Circle $\omega$ lies in the interior of circle $\Omega$, on which a point $X$ moves. The tangents from $X$ to $\omega$ intersect $\Omega$ for the second time at points $A\neq X$ and $B\neq X$. Prove that the lines $AB$ are either all tangent to a fixed circle, or they all pass through a point.

Find the smallest positive integer $G$ such that there exist distinct positive integers $a, b, c$ with the following properties: $\: \bullet \: \gcd(a, b, c) = G$. $\: \bullet \: \text{lcm}(a, b) = \text{lcm}(a, c) = \text{lcm}(b, c)$. $\: \bullet \: \frac{1}{a} + \frac{1}{b}, \frac{1}{a} + \frac{1}{c},$ and $\frac{1}{b} + \frac{1}{c}$ are reciprocals of integers. $\: \bullet \: \gcd(a, b) + \gcd(a, c) + \gcd(b, c) = 16G$.

Let $\vartriangle ABC$ be a triangle with $\angle BAC = 45^o, \angle BCA = 30^o$, and $AB = 1$. Point $D$ lies on segment $AC$ such that $AB = BD$. Find the square of the length of the common external tangent to the circumcircles of triangles $\vartriangle BDC$ and $\vartriangle ABC$.

Find all monotonically increasing or monotonically decreasing functions $f: \mathbb{R}_+\to\mathbb{R}_+$ which satisfy the equation $f\left(xy\right)\cdot f\left(\frac{f\left(y\right)}{x}\right)=1$ for any two numbers $x$ and $y$ from $\mathbb{R}_+$. Hereby, $\mathbb{R}_+$ is the set of all positive real numbers. [i]Note.[/i] A function $f: \mathbb{R}_+\to\mathbb{R}_+$ is called [i]monotonically increasing[/i] if for any two positive numbers $x$ and $y$ such that $x\geq y$, we have $f\left(x\right)\geq f\left(y\right)$. A function $f: \mathbb{R}_+\to\mathbb{R}_+$ is called [i]monotonically decreasing[/i] if for any two positive numbers $x$ and $y$ such that $x\geq y$, we have $f\left(x\right)\leq f\left(y\right)$.

Let $ S$ be the set of all real numbers strictly greater than −1. Find all functions $ f: S \to S$ satisfying the two conditions: (a) $ f(x \plus{} f(y) \plus{} xf(y)) \equal{} y \plus{} f(x) \plus{} yf(x)$ for all $ x, y$ in $ S$; (b) $ \frac {f(x)}{x}$ is strictly increasing on each of the two intervals $ \minus{} 1 < x < 0$ and $ 0 < x$.

Let $ABC$ be an equilateral triangle. Let $P$ be a point inside of it such that the square root of the distance of $P$ to one of the sides is equal to the sum of the square roots of the distances of $P$ to the other two sides. Find the geometric place of $P$.

Let $ABCD$ be a square, such that the length of its sides are integers. This square is divided in $89$ smaller squares, $88$ squares that have sides with length $1$, and $1$ square that has sides with length $n$, where $n$ is an integer larger than $1$. Find all possible lengths for the sides of $ABCD$.

Evaluate $\int_0^{\frac 12} (x+1)\sqrt{1-2x^2}\ dx$. [i]2011 Kyoto University entrance exam/Science, Problem 1B[/i]

Andrew and Blair are bored in class and decide to play a game. They pick a pair $(a, b)$ with $1 \le a, b \le 100$. Andrew says the next number in the geometric series that begins with $a,b$ and Blair says the next number in the arithmetic series that begins with $a,b$. For how many pairs $(a, b)$ is Andrew's number minus Blair's number a positive perfect square?

There are 8 coins, 7 of which are real, which weigh the same, and one is fake, which differs in weight from the rest. Cup scales without weights mean that if you put equal weights on their cups, then any of the cups can outweigh, but if the loads are different in mass, then the cup with a heavier load is definitely overpowered. How to definitely identify a counterfeit coin in four weighings and establish is it lighter or heavier than the others?

Consider the equation \[ x^{2001}=y^x .\] [list] [*] Find all pairs $(x,y)$ of solutions where $x$ is a prime number and $y$ is a positive integer. [*] Find all pairs $(x,y)$ of solutions where $x$ and $y$ are positive integers.[/list] (Remember that $2001=3 \cdot 23 \cdot 29$.)

We say a triple of real numbers $ (a_1,a_2,a_3)$ is [b]better[/b] than another triple $ (b_1,b_2,b_3)$ when exactly two out of the three following inequalities hold: $ a_1 > b_1$, $ a_2 > b_2$, $ a_3 > b_3$. We call a triple of real numbers [b]special[/b] when they are nonnegative and their sum is $ 1$. For which natural numbers $ n$ does there exist a collection $ S$ of special triples, with $ |S| \equal{} n$, such that any special triple is bettered by at least one element of $ S$?

A pentagon has four interior angles each equal to $110^o$. Find the degree measure of the fifth interior angle.

What is the smallest positive integer $n$ such that $\sqrt{n}-\sqrt{n-1}<.01$? $\textbf{(A) }2499\qquad\textbf{(B) }2500\qquad\textbf{(C) }2501\qquad\textbf{(D) }10,000\qquad \textbf{(E) }\text{There is no such integer}$

In triangle $ABC$, angle $A$ is equal to $a$ and the altitude drawn to side $BC$ is equal to $h$. The inscribed circle of the triangle touches the sides of the triangle at points $K$, $M$ and $P$, where $P$ lies on side $BC$. Find the distance from $P$ to $KM$.

A [i]chaisri[/i] figure is a triangle which the three vertices are vertices of a regular $2019$-gon. Two different chaisri figure may be formed by different regular $2019$-gon. A [i]thubkaew[/i] figure is a convex polygon which can be dissected into multiple chaisri figure where each vertex of a dissected chaisri figure does not necessarily lie on the border of the convex polygon. Determine the maximum number of vertices that a thubkaew figure may have.

Let $ABC$ be an acute triangle with orthocenter $H$. Let $G$ be the point such that the quadrilateral $ABGH$ is a parallelogram. Let $I$ be the point on the line $GH$ such that $AC$ bisects $HI$. Suppose that the line $AC$ intersects the circumcircle of the triangle $GCI$ at $C$ and $J$. Prove that $IJ = AH$.

Consider on the coordinate plane all rectangles whose (i) vertices have integer coordinates; (ii) edges are parallel to coordinate axes; (iii) area is $2^k$, where $k = 0,1,2....$ Is it possible to color all points with integer coordinates in two colors so that no such rectangle has all its vertices of the same color?

Let $P$ be the set of all prime numbers. Let $A$ be some subset of $P$ that has at least two elements. Let's say that for every positive integer $n$ the following statement holds: If we take $n$ different elements $p_1,p_2...p_n \in A$, every prime number that divides $p_1 p_2 \cdots p_n-1$ is also an element of $A$. Prove, that $A$ contains all prime numbers.

Farmer Pythagoras has a field in the shape of a right triangle. The right triangle's legs have lengths 3 and 4 units. In the corner where those sides meet at a right angle, he leaves a small unplanted square $S$ so that from the air it looks like the right angle symbol. The rest of the field is planted. The shortest distance from $S$ to the hypotenuse is 2 units. What fraction of the field is planted? [asy] draw((0,0)--(4,0)--(0,3)--(0,0)); draw((0,0)--(0.3,0)--(0.3,0.3)--(0,0.3)--(0,0)); fill(origin--(0.3,0)--(0.3,0.3)--(0,0.3)--cycle, gray); label("$4$", (2,0), N); label("$3$", (0,1.5), E); label("$2$", (.8,1), E); label("$S$", (0,0), NE); draw((0.3,0.3)--(1.4,1.9), dashed); [/asy] $\textbf{(A) } \frac{25}{27} \qquad \textbf{(B) } \frac{26}{27} \qquad \textbf{(C) } \frac{73}{75} \qquad \textbf{(D) } \frac{145}{147} \qquad \textbf{(E) } \frac{74}{75} $

Let $a$, $b$, and $c$ be real numbers such that $0\le a,b,c\le 5$ and $2a + b + c = 10$. Over all possible values of $a$, $b$, and $c$, determine the maximum possible value of $a + 2b + 3c$. [i]Proposed by Andrew Wen[/i]

In a $7 \times 7$ square table, some of the squares are colored black and the others white, such that each white square is adjacent (along an edge) to an edge of the table or to a black square. Find the minimum number of black squares on the table.

In triangle $ABC$, $I$ is the incenter. We have chosen points $P,Q,R$ on segments $IA,IB,IC$ respectively such that $IP\cdot IA=IQ \cdot IB=IR\cdot IC$. Prove that the points $I$ and $O$ belong to Euler line of triangle $PQR$ where $O$ is circumcenter of $ABC$.

Suppose $\alpha_i > 0, \beta_i > 0$ for $1 \leq i \leq n, n > 1$ and that \[ \sum^n_{i=1} \alpha_i = \sum^n_{i=1} \beta_i = \pi. \] Prove that \[ \sum^n_{i=1} \frac{\cos(\beta_i)}{\sin(\alpha_i)} \leq \sum^n_{i=1} \cot(\alpha_i). \]

Let $A_1A_2A_3$ be a triangle and let $\omega_1$ be a circle in its plane passing through $A_1$ and $A_2.$ Suppose there exist circles $\omega_2, \omega_3, \dots, \omega_7$ such that for $k = 2, 3, \dots, 7,$ $\omega_k$ is externally tangent to $\omega_{k-1}$ and passes through $A_k$ and $A_{k+1},$ where $A_{n+3} = A_{n}$ for all $n \ge 1$. Prove that $\omega_7 = \omega_1.$