Let $ABC$ be a triangle with $\angle C=90^\circ$, and $A_0$, $B_0$, $C_0$ be the mid-points of sides $BC$, $CA$, $AB$ respectively. Two regular triangles $AB_0C_1$ and $BA_0C_2$ are constructed outside $ABC$. Find the angle $C_0C_1C_2$.

Let $ABC$ be a triangle with altitude $\overline{AE}$. The $A$-excircle touches $\overline{BC}$ at $D$, and intersects the circumcircle at two points $F$ and $G$. Prove that one can select points $V$ and $N$ on lines $DG$ and $DF$ such that quadrilateral $EVAN$ is a rhombus. [i]Danielle Wang and Evan Chen[/i]

Let $a,b$ be two integers such that their gcd has at least two prime factors. Let $S = \{ x \mid x \in \mathbb{N}, x \equiv a \pmod b \} $ and call $ y \in S$ irreducible if it cannot be expressed as product of two or more elements of $S$ (not necessarily distinct). Show there exists $t$ such that any element of $S$ can be expressed as product of at most $t$ irreducible elements.

In the rectangular table shown below, the number $1$ is written in the upper-left hand corner, and every number is the sum of the any numbers directly to its left and above. The table extends infinitely downwards and to the right. \[ \begin{array}{cccccc} 1 & 1 & 1 & 1 & 1 & \cdots \\ 1 & 2 & 3 & 4 & 5 & \cdots \\ 1 & 3 & 6 & 10 & 15 & \cdots \\ 1 & 4 & 10 & 20 & 35 & \cdots \\ 1 & 5 & 15 & 35 & 70 & \cdots \\ \vdots & \vdots & \vdots & \vdots & \vdots & \ddots \end{array} \] Wanda the Worm, who is on a diet after a feast two years ago, wants to eat $n$ numbers (not necessarily distinct in value) from the table such that the sum of the numbers is less than one million. However, she cannot eat two numbers in the same row or column (or both). What is the largest possible value of $n$? [i]Proposed by Evan Chen[/i]

Let $ x\in \mathbb R$. Find the minimum and maximum values of the expresion: $ E\equal{}\dfrac{(1\plus{}x)^8\plus{}16x^4}{(1\plus{}x^2)^4}$

Let $a,b,c$ be positive numbers with $\sqrt{a}+\sqrt{b}+\sqrt{c}= \frac{\sqrt{3}}{2}$ Prove that the system of equations \[\sqrt{y-a}+\sqrt{z-a}=1\] \[\sqrt{z-b}+\sqrt{x-b}=1\] \[\sqrt{x-c}+\sqrt{y-c}=1\] has exactly one solution $(x,y,z)$ in real numbers. It was proposed by Poland. Have fun! :lol:

On a board there are $n$ nails, each two connected by a rope. Each rope is colored in one of $n$ given distinct colors. For each three distinct colors, there exist three nails connected with ropes of these three colors. a) Can $n$ be $6$ ? b) Can $n$ be $7$ ?

Let $t = (a, b, c)$, and let us define $f^1 (t) = (a + b, b + c, c + a)$ and $f^k (t) = f^{k-1}(f^1(t))$ for all $k > 1$. Furthermore, a permutation of $t$ has the same elements, just in a different order (e.g., $(b, c, a)$). If $f^{2013}(s)$ is a permutation of $s$ for some $s = (k, m, n)$, where $k, m$, and $n$ are integers such that $|k|, |m|, |n|\le 10$, how many possible values of $s$ are there?

Determine all triples of real numbers $(a,b,c)$ that satisfy simultaneously the equations: $$\begin{cases} a(b^2 + c) = c(c + ab) \\ b(c^2 + a) = a(a + bc) \\ c(a^2 + b) = b(b + ca) \end{cases}$$

Sevara writes in red $8$ distinct positive integers and then writes in blue the $28$ sums of each two red numbers. At most how many of the blue numbers can be prime? [i]Marin Hristov, Bulgaria[/i]

Find the greatest positive real number $ k$ such that the inequality below holds for any positive real numbers $ a,b,c$: \[ \frac ab \plus{} \frac bc \plus{} \frac ca \minus{} 3 \geq k \left( \frac a{b \plus{} c} \plus{} \frac b{c \plus{} a} \plus{} \frac c{a \plus{} b} \minus{} \frac 32 \right). \]

Given a positive integer $n\ge 3$, colour each cell of an $n\times n$ square array with one of $\lfloor (n+2)^2/3\rfloor$ colours, each colour being used at least once. Prove that there is some $1\times 3$ or $3\times 1$ rectangular subarray whose three cells are coloured with three different colours. [i](Russia) Ilya Bogdanov, Grigory Chelnokov, Dmitry Khramtsov[/i]

Let $ABC$ be a triangle. The incircle of $ABC$ touches the sides $AB$ and $AC$ at the points $Z$ and $Y$, respectively. Let $G$ be the point where the lines $BY$ and $CZ$ meet, and let $R$ and $S$ be points such that the two quadrilaterals $BCYR$ and $BCSZ$ are parallelogram. Prove that $GR=GS$. [i]Proposed by Hossein Karke Abadi, Iran[/i]

For every ordered pair of integers $(i,j)$, not necessarily positive, we wish to select a point $P_{i,j}$ in the Cartesian plane whose coordinates lie inside the unit square defined by \[ i < x < i+1, \qquad j < y < j+1. \] Find all real numbers $c > 0$ for which it's possible to choose these points such that for all integers $i$ and $j$, the (possibly concave or degenerate) quadrilateral $P_{i,j} P_{i+1,j} P_{i+1,j+1} P_{i,j+1}$ has perimeter strictly less than $c$. [i]Karthik Vedula[/i]

Find all polynomials $P(x)$ , such that $$P(x^3+1)=\left(P (x+1)\right)^3$$

Find the number of root for $\int_0^{\frac{\pi}{2}} e^x\cos (x+a)\ dx=0$ at $0\leq a <2\pi$

Let $ABC$ be an acute triangle with $AB < AC$. The points $A_1$ and $A_2$ are located on the line $BC$ so that $AA_1$ and $AA_2$ are the inner and outer angle bisectors at $A$ for the triangle $ABC$. Let $A_3$ be the mirror image $A_2$ with respect to $C$, and let $Q$ be a point on $AA_1$ such that $\angle A_1QA_3 = 90^o$. Show that $QC // AB$.

$O$ is a point inside triangle $ABC$ such that $OA=OB+OC$. Suppose $B',C'$ be midpoints of arcs $\overarc{AOC}$ and $AOB$. Prove that circumcircles $COC'$ and $BOB'$ are tangent to each other.

An equilateral triangle $ ABC $ is inscribed in a circle; prove that if $ M $ is any point of the circle, then one of the distances $ MA $, $ MB $, $ MC $ is equal to the sum of the other two.

Let $AH$ be an altitude of an acute-angled triangle $ABC$. Points $K$ and $L$ are the projections of $H$ onto sides $AB$ and $AC$. The circumcircle of $ABC$ meets line $KL$ at points $P$ and $Q$, and meets line $AH$ at points $A$ and $T$. Prove that $H$ is the incenter of triangle $PQT$. (M.Plotnikov)

You have $128$ teams in a single elimination tournament. The Engineers and the Crimson are two of these teams. Each of the $128$ teams in the tournament is equally strong, so during each match, each team has an equal probability of winning. Now, the $128$ teams are randomly put into the bracket. What is the probability that the Engineers play the Crimson sometime during the tournament?

[b]9.[/b] Let $w=f(x)$ be regular in $ \left | z \right |\leq 1$. For $0\leq r \leq 1$, denote by c, the image by $f(z)$ of the circle $\left | z \right | = r$. Show that if the maximal length of the chords of $c_{1}$ is $1$, then for every $r$ such that $0\leq r \leq 1$, the maximal length of the chords of c, is not greater than $r$. [b](F. 1)[/b]

a) Let $\{x_n\}$ be a sequence with $x_n \in [0,1]$ for any $n$. Prove that there exists $C > 0$ such that for every positive integer $r$, there exists $m \geq 1$ and $n > m + r$ that satisfy $(n-m)|x_n-x_m| \leq C$. b) Prove that for every $C > 0$, there exists a sequence $\{x_n\}$ with $x_n \in [0,1]$ for all $n$ and an integer $r$ such that, if $m \geq 1$ and $n > m+r$, then $(n-m)|x_n-x_m| > C.$

Let $I$ be the incenter of a triangle $ABC$. $D,E,F$ are the symmetric points of $I$ with respect to $BC,AC,AB$ respectively. Knowing that $D,E,F,B$ are concyclic,find all possible values of $\angle B$.

For each positive integer $n$, let $f(n)$ be a positive integer. Show that if $f(n + 1) > f(f(n))$ for every positive integer n, then $f(x) = x$ for all positive integers $x$.