$ABC$ is an acute-angled triangle. Let $D$ be the point on $BC$ such that $AD$ is the bisector of $\angle A$. Let $E, F$ be the feet of perpendiculars from $D$ to $AC,AB$ respectively. Suppose the lines $BE$ and $CF$ meet at $H$. The circumcircle of triangle $AFH$ meets $BE$ at $G$ (apart from $H$). Prove that the triangle constructed from $BG$, $GE$ and $BF$ is right-angled.

Prove that if $t$ is a natural number then there exists a natural number $n>1$ such that $(n,t)=1$ and none of the numbers $n+t,n^2+t,n^3+t,....$ are perfect powers.

Given any $22$ points in the plane, no three collinear. Show that the points can be divided into $11$ pairs, so that the $11$ line segments defined by the pairs have at least five different intersections

Consider the following two vertex-weighted graphs, and denote them as having vertex sets $V=\{v_1,v_2,\ldots,v_6\}$ and $W=\{w_1,w_2,\ldots,w_6\}$, respectively (numbered in the same direction and way). The weights in the second graph are such that for all $1\le i\le 6$, the weight of $w_i$ is the sum of the weights of the neighbors of $v_i$. Determine the sum of the weights of the original graph.

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 $$ \sup_{z \in D} |f(z)| \geqslant \exp\left(\frac{1-a^2}{4a}\pi\right) .$$