Let $K, L, M$, and $N$ be the midpoints of $CD,DA,AB$ and $BC$ of a square $ABCD$ respectively. Find the are of the triangles $AKB, BLC, CMD$ and $DNA$ if the square $ABCD$ has area $1$.

Solve in the set $R$ the equation $$2 \cdot [x] \cdot \{x\} = x^2 - \frac32 \cdot x - \frac{11}{16}$$ where $[x]$ and $\{x\}$ represent the integer part and the fractional part of the real number $x$, respectively.

Let $\triangle{ABC}$ be a triangle with circumcircle $\tau$. The tangentlines to $\tau$ through $A$ and $B$ intersect at $T$. The circle through $A$, $B$ and $T$ intersects $BC$ and $AC$ again at $D$ and $E$, respectively; $CT$ and $BE$ intersect at $F$. Suppose $D$ is the midpoint of $BC$. Calculate the ratio $BF:BE$. [i](Swiss Mathematical Olympiad 2011, Final round, problem 5)[/i]

Problem: Find all polynomials satisfying the equation $ f(x^2) = (f(x))^2 $ for all real numbers x. I'm not exactly sure where to start though it doesn't look too difficult. Thanks!

In the triangle $ABC$, $| BC | = 1$ and there is exactly one point $D$ on the side $BC$ such that $|DA|^2 = |DB| \cdot |DC|$. Determine all possible values of the perimeter of the triangle $ABC$.

Define a sequence $(x_{n})_{n\geq 1}$ by taking $x_{1}\in\left\{5,7\right\}$; when $k\ge 1$, $x_{k+1}\in\left\{5^{x_{k}},7^{x_{k}}\right\}$. Determine all possible last two digits of $x_{2009}$.

Let $\triangle ABC$ be an acute triangle with circumcircle $\Gamma$ and $D$ the foot of the altitude from $A$. Suppose that $AD=BC$. Point $M$ is the midpoint of $DC$, and the bisector of $\angle ADC$ meets $AC$ at $N$. Point $P$ lies on $\Gamma$ such that lines $BP$ and $AC$ are parallel. Lines $DN$ and $AM$ meet at $F$, and line $PF$ meets $\Gamma$ again at $Q$. Line $AC$ meets the circumcircle of $\triangle PNQ$ again at $E$. Prove that $\angle DQE = 90^{\circ}$.

With $2018$ points, a network composed of hexagons is formed as a sample the figure: [asy] unitsize(1 cm); int i; path hex = dir(30)--(0,1)--dir(150)--dir(210)--(0,-1)--dir(330)--cycle; draw(hex); draw(shift((sqrt(3),0))*(hex)); draw(shift((2*sqrt(3),0))*(hex)); draw(shift((4*sqrt(3),0))*(hex)); draw(shift((5*sqrt(3),0))*(hex)); dot((3*sqrt(3) - 0.3,0)); dot((3*sqrt(3),0)); dot((3*sqrt(3) + 0.3,0)); dot(dir(150)); dot(dir(210)); for (i = 0; i <= 5; ++i) { if (i != 3) { dot((0,1) + i*(sqrt(3),0)); dot(dir(30) + i*(sqrt(3),0)); dot(dir(330) + i*(sqrt(3),0)); dot((0,-1) + i*(sqrt(3),0)); } } dot(dir(150) + 4*(sqrt(3),0)); dot(dir(210) + 4*(sqrt(3),0)); [/asy] A bee and a fly play the following game: initially the bee chooses one of the $2018$ dots and paints it red, then the fly chooses one of the $2017$ unpainted dots and paint it blue. Then the bee chooses an unpainted point and paints it red and then the fly chooses an unpainted one and paints it blue and so they alternate. If at the end of the game there is an equilateral triangle with red vertices, the bee wins, otherwise Otherwise the fly wins. Determine which of the two insects has a winning strategy.

A table tennis tournament has $101$ contestants, where each pair of contestants will play each other exactly once. In each match, the player who gets $11$ points first is the winner, and the other the loser. At the end of the tournament, it turns out that there exist matches with scores $11$ to $0$ and $11$ to $10$. Show that there exists 3 contestants $A,B,C$ such that the score of the losers in the matches between $A,B$ and $A,C$ are equal, but different from the score of the loser in the match between $B,C$.

Prove that if $ n\geq 4$, $ n\in\mathbb Z$ and $ \left \lfloor \frac {2^n}{n} \right\rfloor$ is a power of 2, then $ n$ is also a power of 2.

Non-zero real numbers $a$, $b$ and $c$ are such that any two of the three equations $ax^{11}+bx^4+c=0$, $bx^{11}+cx^4+a=0$, $cx^{11}+ax^4+b=0$ have a common root. Prove that all three equations have a common root. (Author: I. Bogdanov)

Evaluate the integral $$\int^1_0\left(\sqrt{(x-1)^3+1}+x^{2/3}-(1-x)^{3/2}-\sqrt[3]{1-x^2}\right)dx$$

If $ a,b,c$ are three positive real numbers, prove that $ \frac {a^{2}\plus{}1}{b\plus{}c}\plus{}\frac {b^{2}\plus{}1}{c\plus{}a}\plus{}\frac {c^{2}\plus{}1}{a\plus{}b}\ge 3$

Prove that the following statement is true for two natural nos. $m,n$ if and only $v(m) = v(n)$ where $v(k)$ is the highest power of $2$ dividing $k$. $\exists$ a set $A$ of positive integers such that $(i)$ $x,y \in \mathbb{N}, |x-y| = m \implies x \in A $ or $y \in A$ $(ii)$ $x,y \in \mathbb{N}, |x-y| = n \implies x \not\in A $ or $y \not\in A$

Triangle $ABC$ has side-lengths $AB=12$, $BC=24$, and $AC=18$. The line through the incenter of $\triangle ABC$ parallel to $\overline{BC}$ intersects $\overline{AB}$ at $M$ and $\overline{AC}$ at $N$. What is the perimeter of $\triangle AMN$? $ \textbf{(A)}\ 27 \qquad \textbf{(B)}\ 30 \qquad \textbf{(C)}\ 33 \qquad \textbf{(D)}\ 36 \qquad \textbf{(E)}\ 42 $

Prove that there exist infinitely many positive integers $d$ such that we can find a polynomial $P\in\mathbb{Z}[x]$ of degree $d$ and $N\in\mathbb{N}$ such that for all integers $x>N$ and any prime $p$, we have $$\nu_p(P(x)^3+3P(x)^2-3)<\frac{d\cdot\log(x)}{2024^{2024}}.$$ [i]Proposed by Marius Cerlat[/i]

Let $a_1,a_2,a_3,\ldots$ be an infinite sequence where for all positive integers $i$, $a_i$ is chosen to be a random positive integer between $1$ and $2016$, inclusive. Let $S$ be the set of all positive integers $k$ such that for all positive integers $j<k$, $a_j\neq a_k$. (So $1\in S$; $2\in S$ if and only if $a_1\neq a_2$; $3\in S$ if and only if $a_1\neq a_3$ and $a_2\neq a_3$; and so on.) In simplest form, let $\dfrac{p}{q}$ be the expected number of positive integers $m$ such that $m$ and $m+1$ are in $S$. Compute $pq$.

Given $ \triangle{ABC}$, where $ A$ is at $ (0,0)$, $ B$ is at $ (20,0)$, and $ C$ is on the positive $ y$-axis. Cone $ M$ is formed when $ \triangle{ABC}$ is rotated about the $ x$-axis, and cone $ N$ is formed when $ \triangle{ABC}$ is rotated about the $ y$-axis. If the volume of cone $ M$ minus the volume of cone $ N$ is $ 140\pi$, find the length of $ \overline{BC}$.

A book is published in three volumes, the pages being numbered from $1$ onwards. The page numbers are continued from the first volume to the second volume to the third. The number of pages in the second volume is $50$ more than that in the first volume, and the number pages in the third volume is one and a half times that in the second. The sum of the page numbers on the first pages of the three volumes is $1709$. If $n$ is the last page number, what is the largest prime factor of $n$?

For which real values of $m$ does the equation $x^2-\frac{m^2+1}{m -1}x+2m+2=0$ has root $x=-1$?

Let $ABC$ be an acute scalene triangle with incenter $I$. Show that the circumcircle of $BIC$ intersects the Euler line of $ABC$ in two distinct points. (Recall that the [i]Euler line[/i] of a scalene triangle is the line that passes through its circumcenter, centroid, orthocenter, and the nine-point center.) [i]Andrew Gu[/i]

Given that $x, y$, and $z$ are a combination of positive integers such that $xyz = 2(x + y + z)$, compute the sum of all possible values of $x + y + z$.

The lines $a$ and $b$ are parallel and the point $A$ lies on $a$. One chooses one circle $\gamma$ through A tangent to $b$ at $B$. $a$ intersects $\gamma$ for the second time at $T$. The tangent line at $T$ of $\gamma$ is called $t$. Prove that independently of the choice of $\gamma$, there is a fixed point $P$ such that $BT$ passes through $P$. Prove that independently of the choice of $\gamma$, there is a fixed circle $\delta$ such that $t$ is tangent to $\delta$.

Prove that if $m,n$ are relatively prime positive integers, $x^m-y^n$ is irreducible in the complex numbers. (A polynomial $P(x,y)$ is irreducible if there do not exist nonconstant polynomials $f(x,y)$ and $g(x,y)$ such that $P(x,y) = f(x,y)g(x,y)$ for all $x,y$.) [i]David Yang.[/i]

Let $n$ be a positive integer, and let $f_n(z) = n + (n-1)z + (n-2)z^2 + \dots + z^{n-1}$. Prove that $f_n$ has no roots in the closed unit disk $\{z \in \mathbb{C}: |z| \le 1\}$.