$P,Q,R$ are non-zero polynomials that for each $z\in\mathbb C$, $P(z)Q(\bar z)=R(z)$. a) If $P,Q,R\in\mathbb R[x]$, prove that $Q$ is constant polynomial. b) Is the above statement correct for $P,Q,R\in\mathbb C[x]$?

Segment $AB$ of length $13$ is the diameter of a semicircle. Points $C$ and $D$ are located on the semicircle but not on segment $AB$. Segments $AC$ and $BD$ both have length $5$. Given that the length of $CD$ can be expressed as $\frac{a}{b}$ where $a$ and $b$ are relatively prime positive integers, find $a +b$.

Regular hexagon $ABCDEF$ has side length $2$. Points $M$ and $N$ lie on $BC$ and $DE$, respectively. Find the minimum possible value of $(AM + MN + NA)^2$. [i]Lightning 3.4[/i]

Find the least possible value for the fraction $$\frac{lcm(a,b)+lcm(b,c)+lcm(c,a)}{gcd(a,b)+gcd(b,c)+gcd(c,a)}$$ over all distinct positive integers $a, b, c$. By $lcm(x, y)$ we mean the least common multiple of $x, y$ and by $gcd(x, y)$ we mean the greatest common divisor of $x, y$.

Let $ABC$ be a triangle, and let $\omega_1,\omega_2$ be centered at $O_1$, $O_2$ and tangent to line $BC$ at $B$, $C$ respectively. Let line $AB$ intersect $\omega_1$ again at $X$ and let line $AC$ intersect $\omega_2$ again at $Y$. If $Q$ is the other intersection of the circumcircles of triangles $ABC$ and $AXY$, then prove that lines $AQ$, $BC$, and $O_1O_2$ either concur or are all parallel. [i]Advaith Avadhanam[/i]

In a unit square $ABCD$, find the minimum of $\sqrt2 AP + BP + CP$ where $P$ is a point inside $ABCD$.

$\textbf{(a)}$ Let $a,b \in \mathbb{R}$ and $f,g :\mathbb{R}\rightarrow \mathbb{R}$ be differentiable functions. Consider the function $$h(x)=\begin{vmatrix} a &b &x\\ f(a) &f(b) &f(x)\\ g(a) &g(b) &g(x)\\ \end{vmatrix}$$ Prove that $h$ is differentiable and find $h'$. \\ \\ $\textbf{(b)}$ Let $n\in \mathbb{N}$, $n\geq 3$, take $n-1$ pairwise distinct real numbers $a_1<a_2<\dots <a_{n-1}$ with sum $\sum_{i=1}^{n-1}a_i = 0$, and consider $n-1$ functions $f_1,f_2,...f_{n-1}:\mathbb{R}\rightarrow \mathbb{R}$, each of them $n-2$ times differentiable over $\mathbb{R}$. Prove that there exists $a\in (a_1,a_{n-1})$ and $\theta, \theta_1,...,\theta_{n-1}\in \mathbb{R}$, not all zero, such that $$\sum_{k=1}^{n-1} \theta_k a_k=\theta a$$ and, at the same time, $$\sum_{k=1}^{n-1}\theta_kf_i(a_k)=\theta f_i^{(n-2)}(a)$$ for all $i\in\{1,2...,n-1\} $. \\ \\ [i](Sergiu Novac)[/i]

An equilateral triangle is divided into $25$ equal equilateral triangles labelled by $1$ through $25$. Prove that one can find two triangles having a common side whose labels differ by more than $3$.

Let $a,b,c,d$ be positive real numbers with $a+b+c+d=2$. Prove the following inequality: $$\frac{(a+c)^{2}}{ad+bc}+\frac{(b+d)^{2}}{ac+bd}+4\geq 4\left ( \frac{a+b+1}{c+d+1}+\frac{c+d+1}{a+b+1} \right).$$ [i]Proposed by Mohammad Jafari[/i]

Let $ P(x)$ be a nonzero polynomial such that, for all real numbers $ x$, $ P(x^2 \minus{} 1) \equal{} P(x)P(\minus{}x)$. Determine the maximum possible number of real roots of $ P(x)$.

Let $ABC$ be a triangle and $I$ its incenter. The lines $BI$ and $CI$ intersect the circumcircle of $ABC$ again at $M$ and $N$, respectively. Let $C_1$ and $C_2$ be the circumferences of diameters $NI$ and $MI$, respectively. The circle $C_1$ intersects $AB$ at $P$ and $Q$, and the circle $C_2$ intersects $AC$ at $R$ and $S$. Show that $P$, $Q$, $R$ and $S$ are concyclic.

Let $n$ be a given positive integer. In the Cartesian plane, each lattice point with nonnegative coordinates initially contains a butterfly, and there are no other butterflies. The [i]neighborhood[/i] of a lattice point $c$ consists of all lattice points within the axis-aligned $(2n+1) \times (2n+1)$ square entered at $c$, apart from $c$ itself. We call a butterfly [i]lonely[/i], [i]crowded[/i], or [i]comfortable[/i], depending on whether the number of butterflies in its neighborhood $N$ is respectively less than, greater than, or equal to half of the number of lattice points in $N$. Every minute, all lonely butterflies fly away simultaneously. This process goes on for as long as there are any lonely butterflies. Assuming that the process eventually stops, determine the number of comfortable butterflies at the final state.

Bob plants two saplings. Each day, each sapling has a $1/3$ chance of instantly turning into a tree. Given that the expected number of days it takes both trees to grow is $m/n$ , where $m$ and $n$ are relatively prime positive integers, find $m +n$. [i]Proposed by Powell Zhang[/i]

[b]p13.[/b] Suppose $\overline{AB}$ is a radius of a circle. If a point $C$ is chosen uniformly at random inside the circle, what is the probability that triangle $ABC$ has an obtuse angle? [b]p14.[/b] Find the second smallest positive integer $c$ such that there exist positive integers $a$ and $b$ satisfying the following conditions: $\bullet$ $5a = b = \frac{c}{5} + 6$. $\bullet$ $a + b + c$ is a perfect square. [b]p15.[/b] A spotted lanternfly is at point $(0, 0, 0)$, and it wants to land on an unassuming CMU student at point $(2, 3, 4)$. It can move one unit at a time in either the $+x$, $+y$, or $+z$ directions. However, there is another student waiting at $(1, 2, 3)$ who will stomp on the lanternfly if it passes through that point. How many paths can the lanternfly take to reach its target without getting stomped? PS. You should use hide for answers.

Find the number of unordered choices of $ k $ lists, each having $ m $ distinct ordered objects, among a number of $ mn $ objects. [i]Cătălin Țigăeru[/i]

Let $ABCDEF$ be a regular hexagon and let point $O$ be the center of the hexagon. How many ways can you color these seven points either red or blue such that there doesn’t exist any equilateral triangle with vertices of all the same color?

Prove that if two triangles are inscribed in the same circle, then their incircles are not strictly contained one into each other.

In triangle $ABC$, side $AB$ has length $10$, and the $A$- and $B$-medians have length $9$ and $12$, respectively. Compute the area of the triangle. [i]Proposed by Yannick Yao[/i]

Let $a, b,c$ real numbers that are greater than $ 0$ and less than $1$. Show that there is at least one of these three values $ab(1-c)^2$, $bc(1-a)^2$ , $ca(1- b)^2$ which is less than or equal to $\frac{1}{16}$ .

Supppose that there are roads $AB$ and $CD$ but there are no roads $BC$ and $AD$ between four cities $A$, $B$, $C$, and $D$. Define [i]restructing[/i] to be the changing a pair of roads $AB$ and $CD$ to the pair of roads $BC$ and $AD$. Initially there were some cities in a country, some of which were connected by roads and for every city there were exactly $100$ roads starting in it. The minister drew a new scheme of roads, where for every city there were also exactly $100$ roads starting in it. It's known also that in both schemes there were no cities connected by more than one road. Prove that it's possible to obtain the new scheme from the initial after making a finite number of restructings. [i] (Т. Зубов)[/i] [hide=Thanks]Thanks to the user Vlados021 for translating the problem.[/hide]

If the sum of digits in decimal representaion of positive integer $n$ is $111$ and the sum of digits in decimal representation of $7002n$ is $990$, what is the sum of digits in decimal representation of $2003n$? $ \textbf{(A)}\ 309 \qquad\textbf{(B)}\ 330 \qquad\textbf{(C)}\ 550 \qquad\textbf{(D)}\ 555 \qquad\textbf{(E)}\ \text{None of the preceding} $

Albert has a very large bag of candies and he wants to share all of it with his friends. At first, he splits the candies evenly amongst his $20$ friends and himself and he finds that there are five left over. Ante arrives, and they redistribute the candies evenly again. This time, there are three left over. If the bag contains over $500$ candies, what is the fewest number of candies the bag can contain?

Let a sequences: $ x_0\in [0;1],x_{n\plus{}1}\equal{}\frac56\minus{}\frac43 \Big|x_n\minus{}\frac12\Big|$. Find the "best" $ |a;b|$ so that for all $ x_0$ we have $ x_{2009}\in [a;b]$

Find the number of all unordered pairs $\{A,B \}$ of subsets of an $8$-element set, such that $A\cap B \neq \emptyset$ and $\left |A \right | \neq \left |B \right |$.

Let $S$ be the set of all polygonal areas in a plane. Prove that there is a function $f : S \to (0,1)$ which satisfies $f(S_1 \cup S_2) = f(S_1)+ f(S_2)$ for any $S_1,S_2 \in S$ which have common points only on their borders