$x,y\in\left[-\frac{\pi}{4},\frac{\pi}{4}\right],a\in\mathbb{R}$. If $x^3+\sin x-2a=0,4y^3+\sin y \cos y+a=0$, then $\cos (x+2y)=$________.

What is the largest possible radius of a circle contained in a 4-dimensional hypercube of side length 1?

Given triangle $ ABC$. The incircle of triangle $ ABC$ is tangent to $ BC,CA,AB$ at $ D,E,F$ respectively. Construct point $ G$ on $ EF$ such that $ DG$ is perpendicular to $ EF$. Prove that $ \frac{FG}{EG} \equal{} \frac{BF}{CE}$.

Convex quadrilateral $ABCD$ has sides $AB=BC=7$, $CD=5$, and $AD=3$. Given additionally that $m\angle ABC=60^\circ$, find $BD$.

Find all triangles whose side lengths are consecutive integers, and one of whose angles is twice another.

Consider a triangle such that \[\sin^2\alpha+\sin^2\beta+\sin^2\gamma=2.\] Show that the triangle is right.

Given an inscribed quadrilateral $ABCD$, which marked the midpoints of the points $M, N, P, Q$ in this order. Let diagonals $AC$ and $BD$ intersect at point $O$. Prove that the triangle $OMN, ONP, OPQ, OQM$ have the same radius of the circles

Let $ABC$ be a triangle with circumcenter $O$, incenter $I$, $\angle B=45^\circ$, and $OI\parallel BC$. Find $\cos\angle C$.

Find all $x$ for which the inequality holds $$\sqrt{7+8x-16x^2} \ge 2^{\cos^2 \pi x}+2^{\sin ^2 \pi x}$$

Prove that if $f$ is a non-constant real-valued function such that for all real $x$, $f(x+1) + f(x-1) = \sqrt{3} f(x)$, then $f$ is periodic. What is the smallest $p$, $p > 0$ such that $f(x+p) = f(x)$ for all $x$?

$f(x)$ is a monic polynomial of degree $2$ with integer coefficients such that $f(x)$ doesn't have any real roots and also $f(0)$ is a square-free integer (and is not $1$ or $-1$). Prove that for every integer $n$ the polynomial $f(x^n)$ is irreducible over $\mathbb Z[x]$. [i]proposed by Mohammadmahdi Yazdi[/i]

Let $ ABC$ be a triangle, let $ E, F$ be the tangency points of the incircle $ \Gamma(I)$ to the sides $ AC$, respectively $ AB$, and let $ M$ be the midpoint of the side $ BC$. Let $ N \equal{} AM \cap EF$, let $ \gamma(M)$ be the circle of diameter $ BC$, and let $ X, Y$ be the other (than $ B, C$) intersection points of $ BI$, respectively $ CI$, with $ \gamma$. Prove that \[ \frac {NX} {NY} \equal{} \frac {AC} {AB}. \] [i]Cosmin Pohoata[/i]

Let $f:S\to\mathbb{R}$ be the function from the set of all right triangles into the set of real numbers, defined by $f(\Delta ABC)=\frac{h}{r}$, where $h$ is the height with respect to the hypotenuse and $r$ is the inscribed circle's radius. Find the image, $Im(f)$, of the function.

Let $A_1, A_2, \ldots, A_n$ be an $n$ -sided regular polygon. If $\frac{1}{A_1 A_2} = \frac{1}{A_1 A_3} + \frac{1}{A_1A_4}$, find $n$.

Point $O$ is inside triangle $ABC$ such that $\angle AOB = \angle BOC = \angle COA = 120^\circ .$ Prove that \[\frac{AO^2}{BC}+\frac{BO^2}{CA}+\frac{CO^2}{AB} \geq \frac{AO+BO+CO}{\sqrt 3}.\]

Jordi stands 20 m from a wall and Diego stands 10 m from the same wall. Jordi throws a ball at an angle of 30 above the horizontal, and it collides elastically with the wall. How fast does Jordi need to throw the ball so that Diego will catch it? Consider Jordi and Diego to be the same height, and both are on the same perpendicular line from the wall. $\textbf{(A) } 11 \text{ m/s}\\ \textbf{(B) } 15 \text{ m/s}\\ \textbf{(C) } 19 \text{ m/s}\\ \textbf{(D) } 30 \text{ m/s}\\ \textbf{(E) } 35 \text{ m/s}$

The points $D$ and $E$ on the circumcircle of an acute triangle $ABC$ are such that $AD=AE = BC$. Let $H$ be the common point of the altitudes of triangle $ABC$. It is known that $AH^{2}=BH^{2}+CH^{2}$. Prove that $H$ lies on the segment $DE$. [i]Proposed by D. Shiryaev[/i]

If $ \theta$ is an acute angle and $ \sin \frac12 \theta \equal{} \sqrt{\frac{x\minus{}1}{2x}}$, then $ \tan \theta$ equals $ \textbf{(A)}\ x \qquad \textbf{(B)}\ \frac1{x} \qquad \textbf{(C)}\ \frac{\sqrt{x\minus{}1}}{x\plus{}1} \qquad \textbf{(D)}\ \frac{\sqrt{x^2\minus{}1}}{x} \qquad \textbf{(E)}\ \sqrt{x^2\minus{}1}$

Let $ABC$ be a triangle, let $B_{1}$ be the midpoint of the side $AB$ and $C_{1}$ the midpoint of the side $AC$. Let $P$ be the point of intersection, other than $A$, of the circumscribed circles around the triangles $ABC_{1}$ and $AB_{1}C$. Let $P_{1}$ be the point of intersection, other than $A$, of the line $AP$ with the circumscribed circle around the triangle $AB_{1}C_{1}$. Prove that $2AP=3AP_{1}$.

Consider a variable point $P$ inside a given triangle $ABC$. Let $D$, $E$, $F$ be the feet of the perpendiculars from the point $P$ to the lines $BC$, $CA$, $AB$, respectively. Find all points $P$ which minimize the sum \[ {BC\over PD}+{CA\over PE}+{AB\over PF}. \]

Suppose that $\tan \alpha =\frac{p}{q}$, where $p$ and $q$ are integers and $q \neq 0$. Prove the number $\tan \beta$ for which $\tan 2\beta =\tan 3\alpha$ is rational only when $p^2 +q^2$ is the square of an integer.

Let $ABC$ be a triangle and its bisector at $A$ cuts its circumcircle at $D.$ Let $I$ be the incenter of triangle $ABC,$ $M$ be the midpoint of $BC,$ $P$ is the symmetric to $I$ with respect to $M$ (Assuming $P$ is in the circumcircle). Extend $DP$ until it cuts the circumcircle again at $N.$ Prove that among segments $AN, BN, CN$, there is a segment that is the sum of the other two.

Four lighthouses are located at points $ A$, $ B$, $ C$, and $ D$. The lighthouse at $ A$ is $ 5$ kilometers from the lighthouse at $ B$, the lighthouse at $ B$ is $ 12$ kilometers from the lighthouse at $ C$, and the lighthouse at $ A$ is $ 13$ kilometers from the lighthouse at $ C$. To an observer at $ A$, the angle determined by the lights at $ B$ and $ D$ and the angle determined by the lights at $ C$ and $ D$ are equal. To an observer at $ C$, the angle determined by the lights at $ A$ and $ B$ and the angle determined by the lights at $ D$ and $ B$ are equal. The number of kilometers from $ A$ to $ D$ is given by $ \displaystyle\frac{p\sqrt{r}}{q}$, where $ p$, $ q$, and $ r$ are relatively prime positive integers, and $ r$ is not divisible by the square of any prime. Find $ p\plus{}q\plus{}r$,

Find all real $x$ such that $$\sqrt{2+\frac{5}{2}\cos x}\leq\sin x.$$

Evaluate the following sum: $n \choose 1$ $\sin (a) +$ $n \choose 2$ $\sin (2a) +...+$ $n \choose n$ $\sin (na)$ (A) $2^n \cos^n \left(\frac{a}{2}\right)\sin \left(\frac{na}{2}\right)$ (B) $2^n \sin^n \left(\frac{a}{2}\right)\cos \left(\frac{na}{2}\right)$ (C) $2^n \sin^n \left(\frac{a}{2}\right)\sin \left(\frac{na}{2}\right)$ (D) $2^n \cos^n \left(\frac{a}{2}\right)\cos \left(\frac{na}{2}\right)$