Consider the figure (attached), where every small triangle is equilateral with side length 1. Compute the area of the polygon $ AEKS $. (Fun fact: this problem was originally going to ask for the area of $ DANK $, as in "dank memes!")

Let $D$ be a point on side $[BC]$ of $\triangle ABC$ such that $|AB|=3, |CD|=1$ and $|AC|=|BD|=\sqrt{5}$. If the $B$-altitude of $\triangle ABC$ meets $AD$ at $E$, what is $|CE|$? $ \textbf{(A)}\ \dfrac{2}{\sqrt{5}} \qquad\textbf{(B)}\ 1 \qquad\textbf{(C)}\ \dfrac{2}{\sqrt{3}} \qquad\textbf{(D)}\ \dfrac{\sqrt{5}}{2} \qquad\textbf{(E)}\ \dfrac{3}{2} $

[b]a)[/b] Show that the expression $ x^3-5x^2+8x-4 $ is nonegative, for every $ x\in [1,\infty ) . $ [b]b)[/b] Determine $ \min_{a,b\in [1,\infty )} \left( ab(a+b-10) +8(a+b) \right) . $

Let $ABCD$ be a parallelogram which is not a rectangle and $E$ be the point in its plane such that $AE \perp AB$ and $CE \perp CB$. Prove that $\angle DEA = \angle CEB$.

In triangle $ ABC$ we have $ \angle{ACB} \equal{} 2\angle{ABC}$ and there exists the point $ D$ inside the triangle such that $ AD \equal{} AC$ and $ DB \equal{} DC$. Prove that $ \angle{BAC} \equal{} 3\angle{BAD}$.

A convex pentagon $P $ is divided by all its diagonals into ten triangles and one smaller pentagon $P'$. Let $N$ be the sum of areas of five triangles adjacent to the sides of $P$ decreased by the area of $P'$. The same operations are performed with the pentagon $P'$, let $N'$ be the similar difference calculated for this pentagon. Prove that $N > N'$. (A.Belov)

Let an integer $k > 1$ be given. For each integer $n > 1$, we put \[f(n) = k \cdot n \cdot \left(1-\frac{1}{p_1}\right) \cdot \left(1-\frac{1}{p_2}\right) \cdots \left(1-\frac{1}{p_r}\right)\] where $p_1, p_2, \ldots, p_r$ are all distinct prime divisors of $n$. Find all values $k$ for which the sequence $\{x_m\}$ defined by $x_0 = a$ and $x_{m+1} = f(x_m), m = 0, 1, 2, 3, \ldots$ is bounded for all integers $a > 1$.

The vertices of a cube are labeled with the integers $1$ through $8,$ with each used exactly once. Let $s$ be the maximum sum of the labels of two edge-adjacent vertices. Compute the minimum possible value of $s$ over all such labelings.

What is the smallest positive integer $t$ such that there exist integers $x_{1},x_{2}, \cdots, x_{t}$ with \[{x_{1}}^{3}+{x_{2}}^{3}+\cdots+{x_{t}}^{3}=2002^{2002}\;\;?\]

[b]p1.[/b] Let $n$ be the number so that $1 - 2 + 3 - 4 + ... - (n - 1) + n = 2012$. What is $4^{2012}$ (mod $n$)? [b]p2. [/b]Consider three unit squares placed side by side. Label the top left vertex $P$ and the bottom four vertices $A,B,C,D$ respectively. Find $\angle PBA + \angle PCA + \angle PDA$. [b]p3.[/b] Given $f(x) = \frac{3}{x-1}$ , then express $\frac{9(x^2-2x+1)}{x^2-8x+16}$ entirely in terms of $f(x)$. In other words, $x$ should not be in your answer, only $f(x)$. [b]p4.[/b] Right triangle with right angle $B$ and integer side lengths has $BD$ as the altitude. $E$ and $F$ are the incenters of triangles $ADB$ and $BDC$ respectively. Line $EF$ is extended and intersects $BC$ at $G$, and $AB$ at $H$. If $AB = 15$ and $BC = 8$, find the area of triangle $BGH$. [b]p5.[/b] Let $a_1, a_2, ..., a_n$ be a sequence of real numbers. Call a $k$-inversion $(0 < k\le n)$ of a sequence to be indices $i_1, i_2, .. , i_k$ such that $i_1 < i_2 < .. < i_k$ but $a_{i1} > a_{i2} > ...> a_{ik}$ . Calculate the expected number of $6$-inversions in a random permutation of the set $\{1, 2, ... , 10\}$. [b]p6.[/b] Chell is given a strip of squares labeled $1, .. , 6$ all placed side to side. For each $k \in {1, ..., 6}$, she then chooses one square at random in $\{1, ..., k\}$ and places a Weighted Storage Cube there. After she has placed all $6$ cubes, she computes her score as follows: For each square, she takes the number of cubes in the pile and then takes the square (i.e. if there were 3 cubes in a square, her score for that square would be $9$). Her overall score is the sum of the scores of each square. What is the expected value of her score? PS. You had better use hide for answers.

$\{a_n\}$ is an infinite, strictly increasing sequence of positive integers and $a_{a_n}\leq a_n+a_{n+3}$ for all $n\geq 1$. Prove that, there are infinitely many triples $(k,l,m)$ of positive integers such that $k<l<m$ and $a_k+a_m=2a_l$

A sphere is touched by all the four sides of a (space) quadrilateral. Prove that all the four touching points are in the same plane.

As shown in the figure, two circles $\Gamma_1$ and $\Gamma_2$ on the plane intersect at two points $A$ and $B$. The two rays passing through $A$, $\ell_1$ and $\ell_2$ intersect $\Gamma_1$ at points $D$ and $E$ respectively, and $\Gamma_2$ at points $F$ and $C$ respectively (where $E$ and $F$ lie on line segments $AC$ and $AD$ respectively, and neither of them coincides with the endpoints). It is known that the three lines $AB$, $CF$ and $DE$ have a common point, the circumscribed circle of $\vartriangle AEF$ intersects $AB$ at point $G$, the straight line $EG$ intersects the circle $\Gamma_1$ at point $P$, the straight line $FG$ intersects the circle $\Gamma_2$ at point $Q$. Let the symmetric points of $C$ and $D$ wrt the straight line $AB$ be $C'$ and $D'$ respectively. If $PD'$ and $QC'$ intersect at point$ J$, prove that $J$ lies on the straight line $AB$. [img]https://cdn.artofproblemsolving.com/attachments/3/7/eb3acdbad52750a6879b4b6955dfdb7de19ed3.png[/img]

Let $f: (0,+\infty)\rightarrow (0,+\infty)$ be a function satisfying the following condition: for arbitrary positive real numbers $x$ and $y$, we have $f(xy)\le f(x)f(y)$. Show that for arbitrary positive real number $x$ and natural number $n$, inequality $f(x^n)\le f(x)f(x^2)^{\dfrac{1}{2}}\dots f(x^n)^{\dfrac{1}{n}}$ holds.

Each of $A$, $B$, $C$, $D$, $E$, and $F$ knows a piece of gossip. They communicate by telephone via a central switchboard, which can connect only two of them at a time. During a conversation, each side tells the other everything he or she knows at that point. Determine the minimum number of calls for everyone to know all six pieces of gossip.

In a board of $2000\times2001$ squares with integer coordinates $(x,y)$, $0\leq{x}\leq1999$ and $0\leq{y}\leq2000$. A ship in the table moves in the following way: before a move, the ship is in position $(x,y)$ and has a velocity of $(h,v)$ where $x,y,h,v$ are integers. The ship chooses new velocity $(h^\prime,v^\prime)$ such that $h^\prime-h,v^\prime-v\in\{-1,0,1\}$. The new position of the ship will be $(x^\prime,y^\prime)$ where $x^\prime$ is the remainder of the division of $x+h^\prime$ by $2000$ and $y^\prime$ is the remainder of the division of $y+v^\prime$ by $2001$. There are two ships on the board: The Martian ship and the Human trying to capture it. Initially each ship is in a different square and has velocity $(0,0)$. The Human is the first to move; thereafter they continue moving alternatively. Is there a strategy for the Human to capture the Martian, independent of the initial positions and the Martian’s moves? [i]Note[/i]: The Human catches the Martian ship by reaching the same position as the Martian ship after the same move.

Find the least positive integer which is a multiple of $13$ and all its digits are the same. [i](Adapted from Gazeta Matematică 1/1982, Florin Nicolăită)[/i]

On the board are written $n$ natural numbers, $n\in \mathbb{N}$. In one move it is possible to choose two equal written numbers and increase one by $1$ and decrease the other by $1$. Prove that in this the game cannot be played more than $\frac{n^3}{6}$ moves.

Let $0\le a\le b\le c$ be real numbers. Prove that \[(a+3b)(b+4c)(c+2a)\ge 60abc \]

[b]1.[/b] Let $k$ be an arbitrary cardinality. Show that there exists a tournament $T_k = (V_n , E_n)$ such that for any coloring $f: E_n \to k$ of the edge set $E_n$, there are three different vertices $x_0 , x_1 , x_2 \in V_n$ such that $x_0 x_1 , x_1 x_2 , x_2 x_0 \in E_n$ and $\left | \{ f(x_0 x_1), f(x_1 x_2), f(x_2 x_0)\} \right |\leq 2$ (A [i]tounament[/i] is a directed graph such that for any vertices $x, y \in V_n, x \neq y$ exactly one of the relations $xy \in E_n$ holds.) ([b]C.19[/b]) [A. Hajnal]

Consider $4n$ points in the plane such that no three of them are collinear ($n\geq 1$). Show that the set of centroids of all the triangles formed by any three of these points contains at least $4n$ elements. [i]Radu Bumbăcea[/i]

Prove that for any $p>2$ prime number, there exists only one positive number $n$ that makes the equation $n^2-np$ a perfect square of a positive integer

Points $T,P,H$ lie on the side $BC,AC,AB$ respectively of triangle $ABC$, so that $BP$ and $AT$ are angle bisectors and $CH$ is an altitude of $ABC$. Given that the midpoint of $CH$ belongs to the segment $PT,$ find the value of $\cos A + \cos B$ I. Voronovich

For any positive integer $n$, define the subset $S_n$ of natural numbers as follow $$ S_n = \left\{x^2+ny^2 : x,y \in \mathbb{Z} \right\}.$$ Find all positive integers $n$ such that there exists an element of $S_n$ which [u]doesn't belong[/u] to any of the sets $S_1, S_2,\dots,S_{n-1}$. [i]Proposed by Yahya Motevassel[/i]

A rectangle with a diagonal of length $ x$ is twice as long as it is wide. What is the area of the rectangle? $ \textbf{(A)}\ \frac14x^2 \qquad \textbf{(B)}\ \frac25x^2 \qquad \textbf{(C)}\ \frac12x^2 \qquad \textbf{(D)}\ x^2 \qquad \textbf{(E)}\ \frac32x^2$