_{2024}$ the number $a_1\ldots a_{2024}$ is in $M$, then the number $a_1+a_2+\ldots+a_{2024}$ is also in $M$ Prove that all positive integer numbers, starting from $2049$, are in the $M$ [i]M. Zorka[/i]

Given two positive integers $ x$ and $ y$ with $ x < y$. The percent that $ x$ is less than $ y$ is: $ \textbf{(A)}\ \frac {100(y \minus{} x)}{x} \qquad\textbf{(B)}\ \frac {100(x \minus{} y)}{x} \qquad\textbf{(C)}\ \frac {100(y \minus{} x)}{y} \qquad\textbf{(D)}\ 100(y \minus{} x)$ $ \textbf{(E)}\ 100(x \minus{} y)$

Compute the unique ordered pair $(x, y)$ of real numbers satisfying the system of equations $$\frac{x}{\sqrt{x^2 + y^2}}-\frac{1}{x}= 7 \,\,\, \text{and} \,\,\, \frac{y}{\sqrt{x^2 + y^2}}+\frac{1}{y}=4 $$

Let $ABCDE$ be a convex pentagon. Let $P$ be the intersection of the lines $CE$ and $BD$. Assume that $\angle PAD = \angle ACB$ and $\angle CAP = \angle EDA$. Prove that the circumcentres of the triangles $ABC$ and $ADE$ are collinear with $P$.

Shirley went to the store planning to buy $120$ balloons for $10$ dollars. When she arrived, she was surprised to nd that the balloons were on sale for $20$ percent less than expected. How many balloons could Shirley buy for her $10$ dollars?

Let $S=\{1,2,\cdots ,n\}$ where $n$ is an odd integer. Let $f$ be a function defined on $\{(i,j): i\in S, j \in S\}$ taking values in $S$ such that (i) $f(s,r)=f(r,s)$ for all $r,s \in S$ (ii) $\{f(r,s): s\in S\}=S$ for all $r\in S$ Show that $\{f(r,r): r\in S\}=S$

The number of points in the set $\{(x,y)|\lg(x^3+\frac{1}{3}y^3+\frac{1}{9})=\lg x+\lg y)\}$ is $\text{(A)}0\qquad\text{(B)}1\qquad\text{(C)}2\qquad\text{(D)}$more than $2$

Compute the number of ordered pairs $(a,b)$ of positive integers satisfying $a^b=2^{100}$. [i]Proposed by Nathan Xiong[/i]

Show that there exists a polynomial $P(x)$ with integer coefficients, such that the equation $P(x) = 0$ has no integer solutions, but for each positive integer $n$ there is an $x \in \Bbb{Z}$ such that $n \mid P(x).$

Given a cyclic quadrilateral $ABCD$ so that $AB = AD$ and $AB + BC <CD$. Prove that the angle $ABC$ is more than $120$ degrees.

Find the nondecreasing functions $ f:[0,1]\rightarrow\mathbb{R} $ that satisfy $$ \left| \int_0^1 f(x)e^{nx} dx\right|\le 2008 , $$ for any nonnegative integer $ n. $ [i]Mihai Piticari[/i]

An acute triangle $ABC$ has $AB$ as one of its longest sides. The incircle of $ABC$ has center $I$ and radius $r$. Line $CI$ meets the circumcircle of $ABC$ at $D$. Let $E$ be a point on the minor arc $BC$ of the circumcircle of $ABC$ with $\angle ABE > \angle BAD$ and $E\notin \{B,C\}$. Line $AB$ meets $DE$ at $F$ and line $AD$ meets $BE$ at $G$. Let $P$ be a point inside triangle $AGE$ with $\angle APE=\angle AFE$ and $P\neq F$. Let $X$ be a point on side $AE$ with $XP\parallel EG$ and let $S$ be a point on side $EG$ with $PS\parallel AE$. Suppose $XS$ and $GP$ meet on the circumcircle of $AGE$. Determine the possible positions of $E$ as well as the minimum value of $\frac{BE}{r}$.

Let $a_1,a_2,..., a_n$ be real numbers such that $a_1 + a_2 + ... + a_n = 0$ and $|a_1| + |a_2 | + ... + |a_n | = 1$. Prove that $$ |a_1 + 2a_2 + ... + na_n | \le \frac{n-1}{2} $$

Amélia and Beatriz play battleship on a $2n\times2n$ board, using very peculiar rules. Amélia begins by choosing $n$ lines and $n$ columns of the board, placing her $n^2$ submarines on the cells that lie on their intersections. Next, Beatriz chooses a set of cells that will explode. Which is the least number of cells that Beatriz has to choose in order to assure that at least a submarine will explode?

Given that $n> 2$ is a positive integer and a sequence of positive integers $a_1 <a_2 <...<a_n$. In the subsets of the set $\{1,2,..., n\} $, there a subset $X$ such that $| \sum_{i \notin X} a_i -\sum_{i \in X} a_i |$ is the smallest . Prove that there exists a sequence of positive integers $0<b_1 <b_2 <...<b_n$ such that $\sum_{i \notin X} b_i= \sum_{i \in X} b_i$. In case this doesn't make sense, have a look at [url=https://drive.google.com/file/d/1xoBhJlG0xHwn6zAAA7AZDoaAqzZue-73/view]original wording in Vietnamese[/url].

Let $ \left( c_n \right)_{n\ge 1} $ be a sequence of real numbers. Prove that the sequences $ \left( c_n\sin n \right)_{n\ge 1} ,\left( c_n\cos n \right)_{n\ge 1} $ are both convergent if and only if $ \left( c_n \right)_{n\ge 1} $ converges to $ 0. $ [i]Mihai Piticari[/i] and [i]Vladimir Cerbu[/i]

The center of the circumcircle of the acute triangle $ABC$ is $M$, and the circumcircle of $ABM$ meets $BC$ and $AC$ at $P$ and $Q$ ($P\ne B$). Show that the extension of the line segment $CM$ is perpendicular to $PQ$.