Let $ \Gamma$ be the circumcircle of a triangle $ ABC$. A circle passing through points $ A$ and $ C$ meets the sides $ BC$ and $ BA$ at $ D$ and $ E$, respectively. The lines $ AD$ and $ CE$ meet $ \Gamma$ again at $ G$ and $ H$, respectively. The tangent lines of $ \Gamma$ at $ A$ and $ C$ meet the line $ DE$ at $ L$ and $ M$, respectively. Prove that the lines $ LH$ and $ MG$ meet at $ \Gamma$.

Two natural numbers have the same decimal digits in different order and have the sum $999\cdots 999$. Is this possible when each of the numbers consists of $1999$ digits?

A positive integer $n$ is called [i]naughty[/i] if it can be written in the form $n=a^b+b$ with integers $a,b \geq 2$. Is there a sequence of $102$ consecutive positive integers such that exactly $100$ of those numbers are naughty?

A triangle $ABC$ is inscribed in a circle $S$. Let $A_0$ and $C_0$ be the midpoints of the arcs $BC$ and $AB$ on $S$, not containing the opposite vertex, respectively. The circle $S_1$ centered at $A_0$ is tangent to $BC$, and the circle $S_2$ centered at $C_0$ is tangent to $AB$. Prove that the incenter $I$ of $\triangle ABC$ lies on a common tangent to $S_1$ and $S_2$.

Let $A$ and $B$ be distinct positive integers such that each has the same number of positive divisors that 2013 has. Compute the least possible value of $\left| A - B \right|$.

For a positive integer $n$, let $f(n)$ be the number of factors of $n^2+n+1$. Show that there are infinitely many integers $n$ which satisfy $f(n) \geq f(n+1)$.

Determine all functions $f:\mathbb R\rightarrow\mathbb R$ satisfying the condition \[f(y^2+2xf(y)+f(x)^2)=(y+f(x))(x+f(y))\] for all real numbers $x$ and $y$.

Let $ABC$ be a right-angled triangle with $\angle B = 90^\circ$ and let $BD$ be the altitude from $B$ on to $AC$. Draw $DE \perp AB$ and $DF \perp BC$. Let $P, Q, R$ and $S$ be respectively the incentres of triangle $DF C, DBF, DEB$ and $DAE$. Suppose $S, R, Q$ are collinear. Prove that $P, Q, R, D$ lie on a circle.

For any positive integer $n$, let $f(n)$ be the maximum number of groups formed by a total of $n$ people such that the following holds: every group consists of an even number of members, and every two groups share an odd number of members. Compute $\textstyle\sum_{n=1}^{2022}f(n)\text{ mod }1000$.

$\triangle ABC$ has a right angle at $B$, $AB = 12$, and $BC = 16$. Let $M$ be the midpoint of $AC$. Let $\omega_1$ be the incircle of $\triangle ABM$ and $\omega_2$ be the incircle of $\triangle BCM$. The line externally tangent to $\omega_1$ and $\omega_2$ that is not $AC$ intersects $AB$ and $BC$ at $X$ and $Y$, respectively. If the area of $\triangle BXY$ can be expressed as $\frac{m}{n}$, compute is $m+n$. [i]Proposed by Alex Li[/i]

In the Cartesian plane, let $S_{i,j} = \{(x,y)\mid i \le x \le j\}$. For $i=0,1,\ldots,2012$, color $S_{i,i+1}$ pink if $i$ is even and gray if $i$ is odd. For a convex polygon $P$ in the plane, let $d(P)$ denote its pink density, i.e. the fraction of its total area that is pink. Call a polygon $P$ [i]pinxtreme[/i] if it lies completely in the region $S_{0,2013}$ and has at least one vertex on each of the lines $x=0$ and $x=2013$. Given that the minimum value of $d(P)$ over all non-degenerate convex pinxtreme polygons $P$ in the plane can be expressed in the form $\frac{(1+\sqrt{p})^2}{q^2}$ for positive integers $p,q$, find $p+q$. [i]Victor Wang.[/i]

How many solutions has the boundary-value problem \[u_{xx}+\lambda u=\sin x,\;u(0)=u(\pi)=0\]

Show that all terms of the sequence $1,11,111,1111,...$ in base $9$ are triangular numbers, i.e. of the form $\frac{m(m+1)}{2} $for an integer $m$

Consider the sequence $(a_n)_{n\ge 1}$ such that $a_1=1$ and $a_{n+1}=\sqrt{a_n+n^2}$, $\forall n\ge 1$. $\textbf{(a)}$ Prove that there is exactly one rational number among the numbers $a_1,a_2,a_3,\dots$. $\textbf{(b)}$ Consider the sequence $(S_n)_{n\ge 1}$ such that $$S_n=\sum_{i=1}^n\frac{4}{\left (\left \lfloor a_{i+1}^2\right \rfloor-\left \lfloor a_i^2\right \rfloor\right)\left(\left \lfloor a_{i+2}^2\right \rfloor-\left \lfloor a_{i+1}^2\right \rfloor\right)}.$$ Prove that there exists an integer $N$ such that $S_n>0.9$, $\forall n>N$. [i] (Stefan Obadă)[/i]

Given are $n$ integers, not necessarily distinct, and two positive integers $p$ and $q$. If the $n$ numbers are not all distinct, choose two equal ones. Add $p$ to one of them and subtract $q$ from the other. If there are still equal ones among the $n$ numbers, repeat this procedure. Prove that after a finite number of steps, all $n$ numbers are distinct.

Find all natural numbers $n$, such that $\min_{k\in \mathbb{N}}(k^2+[n/k^2])=1991$. ($[n/k^2]$ denotes the integer part of $n/k^2$.)

Given natural $n$. Consider all the fractions $1/(pq)$, where $p$ and $q$ are relatively prime, $0<p<q\le n , p+q>n$. Prove that the sum of all such a fractions equals to $1/2$.

Let $n > 3$ be a natural number. Prove that $4^n + 1$ has a prime divisor $> 20$.

Let $x,y,z$ be non-negative numbers such that $x+y+z \leq 3$. Prove that $$\frac{2}{1+x}+\frac{2}{1+y}+\frac{2}{1+z} \geq 3$$

Solve the following Diophantines. [b]a)[/b] $ x^2+y^2=6z^2 $ [b]b)[/b] $ x^2+y^2-2x+4y-1=0 $ [i]Dan Negulescu[/i]

Cindy and Neil wanted to paint the side of a staircase in the six-square pattern shown below so that each of the six squares is painted a solid color, and no two squares that share an edge are the same color. Cindy draws all n patterns that can be colored using the four colors red, white, blue, and green. Neil looked at these patterns and claimed that k of the patterns Cindy drew were incorrect because two adjacent squares were colored with the same color. This is because Neil is color-blind and cannot distinguish red from green. Find $n + k$. For picture go to http://www.purplecomet.org/welcome/practice

In quadrilateral $ABCD$ sides $BC$ and $AD$ are parallel. In each of the four vertices we draw an angular bisector. The angular bisectors of angles $A$ and $B$ intersect in point $P$, those of angles $B$ and $C$ intersect in point $Q$, those of angles $C$ and $D$ intersect in point $R$, and those of angles $D$ and $A$ intersect in point S. Suppose that $PS$ is parallel to $QR$. Prove that $|AB| =|CD|$. [asy] unitsize(1.2 cm); pair A, B, C, D, P, Q, R, S; A = (0,0); D = (3,0); B = (0.8,1.5); C = (3.2,1.5); S = extension(A, incenter(A,B,D), D, incenter(A,C,D)); Q = extension(B, incenter(A,B,C), C, C + incenter(A,B,D) - A); P = extension(A, S, B, Q); R = extension(D, S, C, Q); draw(A--D--C--B--cycle); draw(B--Q--C); draw(A--S--D); dot("$A$", A, SW); dot("$B$", B, NW); dot("$C$", C, NE); dot("$D$", D, SE); dot("$P$", P, dir(90)); dot("$Q$", Q, dir(270)); dot("$R$", R, dir(90)); dot("$S$", S, dir(90)); [/asy] Attention: the figure is not drawn to scale.

Pawns are arranged on an $8 \times 8$ chessboard such that: Each $2 \times 1$ or $1 \times 2$ rectangle has at least $1$ pawn; Each $7 \times 1$ or $1 \times 7$ rectangle has at least $1$ pair of adjacent pawns. What is the minimum number of pawns in such an arrangement?

Let $C$ be a circle, $A_1 , A_2,\ldots ,A_n$ be distinct points inside $C$ and $B_1 , B_2 ,\ldots ,B_n$ be distinct points on $C$ such that no two of the segments $A_1B_1 , A_2 B_2 ,\ldots ,A_n B_n$ intersect. A grasshopper can jump from $A_r$ to $A_s$ if the line segment $A_r A_s$ does not intersect any line segment $A_t B_t (t \neq r, s)$. Prove that after a certain number of jumps, the grasshopper can jump from any $A_u$ to any $A_v$ .

$ABCDEF$ is a regular hexagon. A frog sarts jumping at $A$, each time it can jump to one of the two adjacent points. If the frog jump to $D$ in no more than five times, it stops. After five jumpings, if the frog hasn't jumped to $D$ yet, it will stop as well. Then the number of different ways to jump is________.