In the diagram below, the circle with center $A$ is congruent to and tangent to the circle with center $B$. A third circle is tangent to the circle with center $A$ at point $C$ and passes through point $B$. Points $C$, $A$, and $B$ are collinear. The line segment $\overline{CDEFG}$ intersects the circles at the indicated points. Suppose that $DE = 6$ and $FG = 9$. Find $AG$. [asy] unitsize(5); pair A = (-9 sqrt(3), 0); pair B = (9 sqrt(3), 0); pair C = (-18 sqrt(3), 0); pair D = (-4 sqrt(3) / 3, 10 sqrt(6) / 3); pair E = (2 sqrt(3), 4 sqrt(6)); pair F = (7 sqrt(3), 5 sqrt(6)); pair G = (12 sqrt(3), 6 sqrt(6)); real r = 9sqrt(3); draw(circle(A, r)); draw(circle(B, r)); draw(circle((B + C) / 2, 3r / 2)); draw(C -- D); draw("$6$", E -- D); draw(E -- F); draw("$9$", F -- G); dot(A); dot(B); label("$A$", A, plain.E); label("$B$", B, plain.E); label("$C$", C, W); label("$D$", D, dir(160)); label("$E$", E, S); label("$F$", F, SSW); label("$G$", G, N); [/asy]

Let $ABCD$ be a cyclic quadrilateral. Suppose that $E$ is the intersection of $AB$ and $CD, F$ is the intersection of $AD$ and $CB, I$ is the intersection of $AC$ and $BD$. The circumcircles $(FAB), (FCD)$ meet $FI$ at $K, L$. Prove that $EK = EL$ Nguyễn Minh Hà

Suppose there are 18 lighthouses on the Persian Gulf. Each of the lighthouses lightens an angle with size 20 degrees. Prove that we can choose the directions of the lighthouses such that whole of the blue Persian (always Persian) Gulf is lightened.

For a group of 5 girls, denoted as $G_1,G_2,G_3,G_4,G_5$ and $12$ boys. There are $17$ chairs arranged in a row. The students have been grouped to sit in the seats such that the following conditions are simultaneously met: (a) Each chair has a proper seat. (b) The order, from left to right, of the girls seating is $G_1; G_2; G_3; G_4; G_5.$ (c) Between $G_1$ and $G_2$ there are at least three boys. (d) Between $G_4$ and $G_5$ there are at least one boy and most four boys. How many such arrangements are possible?

There are $n$ students sitting on a round table. You collect all of $ n $ name tags and give them back arbitrarily. Each student gets one of $n$ name tags. Now $n$ students repeat following operation: The students who have their own name tags exit the table. The other students give their name tags to the student who is sitting right to him. Find the number of ways giving name tags such that there exist a student who don't exit the table after 4 operations.

Georg has bought lots of filled chocolates for a party, and when he counts how many he has, he discovers that the number is a prime number. He distributes so many of the chocolates as possible on $60$ trays with an equal number on each. He notes then that he has more than one piece left and that the number left pieces is not a prime number. How many pieces of chocolate does Georg have left?

Determine all triples $(x,y,z)$ of positive rational numbers with $x\le y\le z$ such that $x+y+z,\frac1x+\frac1y+\frac1z$, and xyz are natural numbers.

There are ten cards with the number $a$ on each, ten with $b$ and ten with $c$, where $a, b$ and $c$ are distinct real numbers. For every five cards, it is possible to add another five cards so that the sum of the numbers on these ten cards is $0$. Prove that one of $a, b$ and $c$ is $0$.

For an arbitrary positive integer $ n$, define $ p(n)$ as the product of the digits of $ n$ (in decimal). Find all positive integers $ n$ such that $ 11p(n)\equal{}n^2\minus{}2005$.

Jacob and Aaron are playing a game in which Aaron is trying to guess the outcome of an unfair coin which shows heads $\tfrac{2}{3}$ of the time. Aaron randomly guesses ``heads'' $\tfrac{2}{3}$ of the time, and guesses ``tails'' the other $\tfrac{1}{3}$ of the time. If the probability that Aaron guesses correctly is $p$, compute $9000p$. [i]Proposed by Aaron Lin[/i]

On a table there are $1999$ counters, red on one side and black on the other side, arranged arbitrarily. Two people alternately make moves, where each move is of one of the following two types: (1) Remove several counters which all have the same color up; (2) Reverse several counters which all have the same color up. The player who takes the last counter wins. Decide which of the two players (the one playing first or the other one) has a wining strategy.

Let $ABC$ be an acutangle triangle inscribed in a circle $\Gamma$ of center $O$. Let $D$ be the height of the vertex $A$. Let E and F be points over $\Gamma$ such that $AE = AD = AF$. Let $P$ and $Q$ be the intersection points of the $EF $ with sides $AB$ and $AC$ respectively. Let $X$ be the second intersection point of $\Gamma$ with the circle circumscribed to the triangle $AP Q$. Show that the lines $XD$ and $AO $ meet at a point above sobre

The diagram below shows a rectangle with side lengths $36$ and $48$. Each of the sides is trisected and edges are added between the trisection points as shown. Then the shaded corner regions are removed, leaving the octagon which is not shaded in the diagram. Find the perimeter of this octagon. [asy] size(4cm); dotfactor=3.5; pair A,B,C,D,E,F,G,H,W,X,Y,Z; A=(0,12); B=(0,24); C=(16,36); D=(32,36); E=(48,24); F=(48,12); G=(32,0); H=(16,0); W=origin; X=(0,36); Y=(48,36); Z=(48,0); filldraw(W--A--H--cycle^^B--X--C--cycle^^D--Y--E--cycle^^F--Z--G--cycle,rgb(.76,.76,.76)); draw(W--X--Y--Z--cycle,linewidth(1.2)); dot(A); dot(B); dot(C); dot(D); dot(E); dot(F); dot(G); dot(H); [/asy]

For $f(x) = \cos\left(\frac{3\sqrt{3}\pi}{8}(x-x^3 ) \right)$, find the value of $$\lim_{t\to\infty} \left( \int_0^1 f(x)^t dx \right)^\frac{1}{t} + \lim_{t\to-\infty} \left( \int_0^1 f(x)^t dx \right)^\frac{1}{t} $$

Let $ABCD$ be a circumscribed quadrilateral with center $O$. Assume the incenters of $\vartriangle AOC, \vartriangle BOD$ are $I_1, I_2$, respectively. If circumcircles of $\vartriangle AI_1C$ and $\vartriangle BI_2D$ intersect at $X$, prove the following identity: $(AB \cdot CX \cdot DX)^2 + (CD\cdot AX \cdot BX)^2 = (AD\cdot BX \cdot CX)^2 + (BC \cdot AX \cdot DX)^2$

If in a convex quadrilateral $ ABCD, E$ and $ F$ are the midpoints of the sides $ BC$ and $ DA$ respectively. Show that the sum of the areas of the triangles $ EDA$ and $ FBC$ is equal to the area of the quadrangle.

Let $n$ be a positive integer. For each $4n$-tuple of nonnegative real numbers $a_1,\ldots,a_{2n}$, $b_1,\ldots,b_{2n}$ that satisfy $\sum_{i=1}^{2n}a_i=\sum_{j=1}^{2n}b_j=n$, define the sets \[A:=\left\{\sum_{j=1}^{2n}\frac{a_ib_j}{a_ib_j+1}:i\in\{1,\ldots,2n\} \textup{ s.t. }\sum_{j=1}^{2n}\frac{a_ib_j}{a_ib_j+1}\neq 0\right\},\] \[B:=\left\{\sum_{i=1}^{2n}\frac{a_ib_j}{a_ib_j+1}:j\in\{1,\ldots,2n\} \textup{ s.t. }\sum_{i=1}^{2n}\frac{a_ib_j}{a_ib_j+1}\neq 0\right\}.\] Let $m$ be the minimum element of $A\cup B$. Determine the maximum value of $m$ among those derived from all such $4n$-tuples $a_1,\ldots,a_{2n},b_1,\ldots,b_{2n}$. [I]Proposed by usjl.[/i]

Given a circle of radius $R$. Find the ratio of the largest area of ​​the circumscribed quadrilateral to the smallest area of ​​the inscribed one.

Before district play, the Unicorns had won $45\%$ of their basketball games. During district play, they won six more games and lost two, to finish the season having won half their games. How many games did the Unicorns play in all? $\textbf{(A)}\ 48 \qquad \textbf{(B)}\ 50 \qquad \textbf{(C)}\ 52 \qquad \textbf{(D)}\ 54 \qquad \textbf{(E)}\ 60$

For a positive integer $n$, let $f_n(x)=\cos (x) \cos (2 x) \cos (3 x) \cdots \cos (n x)$. Find the smallest $n$ such that $\left|f_n^{\prime \prime}(0)\right|>2023$.

What are the possible areas of a hexagon with all angles equal and sides $1, 2, 3, 4, 5$, and $6$, in some order?

Let $I$, $I_{a}$ be the incenter and the $A$-excenter of a triangle $ABC$; $E$, $F$ be the touching points of the incircle with $AC$, $AB$ respectively; $G$ be the common point of $BE$ and $CF$. The perpendicular to $BC$ from $G$ meets $AI$ at point $J$. Prove that $E$, $F$, $J$, $I_{a}$ are concyclic. Proposed by:Y.Shcherbatov

Given positive numbers $a_1$ and $b_1$, consider the sequences defined by \[a_{n+1}=a_n+\frac{1}{b_n},\quad b_{n+1}=b_n+\frac{1}{a_n}\quad (n \ge 1)\] Prove that $a_{25}+b_{25} \geq 10\sqrt{2}$.

A positive integer is called ascending if, in its decimal representation, there are at least two digits and each digit is less than any digit to its right. How many ascending positive integers are there?

Prove that \[\tan 7 30^{\prime }=\sqrt{6}+\sqrt{2}-\sqrt{3}-2.\]