Consider a circle with diameter $AB$ and let $C,D$ be points on its circumcircle such that $C,D$ are not in the same side of $AB$.Consider the parallel line to $AC$ passing from $D$ and let it intersect $AB$ at $E$.Similarly consider the paralell line to $AD$ passing from $C$ and let it intersect $AB$ at $F$.The perpendicular line to $AB$ at $E$ intersects $BC$ at $X$ and the perpendicular line to $AB$ at $F$ intersects $DB$ at $Y$.Prove that the permiter of triangle $AXY$ is twice $CD$. [b]Remark:[/b]This problem is proved to be wrong due to a typo in the exam papers you can find the correct version [url=https://artofproblemsolving.com/community/c6h1832731_geometry__iran_mo_2019]here[/url].

On the plane there are two cylindrical towers with radii of bases $r$ and $R$. Find the set of all those points of the plane from which these towers are visible at the same angle. Consider the case of more towers.

Consider the sequence $(a_n)_{n\geq 1}$ given by $a_1=1$ and $a_{n+1}=\frac{a_n}{1+\sqrt{1+a_n}}$, for all $n\geq 1$. Show that $$\lim_{n\rightarrow\infty}\frac{a_{n+1}}{a_n} = \lim_{n\rightarrow\infty}\sum_{k=1}^n \log_2(1+a_k)=2.$$ [i]Mathematical Gazette[/i]

NASA’s Perseverance Rover was launched on July $30,$ $2020.$ After traveling $292{,}526{,}838$ miles, it landed on Mars in Jezero Crater about $6.5$ months later. Which of the following is closest to the Rover’s average interplanetary speed in miles per hour? $\textbf{(A) } 6{,}000 \qquad \textbf{(B) } 12{,}000 \qquad \textbf{(C) } 60{,}000 \qquad \textbf{(D) } 120{,}000 \qquad \textbf{(E) } 600{,}000$

Six assassins, numbered 1-6, stand in a circle. Each assassin is randomly assigned a target such that each assassin has a different target and no assassin is their own target. In increasing numerical order, each assassin, if they are still alive, kills their target. Find the expected number of assassins still alive at the end of this process. [i]Proposed by Allen Yang[/i]

Let $\mathcal{G}$ be a simple graph with $n$ vertices and $m$ edges such that no two cycles share an edge. Prove that $2m < 3n$. [i]Note[/i]: A [i]simple graph[/i] is a graph with at most one edge between any two vertices and no edges from any vertex to itself. A [i]cycle[/i] is a sequence of distinct vertices $v_1, \dots, v_n$ such that there is an edge between any two consecutive vertices, and between $v_n$ and $v_1$. [i]Proposed by Ethan Tan[/i]

Adithya and Bill are playing a game on a connected graph with $n > 2$ vertices, two of which are labeled $A$ and $B$, so that $A$ and $B$ are distinct and non-adjacent and known to both players. Adithya starts on vertex $A$ and Bill starts on $B$. Each turn, both players move simultaneously: Bill moves to an adjacent vertex, while Adithya may either move to an adjacent vertex or stay at his current vertex. Adithya loses if he is on the same vertex as Bill, and wins if he reaches $B$ alone. Adithya cannot see where Bill is, but Bill can see where Adithya is. Given that Adithya has a winning strategy, what is the maximum possible number of edges the graph may have? (Your answer may be in terms of $n$.) [i]Proposed by Steven Liu[/i]

Prove that among $18$ consecutive three digit numbers there must be at least one which is divisible by the sum of its digits.

The numbers from $1$ to $2010$ inclusive are placed along a circle so that if we move along the circle in clockwise order, they increase and decrease alternately. Prove that the difference between some two adjacent integers is even.

Let $S$ denote the sum of the 2011th powers of the roots of the polynomial $(x-2^0)(x-2^1) \cdots (x-2^{2010}) - 1$. How many ones are in the binary expansion of $S$? [i]Author: Alex Zhu[/i]

Find all integral ordered triples $(x,y,z)$ such that $\displaystyle\sqrt{\frac{2015}{x+y}}+\sqrt{\frac{2015}{y+z}}+\sqrt{\frac{2015}{x+z}}$ are positive integers

How many diagonals can you draw in a convex $2009$-gon if in the finished drawing, every drawn diagonal inside the $2009$-gon may cut at most another drawn diagonal?

How many numbers are there that appear both in the arithmetic sequence $10, 16, 22, 28, ... 1000$ and the arithmetic sequence $10, 21, 32, 43, ..., 1000?$

If $ x < a < 0$ means that $ x$ and $ a$ are numbers such that $ x$ is less than $ a$ and $ a$ is less than zero, then: $ \textbf{(A)}\ x^2 < ax < 0 \qquad\textbf{(B)}\ x^2 > ax > a^2 \qquad\textbf{(C)}\ x^2 < a^2 < 0$ $ \textbf{(D)}\ x^2 > ax\text{ but }ax < 0 \qquad\textbf{(E)}\ x^2 > a^2\text{ but }a^2 < 0$

Given a $n \times n$ square board. Two players by turn remove some side of unit square if this side is not a bound of $n \times n$ square board. The player lose if after his move $n \times n$ square board became broken into two parts. Who has a winning strategy?

On a circle with diameter $AC$, let $B$ be an arbitrary point distinct from $A$ and $C$. Points $M, N$ are the midpoints of chords $AB, BC$, and points $P, Q$ are the midpoints of smaller arcs restricted by these chords. Lines $AQ$ and $BC$ meet at point $K$, and lines $CP$ and $AB$ meet at point $L$. Prove that lines $MQ, NP$ and $KL$ concur.

Isosceles triangle $ABC$, with $AB=AC$, is inscribed in circle $\omega$. Point $D$ lies on arc $\frown{BC}$ not containing $A$. Let $E$ be the foot of perpendicular from $A$ to line $CD$. Prove that $BC+DC=2DE$.

In a coordinate system there are drawn the graphs of the functions $y=ax+b$ and $y=bx+a, (a\neq b)$. Their intersection is marked with red and their intersections with the $Oy$ axis are marked with blue. Everything is erased except the marked points. Using only a ruler and a compass, find the origin of the coordinate system.

An ant walks around on the coordinate plane. It moves from the origin to $(3,4)$, then to $(-9, 9)$, then back to the origin. How many units did it walk? Express your answer as a decimal rounded to the nearest tenth.

Find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that for all reals $x,y,z$, we have $$(f(x)+1)(f(y)+f(z))=f(xy+z)+f(xz-y)$$

How many $7$ digit numbers are there that satisfy the following? [list] [*] All digits are distinct from $1-7.$ [*] The first digit (from the left) is divisible by $1.$ [*] The two-digit number formed by the first two digits is divisible by $2.$ [*] The three-digit number formed by the first three digits is divisible by $3.$ [*] The four-digit number formed by the first four digits is divisible by $4.$ [*] The five-digit number formed by the first five digits is divisible by $5.$ [*] The six-digit number formed by the first six digits is divisible by $6.$ [/list]

Find the base of the number system less than $100$, in which $2101$ is a perfect square.

Alice and Bob play a game, in which they take turns drawing segments of length $1$ in the Euclidean plane. Alice begins, drawing the first segment, and from then on, each segment must start at the endpoint of the previous segment. It is not permitted to draw the segment lying over the preceding one. If the new segment shares at least one point - except for its starting point - with one of the previously drawn segments, one has lost. a) Show that both Alice and Bob could force the game to end, if they don’t care who wins. b) Is there a winning strategy for one of them?

Triangle $ABC$ is an isosceles right triangle with $AB=AC=3$. Let $M$ be the midpoint of hypotenuse $\overline{BC}$. Points $I$ and $E$ lie on sides $\overline{AC}$ and $\overline{AB}$, respectively, so that $AI>AE$ and $AIME$ is a cyclic quadrilateral. Given that triangle $EMI$ has area $2$, the length $CI$ can be written as $\frac{a-\sqrt{b}}{c}$, where $a$, $b$, and $c$ are positive integers and $b$ is not divisible by the square of any prime. What is the value of $a+b+c$? $ \textbf{(A) }9 \qquad \textbf{(B) }10 \qquad \textbf{(C) }11 \qquad \textbf{(D) }12 \qquad \textbf{(E) }13 \qquad $

If $f(x)=x^n-7x^{n-1}+17x^{n-2}+a_{n-3}x^{n-3}+\ldots+a_0$ is a real-valued function of degree $n>2$ with all real roots, prove that no root has value greater than $4$ and at least one root has value less than $0$ or greater than $2$.