Does there exist a $2n$-digit number $\overline{a_{2n}a_{2n-1}\cdots a_1}$(for an arbitrary $n$) for which the following equality holds: \[\overline{a_{2n}\cdots a_1}= (\overline{a_n \cdots a_1})^2?\]

Find all positive integer solutions of the equation $10(m+n)=mn$.

Let $ABC$ be a scalene acute triangle with circumcenter $O$. Let $K$ be a point on the side $\overline{BC}$. Define $M$ as the second intersection of $\overleftrightarrow{OK}$ with the circumcircle of $BOC$. Let $L$ be the reflection of $K$ by $\overleftrightarrow{AC}$. Show that the circumcircles of the triangles $LCM$ and $ABC$ are tangent if, and only if, $\overline{AK} \perp \overline{BC}$.

Find all polynomials $p(x)$ with real coeffcients such that \[p(a + b - 2c) + p(b + c - 2a) + p(c + a - 2b) = 3p(a - b) + 3p(b - c) + 3p(c - a)\] for all $a, b, c\in\mathbb{R}$. [i](2nd Benelux Mathematical Olympiad 2010, Problem 2)[/i]

In a grid of $100\times 100$ squares, there is a mouse on the top-left square, and there is a piece of cheese in the bottom-right square. The mouse wants to move to the bottom-right square to eat the cheese. For each step, the mouse can move from one square to an adjacent square (two squares are considered adjacent if they share a common edge). Now, any divider can be placed on the common edge of two adjacent squares such that the mouse cannot directly move between these two adjacent squares. A placement of dividers is called "kind" if the mouse can still reach the cheese after the dividers are placed. Find the smallest positive integer $n$ such that, regardless of any "kind" placement of $2023$ dividers, the mouse can reach the cheese in at most $n$ steps.

Let $a.b,c,d$ be positive real numbers such that $cd = 1$. Prove that there is an integer $n$ such that $ab\le n^2\le (a + c)(b + d)$.

Let $M$ be the midpoint of the arc $BAC$ of the circumcircle of the triangle $ABC,$ $I$ the incenter of the triangle $ABC$ and $L$ a point on the side $BC$ such that $AL$ is bisector. The line $MI$ cuts the circumcircle again at $K.$ The circumcircle of the triangle $AKL$ cuts the line $BC$ again at $P.$ Prove that $\angle AIP = 90^{\circ}.$

Let $ABCD$ be a rectangle and let $M, N$ and $P, Q$ be the points of intersections of some line $e$ with the sides $AB, CD$ and $AD, BC$, respectively (or their extensions). Given the points $M, N, P, Q$ and the length $p$ of side $AB$, construct the rectangle. Under what conditions can this problem be solved, and how many solutions does it have?

Consider a very simple model for an open self-replicative system such as a cell, or an economy. A system $\mathbf{S}$ is comprised of two kinds of mass: one kind is $\mathbf{S}_R$ that is capable of of taking raw material that comes from outside the system, and converting it into components of the system, and the other is $\mathbf{S}_O$ that takes care of other things. Think of $\mathbf{S}_R$ like factories and $\mathbf{S}_O$ like street sweepers. One part is creating new things and the other is doing maintenance on what already exists. The catch is, the material in $\mathbf{S}_R$ must not only make the rest of the system, but also itself! Suppose that the materials in $\mathbf{S}_R$ and the materials in $\mathbf{S}_O$ cost the same amount of energy for $\mathbf{S}_R$ to make per unit amount. Suppose the material in $\mathbf{S}_R$ can convert raw material from the environment into system mass at the rate $\gamma_R = 3 \mbox{ kg } \mathbf{S}/\mbox{hr}/\mbox{kg }\mathbf{S}_R$. If the system doubles in size once every 2 $\mbox{hrs}$, what fraction of the material in $\mathbf{S}$ is devoted to $\mathbf{S}_O$? $\textbf{Assumptions:}$ The fact that the system continuously doubles in size in a fixed time means that this system is in exponential growth, i.e. $\dot{\mathbf{S}} = \lambda \mathbf{S}$. [i]Problem proposed by Josh Silverman[/i]

Let $n\ge2$ be a natural number. Prove that $$\prod_{k=2}^n\ln k<\frac{\sqrt{n!}}n.$$

Let $ABC$ be a triangle, right-angled at point $ A$ and with $AB>AC$. The tangent through $ A$ of the circumcircle $G$ of $ABC$ cuts $BC$ at $D$. $E$ is the reflection of $ A$ over line $BC$. $X$ is the foot of the perpendicular from $ A$ over $BE$. $Y$ is the midpoint of $AX$, $Z$ is the intersection of $BY$ and $G$ other than $ B$, and $F$ is the intersection of $AE$ and $BC$. Prove $D, Z, F, E$ are concyclic.

How many $f : A \rightarrow A$ are there satisfying $f(f(a)) = a$ for every $a \in A=\{1,2,3,4,5,6,7\}$? $ \textbf{(A)}\ 1 \qquad \textbf{(B)}\ 106 \qquad \textbf{(C)}\ 127 \qquad \textbf{(D)}\ 232 \qquad \textbf{(E)}\ \text{None}$

Two distinct numbers are selected from the set $\{1,2,3,4,\dots,36,37\}$ so that the sum of the remaining $35$ numbers is the product of these two numbers. What is the difference of these two numbers? $\textbf{(A) }5 \qquad \textbf{(B) }7 \qquad \textbf{(C) }8\qquad \textbf{(D) }9 \qquad \textbf{(E) }10$

Twelve balls are numbered by the numbers $1,2,3,\cdots,12$. Each ball is colored either red or green, so that the following two conditions are satisfied: (i) If two balls marked by different numbers $a$ and $b$ are colored red and $a+b<13$, then the ball marked by the number $a+b$ is colored red, too. (ii) If two balls marked by different numbers $a$ and $b$ are colored green and $a+b<13$, then the ball marked by the number $a+b$ is also colored green. How many ways are there of coloring the balls? Please remember to hide your solution. (by using the hide tags of course.. I don't literally mean that you should hide it :ninja: )

Along a circle-shaped path are $100$ boys and $100$ girls. The distance between two points on the path is defined as the length of the smaller arc through which it is possible to get from one point to the other. Prove that the sum of distances between pairs of the same gender is always less than or equal to the sum of distances between all pairs of a boy and a girl.

Two trapezoid angles and diagonals are respectively equal. Is it true that such are the trapezoid equal?

For points $ A_1, \ldots ,A_5$ on the sphere of radius 1, what is the maximum value that $ min_{1 \leq i,j \leq 5} A_iA_j$ can take? Determine all configurations for which this maximum is attained. (Or: determine the diameter of any set $ \{A_1, \ldots ,A_5\}$ for which this maximum is attained.)