Points $A$, $B$, $C$, and $D$ lie on a circle such that chords $\overline{AC}$ and $\overline{BD}$ intersect at a point $E$ inside the circle. Suppose that $\angle ADE =\angle CBE = 75^\circ$, $BE=4$, and $DE=8$. The value of $AB^2$ can be written in the form $a+b\sqrt{c}$ for positive integers $a$, $b$, and $c$ such that $c$ is not divisible by the square of any prime. Find $a+b+c$. [i]Proposed by Tony Kim[/i]

An isosceles trapezoid $ABCD$ ($AB = CD$) is given. A point $P$ on its circumcircle is such that segments $CP$ and $AD$ meet at point $Q$. Let $L$ be tha midpoint of$ QD$. Prove that the diagonal of the trapezoid is not greater than the sum of distances from the midpoints of the lateral sides to ana arbitrary point of line $PL$.

Let $A = (1, 0)$, $B = (0, 1)$, and $C = (0, 0)$. There are three distinct points, $P, Q, R$, such that $\{A, B, C, P\}$, $\{A, B, C, Q\}$, $\{A, B, C, R\}$ are all parallelograms (vertices unordered). Find the area of $\vartriangle PQR$.

a) Does there exist a function $f: N \to N$ such that $f(f(n))=f(n+1) - f(n)$ for all $n \in N$? b) Does there exist a function $f: N \to N$ such that $f(f(n))=f(n+2) - f(n)$ for all $n \in N$? I. Voronovich

In a cardboard box are a large number of loose socks. Some of the socks are red, the others are blue. It is stated that the total number of socks does not exceed $1993$. Furthermore, it is stated that the probability of pulling two socks from the same color when two socks are randomly drawn from the box is $1/2$. What is according to the available information, the largest number of red socks that can exist in the box?

How many ordered triples of integers $(a, b, c)$ satisfy the following system? $$ \begin{cases} ab + c &= 17 \\ a + bc &= 19 \end{cases} $$ $$ \mathrm a. ~ 2\qquad \mathrm b.~3\qquad \mathrm c. ~4 \qquad \mathrm d. ~5 \qquad \mathrm e. ~6 $$

Find all functions $f:(0,\infty) \rightarrow (0,\infty)$ such that \[f\left(x+\frac{1}{y}\right)+f\left(y+\frac{1}{z}\right) + f\left(z+\frac{1}{x}\right) = 1\] for all $x,y,z >0$ with $xyz =1$.

A straight line connects City A at $(0, 0)$ to City B, 300 meters away at $(300, 0)$. At time $t=0$, a bullet train instantaneously sets out from City A to City B while another bullet train simultaneously leaves from City B to City A going on the same train track. Both trains are traveling at a constant speed of $50$ meters/second. Also, at $t=0$, a super y stationed at $(150, 0)$ and restricted to move only on the train tracks travels towards City B. The y always travels at 60 meters/second, and any time it hits a train, it instantaneously reverses its direction and travels at the same speed. At the moment the trains collide, what is the total distance that the y will have traveled? Assume each train is a point and that the trains travel at their same respective velocities before and after collisions with the y

Let $ C_1,C_2 $ be two distinct concentric circles, and $ BA $ be a diameter of $ C_1. $ Choose the points $ M,N $ on $ C_1,C_2, $ respectively, but not on the line $ BA. $ [b]a)[/b] Show that there are unique points $ P,Q $ on $ MA,MB, $ respectively, so that $ N $ is the middle of $ PQ. $ [b]b)[/b] Prove that the value $ AP^2+BQ^2 $ does not depend on $ M,N. $ [i]Mihai Piticari, Mihail Bălună[/i]

Let $ABC$ be a triangle with circumcircle $\Omega$ and incentre $I$. A line $\ell$ intersects the lines $AI$, $BI$, and $CI$ at points $D$, $E$, and $F$, respectively, distinct from the points $A$, $B$, $C$, and $I$. The perpendicular bisectors $x$, $y$, and $z$ of the segments $AD$, $BE$, and $CF$, respectively determine a triangle $\Theta$. Show that the circumcircle of the triangle $\Theta$ is tangent to $\Omega$.

The number halfway between $ \frac {1}{8}$ and $ \displaystyle \frac {1}{10}$ is $ \textbf{(A)}\ \frac {1}{80} \qquad \textbf{(B)}\ \frac {1}{40} \qquad \textbf{(C)}\ \frac {1}{18} \qquad \textbf{(D)}\ \frac {1}{9} \qquad \textbf{(E)}\ \frac {9}{80}$

Let $ x$ and $ y$ be two-digit integers such that $ y$ is obtained by reversing the digits of $ x$. The integers $ x$ and $ y$ satisfy $ x^2 \minus{} y^2 \equal{} m^2$ for some positive integer $ m$. What is $ x \plus{} y \plus{} m$? $ \textbf{(A)}\ 88\qquad \textbf{(B)}\ 112\qquad \textbf{(C)}\ 116\qquad \textbf{(D)}\ 144\qquad \textbf{(E)}\ 154$

$ S$ is a non-empty subset of the set $ \{ 1, 2, \cdots, 108 \}$, satisfying: (1) For any two numbers $ a,b \in S$ ( may not distinct), there exists $ c \in S$, such that $ \gcd(a,c)\equal{}\gcd(b,c)\equal{}1$. (2) For any two numbers $ a,b \in S$ ( may not distinct), there exists $ c' \in S$, $ c' \neq a$, $ c' \neq b$, such that $ \gcd(a, c') > 1$, $ \gcd(b,c') >1$. Find the largest possible value of $ |S|$.

Is it possible to tile $2003 \times 2003$ board by $1 \times 2$ dominoes placed horizontally and $1 \times 3$ rectangles placed vertically?

Let $P$ be a point on the circumcircle of a triangle $A_{1}A_{2}A_{3}$, and let $H$ be the orthocenter of the triangle. The feet $B_{1},B_{2},B_{3}$ of the perpendiculars from $P$ to $A_{2}A_{3},A_{3}A_{1},A_{1}A_{2}$ lie on a line. Prove that this line bisects the segment $PH$.

A [i]synogon [/i] is a convex $2n$-gon with all sides of the same length and all opposite sides are parallel. Show that every synogon can be broken down into a finite number of rhombuses.

Denmark has played an international football match against Georgia. the fight ended $5-5$, and between the first and the last goal the game has justnever stood . No country has scored three goals in a row, and Denmark scored the sixth goal. Can you use this information to determine which country scored the fifth goal?

Consider a regular heptagon ( polygon of $7$ equal sides and angles) $ABCDEFG$ as in the figure below:- $(a).$ Prove $\frac{1}{\sin\frac{\pi}{7}}=\frac{1}{\sin\frac{2\pi}{7}}+\frac{1}{\sin\frac{3\pi}{7}}$ $(b).$ Using $(a)$ or otherwise, show that $\frac{1}{AG}=\frac{1}{AF}+\frac{1}{AE}$ [asy] draw(dir(360/7)..dir(2*360/7),blue); draw(dir(2*360/7)..dir(3*360/7),blue); draw(dir(3*360/7)..dir(4*360/7),blue); draw(dir(4*360/7)..dir(5*360/7),blue); draw(dir(5*360/7)..dir(6*360/7),blue); draw(dir(6*360/7)..dir(7*360/7),blue); draw(dir(7*360/7)..dir(360/7),blue); draw(dir(2*360/7)..dir(4*360/7),blue); draw(dir(4*360/7)..dir(1*360/7),blue); label("$A$",dir(4*360/7),W); label("$B$",dir(5*360/7),S); label("$C$",dir(6*360/7),S); label("$D$",dir(7*360/7),E); label("$E$",dir(1*360/7),E); label("$F$",dir(2*360/7),N); label("$G$",dir(3*360/7),W); [/asy]

Let $n \ge 2$ be an integer and $p_1 < p_2 < ... < p_n$ prime numbers. Prove that there exists an integer $k$ relatively prime with $p_1p_2... p_n$ and such that $gcd (k + p_1p_2...p_i, p_1p_2...p_n) = 1$ for all $i = 1, 2,..., n - 1$. Malik Talbi

A ladder style tournament is held with $2016$ participants. The players begin seeded $1,2,\cdots 2016$. Each round, the lowest remaining seeded player plays the second lowest remaining seeded player, and the loser of the game gets eliminated from the tournament. After $2015$ rounds, one player remains who wins the tournament. If each player has probability of $\tfrac{1}{2}$ to win any game, then the probability that the winner of the tournament began with an even seed can be expressed has $\tfrac{p}{q}$ for coprime positive integers $p$ and $q$. Find the remainder when $p$ is divided by $1000$. [i]Proposed by Nathan Ramesh

How many positive integers $n$ satisfy$$\dfrac{n+1000}{70} = \lfloor \sqrt{n} \rfloor?$$(Recall that $\lfloor x\rfloor$ is the greatest integer not exceeding $x$.) $\textbf{(A) } 2 \qquad\textbf{(B) } 4 \qquad\textbf{(C) } 6 \qquad\textbf{(D) } 30 \qquad\textbf{(E) } 32$

Let $a,b,c>0$ and $abc\ge1.$ Prove that a) $\left(a+\frac{1}{a+1}\right)\left(b+\frac{1}{b+1}\right) \left(c+\frac{1}{c+1}\right)\ge\frac{27}{8}.$ b)$27(a^{3}+a^{2}+a+1)(b^{3}+b^{2}+b+1)(c^{3}+c^{2}+c+1)\ge$ $\ge 64(a^{2}+a+1)(b^{2}+b+1)(c^{2}+c+1).$ [hide="Generalization"]$n^{3}(a^{n}+\ldots+a+1)(b^{n}+\ldots+b+1)(c^{n}+\ldots+c+1)\ge$ $\ge (n+1)^{3}(a^{n-1}+\ldots+a+1)(b^{n-1}+\ldots+b+1)(c^{n-1}+\ldots+c+1),\ n\ge1.$ [/hide]

Compute the only element of the set \[\{1, 2, 3, 4, \ldots\} \cap \left\{\frac{404}{r^2-4} \;\bigg| \; r \in \mathbb{Q} \backslash \{-2, 2\}\right\}.\] [i]Proposed by Michael Tang[/i]

Let $m, n, k > 1$ be positive integers. For a set $S$ of positive integers, define $S(i,j)$ for $i<j$ to be the number of elements in $S$ strictly between $i$ and $j$. We say two sets $(X,Y)$ are a [i]fat[/i] pair if \[ X(i,j)\equiv Y(i,j) \pmod{n} \] for every $i,j \in X \cap Y$. (In particular, if $\left\lvert X \cap Y \right\rvert < 2$ then $(X,Y)$ is fat.) If there are $m$ distinct sets of $k$ positive integers such that no two form a fat pair, show that $m<n^{k-1}$. [i]Proposed by Allen Liu[/i]

Let $ABCD$ be a cyclic quadrilateral such that the ratio of its diagonals is $AC:BD=7:5.$ Let $E$ and $F$ be the intersections of lines $AB$ and $CD$ and lines $BC$ and $AD$, respectively. Let $L$ and $M$ be the midpoints of diagonals $AC$ and $BD$, respectively. Given that $EF=2020,$ the length of $LM$ can be written as $\frac{p}{q}$ where $p,q$ are relatively prime positive integers. Compute $p+q.$