Let $n>1$ be an integer. Find all non-constant real polynomials $P(x)$ satisfying , for any real $x$ , the identy \[P(x)P(x^2)P(x^3)\cdots P(x^n)=P(x^{\frac{n(n+1)}{2}})\]

Farmer James wishes to cover a circle with circumference $10\pi$ with six different types of colored arcs. Each type of arc has radius $5$, has length either $\pi$ or $2\pi$, and is colored either red, green, or blue. He has an unlimited number of each of the six arc types. He wishes to completely cover his circle without overlap, subject to the following conditions: [list][*] Any two adjacent arcs are of different colors. [*] Any three adjacent arcs where the middle arc has length $\pi$ are of three different colors. [/list] Find the number of distinct ways Farmer James can cover his circle. Here, two coverings are equivalent if and only if they are rotations of one another. In particular, two colorings are considered distinct if they are reflections of one another, but not rotations of one another. [i]Proposed by James Lin.[/i]

Let $ABC$ be a triangle with $AB=13,BC=15,AC=14$, circumcenter $O$, and orthocenter $H$, and let $M,N$ be the midpoints of minor and major arcs $BC$ on the circumcircle of $ABC$. Suppose $P\in AB, Q\in AC$ satisfy that $P,O,Q$ are collinear and $PQ||AN$, and point $I$ satisfies $IP\perp AB,IQ\perp AC$. Let $H'$ be the reflection of $H$ over line $PQ$, and suppose $H'I$ meets $PQ$ at a point $T$. If $\frac{MT}{NT}$ can be written in the form $\frac{\sqrt{m}}{n}$ for positive integers $m,n$ where $m$ is not divisible by the square of any prime, then find $100m+n$. [i]Proposed by Vincent Huang[/i]

Let $ABCDE$ be a right quadrangular pyramid with vertex $E$ and height $EO$. Point $S$ divides this height in the ratio $ES: SO=m$. In which ratio does the plane $(ABC)$ divide the lateral area of the pyramid.

The following sum of three four digits numbers is divisible by $75$, $7a71 + 73b7 + c232$, where $a, b, c$ are decimal digits. Find the necessary conditions in $a, b, c$.

Let $n$ be an integer greater than or equal to 3. Prove that there is a set of $n$ points in the plane such that the distance between any two points is irrational and each set of three points determines a non-degenerate triangle with a rational area.

In the isosceles triangle $ABC$, $ (AB = BC)$ the bisector $AD$ was drawn, and in the triangle $ABD$ the bisector $DE$ was drawn. Find the values of the angles of the triangle $ABC$, if it is known that the bisectors of the angles $ABD$ and $AED$ intersect on the line $AD$. (Fedak Ivan)

Given an acute triangle $ABC$. The incircle with center $I$ touches $BC,CA,AB$ at $D,E,F$ respectively. Let $M,N$ be the midpoint of the minor arc of $AB$ and $AC$ respectively. Prove that $M,F,E,N$ are collinear if and only if $\angle BAC =90$$^{\circ}$

Prove that there exists an infinite sequence of perfect squares with the following properties: (i) The arithmetic mean of any two consecutive terms is a perfect square, (ii) Every two consecutive terms are coprime, (iii) The sequence is strictly increasing.

Determine the angular acceleration of the disk when $t=\text{2.0 s}$. (A) $\text{-12 rad/s}^2$. (B) $\text{-8 rad/s}^2$. (C) $\text{-4 rad/s}^2$. (D) $\text{-2 rad/s}^2$. (E) $\text{0 rad/s}^2$.

For all positive integers $r$ and $s,$ let $\text{Top}(r,s)$ denote the top number (i.e., numerator) when $r$ $s$ is written in simplified form. For instance, $\text{Top}(20,24) = 5.$ Compute the number of ordered pairs of positive integers $(a,z)$ such that $200 \le a \le 300$ and $\text{Top}(a,z) = \text{Top}(z,a-1).$

In a triangle $ABC$, let $D$ be the point on the segment $BC$ such that $AB+BD=AC+CD$. Suppose that the points $B$, $C$ and the centroids of triangles $ABD$ and $ACD$ lie on a circle. Prove that $AB=AC$.

Let $p>2$ be a prime number and $n$ a positive integer. Prove that $pn^2$ has at most one positive divisor $d$ for which $n^2+d$ is a square number.

Let $A$ be the set of the $16$ first positive integers. Find the least positive integer $k$ satisfying the condition: In every $k$-subset of $A$, there exist two distinct $a, b \in A$ such that $a^2 + b^2$ is prime.

Function $f$ satisfies the equation $f(\cos x) = \cos (17x)$. Prove that it also satisfies the equation $f(\sin x) = \sin (17x)$.

Let $\mathbb{Q}^+$ denote the set of all positive rational numbers. Determine all functions $f:\mathbb{Q}^+\to \mathbb{Q}^+$ satisfying$$f(x^2f(y)^2)=f(x^2)f(y)$$for all $x,y\in\mathbb{Q}^+$

The number of cubic feet in the volume of a cube is the same as the number of square inches in its surface area. The length of the edge expressed as a number of feet is $\textbf{(A) }6\qquad\textbf{(B) }864\qquad\textbf{(C) }1728\qquad\textbf{(D) }6\times 1728\qquad \textbf{(E) }2304$

Let $ABC$ be an acute triangle. Denote by $H$ and $M$ the orthocenter of $ABC$ and the midpoint of side $BC,$ respectively. Let $Y$ be a point on $AC$ such that $YH$ is perpendicular to $MH$ and let $Q$ be a point on $BH$ such that $QA$ is perpendicular to $AM.$ Let $J$ be the second point of intersection of $MQ$ and the circle with diameter $MY.$ Prove that $HJ$ is perpendicular to $AM.$ (Steve Dinh)

Suppose that $\displaystyle{{v_1},{v_2},...,{v_d}}$ are unit vectors in $\displaystyle{{{\Bbb R}^d}}$. Prove that there exists a unitary vector $\displaystyle{u}$ such that $\displaystyle{\left| {u \cdot {v_i}} \right| \leq \frac{1}{{\sqrt d }}}$ for $\displaystyle{i = 1,2,...,d}$. [b]Note.[/b] Here $\displaystyle{ \cdot }$ denotes the usual scalar product on $\displaystyle{{{\Bbb R}^d}}$. [i]Proposed by Tomasz Tkocz, University of Warwick.[/i]

Let $P_n(x)=a_0 + a_1x + \cdots + a_nx^n$, with $n \geq 2$, be a real-coefficient polynomial. Prove that if there exists $a > 0$ such that \begin{align*} P_n(x) = (x + a)^2 \left( \sum_{i=0}^{n-2} b_i x^i \right), \end{align*} where $b_i$ are positive real numbers, then there exists some $i$, with $1 \leq i \leq n-1$, such that \[a_i^2 - 4a_{i-1}a_{i+1} \leq 0.\]

Find all two-digit numbers $\overline {ab}$ such that $\overline {ab} + \overline {ba}$ is a perfect square.

$HOW,BOW,$ and $DAH$ are equilateral triangles in a plane such that $WO=7$ and $AH=2$. Given that $D,A,B$ are collinear in that order, find the length of $BA$.

Let $ ABC$ be an equilateral triangle and $ \Gamma$ the semicircle drawn exteriorly to the triangle, having $ BC$ as diameter. Show that if a line passing through $ A$ trisects $ BC,$ it also trisects the arc $ \Gamma.$

Two non-rolling circles $C_1$ and $C_2$ with centers $O_1$ and $O_2$ and radii $2R$ and $R$, respectively, are given on the plane. Find the locus of the centers of gravity of triangles in which one vertex lies on $C_1$ and the other two lie on $C_2$. (B. Frenkin)

Eight children, all of different heights, must form an orderly line from smallest to largest. We will say that the row has exactly one error if there is a child that is immediately behind another taller than it, and everyone else (except the first in line) is immediately behind a shorter one. of how many ways the eight children can line up with exactly one mistake?