Let $K$ and $M$ be points on the side $AB$ of a triangle $\triangle{ABC}$, and let $L$ and $N$ be points on the side $AC$. The point $K$ is between $M$ and $B$, and the point $L$ is between $N$ and $C$. If $\frac{BK}{KM}=\frac{CL}{LN}$, then prove that the orthocentres of the triangles $\triangle{ABC}$, $\triangle{AKL}$ and $\triangle{AMN}$ lie on one line.

Find the number of permutations $(a_1, a_2, . . . , a_{2013})$ of $(1, 2, \dots , 2013)$ such that there are exactly two indices $i \in \{1, 2, \dots , 2012\}$ where $a_i < a_{i+1}$.

Let $ a_1, a_2, ..., a_k $ be distinct positive integers such that the $2^k$ sums $\displaystyle\sum\limits_{i=1}^{k}{\epsilon_i a_i}$, $\epsilon_i\in\left\{0,1\right\}$ are distinct. a) Show that $ \dfrac{1}{a_1}+\dfrac{1}{a_2}+...+\dfrac{1}{a_k}\le2(1-2^{-k}) $; b) Find the sequences $(a_1,a_2,...,a_k)$ for which the equality holds. [i]Șerban Buzețeanu[/i]

Let $r_n$ be the $n$th smallest positive solution to $\tan x=x$, where the argument of tangent is in radians. Prove that \[ 0<r_{n+1}-r_n-\pi<\frac{1}{(n^2+n)\pi} \] for $n\geq 1$.

It is known that $a_1, a_2,...a_{108}$ are $108$ different positive integers not exceeding $2015$. Prove that there is a positive integer $k$ such that there are at least four different pairs $(i, j) $satisfying $a_i-a_j =k$.

A square is inscribed in an ellipse such that two sides of the square respectively pass through the two foci of the ellipse. The square has a side length of $4$. The square of the length of the minor axis of the ellipse can be written in the form $a + b\sqrt{c}$ where $a, b$, and $c$ are integers, and $c$ is not divisible by the square of any prime. Find the sum $a + b + c$.

The sequence $ \{a_n\}$ of integers is defined by \[ a_1 \equal{} 2, a_2 \equal{} 7 \] and \[ \minus{} \frac {1}{2} < a_{n \plus{} 1} \minus{} \frac {a^2_n}{a_{n \minus{} 1}} \leq \frac {}{}, n \geq 2. \] Prove that $ a_n$ is odd for all $ n > 1.$

On the sides $AB,BC$ and $CA$ of the triangle $ABC$ consider the points $Z,X$ and $Y$ respectively such that \[AZ-AY=BX-BZ=CY-CX.\]Let $P,M$ and $N$ be the circumcenters of the triangles $AYZ, BZX$ and $CXY$ respectively. Prove that the incenters of the triangle $ABC$ coincides with that of the triangle $MNP$.

Call a polynomial with real roots [i]n-local[/i] if the greatest difference between any pair of its roots is $n$. Let $f(x)=x^2+ax+b$ be a 1-[i]local[/i] polynomial with distinct roots such that $a$ and $b$ are non-zero integers. If $f(f(x))$ is a 23-[i]local[/i] polynomial, find the sum of the roots of $f(x)$. [i]Proposed by Anthony Yang[/i]

Consider $2n+1$ points in space, no four of which are coplanar where $n>1$. Each line segment connecting any two of these points is either colored red, white or blue. A subset $M$ of these points is called a [i]connected monochromatic[/i] subset, if for each $a,b \in M$, there are points $a=x_0,x_1, \dots, x_l = b$ that belong to $M$ such that the line segments $x_0x_1, x_1x_2, \dots, x_{l-1}x_1$ are all have the same color. No matter how the points are colored, if there always exists a connected monochromatic $k-$subset, find the largest value of $k$. ($l > 1$)

Let $ABCD$ be a convex quadrilateral. $O$ is the intersection of $AC$ and $BD$. $AO=3$ ,$BO=4$, $CO=5$, $DO=6$. $X$ and $Y$ are points in segment $AB$ and $CD$ respectively, such that $X,O,Y$ are collinear. The minimum of $\frac{XB}{XA}+\frac{YC}{YD}$ can be written as $\frac{a\sqrt{c}}{b}$ , where $\frac{a}{b}$ is in lowest term and $c$ is not divisible by any square number greater then $1$. What is the value of $10a+b+c$?

Show that a line passing through the feet of two altitudes of an acute-angled triangle cuts off a similar triangle.

Prove that $ \forall n > 1, n \in \mathbb{N}$ the equation \[ \sum^n_{k\equal{}1} \frac{x^k}{k!} \plus{} 1 \equal{} 0\] has no rational roots.

Kritesh manages traffic on a $45 \times 45$ grid consisting of 2025 unit squares. Within each unit square is a car, facing either up, down, left, or right. If the square in front of a car in the direction it is facing is empty, it can choose to move forward. Each car wishes to exit the $45 \times 45$ grid. Kritesh realizes that it may not always be possible for all the cars to leave the grid. Therefore, before the process begins, he will remove $k$ cars from the $45 \times 45$ grid in such a way that it becomes possible for all the remaining cars to eventually exit the grid. What is the minimum value of $k$ that guarantees that Kritesh's job is possible? $\textbf{Proposed by Shining Sun. USA}$

In a certain country $10\%$ of the employees get $90\%$ of the total salary paid in this country. Supposing that the country is divided in several regions, is it possible that in every region the total salary of any 10% of the employees is no greater than $11\%$ of the total salary paid in this region?

In the triangle $ABC$, it is known that $AB = BC = 20$ cm, and $AC = 24$ cm. The point $M$ lies on the side $BC$ and is equidistant from sides $AB$ and $AC$. Find this distance. (Alexander Shkolny)