Point $A$ is located in this circle of radius $1$. An arbitrary chord is drawn through it, and then a circle of radius $2$ is drawn through the ends of this chord. Prove that all such circles touch some fixed circle, not depending from the initial choice of the chord.

Triangle $JHT$ has side lengths $JH = 14$, $HT = 10$, and $TJ = 16$. Points $I$ and $U$ lie on $\overline{JH}$ and $\overline{JT},$ respectively, so that $HI = TU = 1.$ Let $M$ and $N$ be the midpoints of $\overline{HT}$ and $\overline{IU},$ respectively. Line $MN$ intersects another side of $\triangle JHT$ at a point $P$ other than $M.$ Compute $MP^2.$

Let $ABC$ be a triangle. A circle passing through $A$ and $B$ intersects segments $AC$ and $BC$ at $D$ and $E$, respectively. Lines $AB$ and $DE$ intersect at $F$, while lines $BD$ and $CF$ intersect at $M$. Prove that $MF = MC$ if and only if $MB\cdot MD = MC^2$.

The Euler line of a scalene triangle touches its incircle. Prove that this triangle is obtuse-angled.

Let $\omega$ be the incircle of triangle $ABC$ touches $BC$ at point $K$ . Draw a circle passing through points $B$ and $C$ , and touching $\omega$ at the point $S$ . Prove that $S K$ passes through the center of the exscribed circle of triangle $A B C$ , tangent to side $B C$ .

Find all functions $f : \mathbb{R} \longrightarrow \mathbb{R}$ such that \[ f(x^2 + yf(z)) = xf(x) + zf(y) , \] for all $x, y, z \in \mathbb{R}$.

Find all pairs $(x,n)$ of positive integers for which $x^n + 2^n + 1$ divides $x^{n+1} +2^{n+1} +1$.

Determine all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that for all real numbers $x, y$ \[f(x + y)=f(f(x)) + y + 2022\]

A right circular cone stands on plane $P$. The radius of the cone’s base is $r$, its height is $h$. A source of light is placed at distance $H$ from the plane, and distance $1$ from the axis of the cone. What is the illuminated part of the disc of radius $R$, that belongs to $P$ and is concentric with the disc forming the base of the cone?

Find the sum of all prime numbers $p$ which satisfy \[p = a^4 + b^4 + c^4 - 3\] for some primes (not necessarily distinct) $a$, $b$ and $c$.

[i]Centered hexagonal numbers[/i] are the numbers of dots used to create hexagonal arrays of dots. The first four centered hexagonal numbers are 1, 7, 19, and 37 as shown below: [asy] size(250);defaultpen(linewidth(0.4)); dot(origin^^shift(-12,0)*origin^^shift(-24,0)*origin^^shift(-36,0)*origin); int i; for(i=0; i<360; i=i+60) { dot(1*dir(i)^^2*dir(i)^^3*dir(i)); dot(shift(1/2, sqrt(3)/2)*1*dir(i)^^shift(1/2, sqrt(3)/2)*2*dir(i)); dot(shift(1, sqrt(3))*1*dir(i)); dot(shift(-12,0)*origin+1*dir(i)^^shift(-12,0)*origin+2*dir(i)); dot(shift(-12,0)*origin+sqrt(3)*dir(i+30)); dot(shift(-24,0)*origin+1*dir(i)); } label("$1$", (-36, -5), S); label("$7$", (-24, -5), S); label("$19$", (-12, -5), S); label("$37$", (0, -5), S); label("Centered Hexagonal Numbers", (-18,-10), S);[/asy] Consider an arithmetic sequence 1, $a$, $b$ and a geometric sequence 1,$c$,$d$, where $a$,$b$,$c$, and $d$ are all positive integers and $a+b=c+d$. Prove that each centered hexagonal number is a possible value of $a$, and prove that each possible value of $a$ is a centered hexagonal number.

Let us define $f$ and $g$ by $f(x) = x^5 +5x^4 +5x^3 +5x^2 +1$, $g(x) = x^5 +5x^4 +3x^3 -5x^2 -1$. Determine all prime numbers $p$ such that, for at least one integer $x, 0 \le x < p-1$, both $f(x)$ and $g(x)$ are divisible by $p$. For each such $p$, find all $x$ with this property.

For any positive integer $n$ , let $s(n)$ denote the number of ordered pairs $(x,y)$ of positive integers for which $\dfrac{1}{x} + \dfrac{1}{y} = \dfrac{1}{n}$ . Determine the set of positive integers for which $s(n) = 5$

The following process is applied to each positive integer: the sum of its digits is subtracted from the number, and the result is divided by $9$. For example, the result of the process applied to $938$ is $102$, since $\frac{938-(9 + 3 + 8)}{9} = 102.$ Applying the process twice to $938$ the result is $11$, applied three times the result is $1$, and applying it four times the result is $0$. When the process is applied one or more times to an integer $n$, the result is eventually $0$. The number obtained before obtaining $0$ is called the [i]house[/i] of $n$. How many integers less than $26000$ share the same [i]house[/i] as $2012$?

Let $a_0, a_1, . . . , a_N$ be real numbers satisfying $a_0 = a_N = 0$ and \[a_{i+1} - 2a_i + a_{i-1} = a^2_i\] for $i = 1, 2, . . . , N - 1.$ Prove that $a_i\leq 0$ for $i = 1, 2, . . . , N- 1.$

Given a number $\overline{abcd}$, where $a$, $b$, $c$, and $d$, represent the digits of $\overline{abcd}$, find the minimum value of \[\frac{\overline{abcd}}{a+b+c+d}\] where $a$, $b$, $c$, and $d$ are distinct [hide=Answer]$\overline{abcd}=1089$, minimum value of $\dfrac{\overline{abcd}}{a+b+c+d}=60.5$[/hide]

Find all integers $ n \ge 3$ such that there are $ n$ points in space, with no three on a line and no four on a circle, such that all the circles pass through three points between them are congruent.

Which one does not divide the numbers of $500$-subset of a set with $1000$ elements? $ \textbf{(A)}\ 3 \qquad\textbf{(B)}\ 5 \qquad\textbf{(C)}\ 11 \qquad\textbf{(D)}\ 13 \qquad\textbf{(E)}\ 17 $

Let $n, k \geq 3$ be integers, and let $S$ be a circle. Let $n$ blue points and $k$ red points be chosen uniformly and independently at random on the circle $S$. Denote by $F$ the intersection of the convex hull of the red points and the convex hull of the blue points. Let $m$ be the number of vertices of the convex polygon $F$ (in particular, $m=0$ when $F$ is empty). Find the expected value of $m$.

Let $A$ be a point outside the circle $\omega$. Two points $B, C$ lie on $\omega$ such that $AB, AC$ are tangent to $\omega$. Let $D$ be any point on $\omega$ ($D$ is neither $B$ nor $C$) and $M$ the foot of perpendicular from $B$ to $CD$. The line through $D$ and the midpoint of $BM$ meets $\omega$ again at $P$. Prove that $AP \perp CP$

Let $D$ be the complex unit disk $D=\{z \in \mathbb{C}: |z|<1\}$, and $0<a<1$ a real number. Suppose that $f:D \to \mathbb{C}\setminus \{0\}$ is a holomorphic function such that $f(a)=1$ and $f(-a)=-1$. Prove that $$ \sup_{z \in D} |f(z)| \geqslant \exp\left(\frac{1-a^2}{4a}\pi\right) .$$

On a board, the numbers from 1 to 2009 are written. A couple of them are erased and instead of them, on the board is written the remainder of the sum of the erased numbers divided by 13. After a couple of repetition of this erasing, only 3 numbers are left, of which two are 9 and 999. Find the third number.

Let $\triangle ABC$ be a triangle, and let $l$ be the line passing through its incenter and centroid. Assume that $B$ and $C$ lie on the same side of $l$, and that the distance from $B$ to $l$ is twice the distance from $C$ to $l$. Suppose also that the length $BA$ is twice that of $CA$. If $\triangle ABC$ has integer side lengths and is as small as possible, what is $AB^2+BC^2+CA^2$? [i]Proposed by Thomas Lam[/i]

Let $P$ be a point inside an acute triangle $ABC$. Let the lines $BP$ and $CP$ intersect the sides $AC$ and $AB$ at $D$ and $E$, respectively. Let the circles with diameters $BD$ and $CE$ intersect at points $S$ and $T$. Prove that if the points $A$, $S$, and $T$ are colinear, then $P$ lies on a median of $\triangle ABC$.

The two digits in Jack's age are the same as the digits in Bill's age, but in reverse order. In five years Jack will be twice as old as Bill will be then. What is the difference in their current ages? $ \textbf{(A)}\ 9 \qquad \textbf{(B)}\ 18 \qquad \textbf{(C)}\ 27 \qquad \textbf{(D)}\ 36 \qquad \textbf{(E)}\ 45$