Let $AB=AC$ in $\triangle ABC$, and let $D$ be a point on segment $AB$. The tangent at $D$ to the circumcircle $\omega$ of $BCD$ hits $AC$ at $E$. The other tangent from $E$ to $\omega$ touches it at $F$, and $G=BF \cap CD$, $H=AG \cap BC$. Prove that $BH=2HC$. [i]Proposed by David Stoner[/i]

Let $ABC$ be a triangle with $\angle C = 90^\circ$ and $CA \neq CB$. Let $CH$ be an altitude and $CL$ be an interior angle bisector. Show that for $X \neq C$ on the line $CL$, we have $\angle XAC \neq \angle XBC$. Also show that for $Y \neq C$ on the line $CH$ we have $\angle YAC \neq \angle YBC$. [i]Bulgaria[/i]

Square $ABCD$ is divided into four rectangles by $EF$ and $GH$. $EF$ is parallel to $AB$ and $GH$ parallel to $BC$. $\angle BAF = 18^\circ$. $EF$ and $GH$ meet at point $P$. The area of rectangle $PFCH$ is twice that of rectangle $AGPE$. Given that the value of $\angle FAH$ in degrees is $x$, find the nearest integer to $x$. [asy] size(100); defaultpen(linewidth(0.7)+fontsize(10)); pair D2(pair P) { dot(P,linewidth(3)); return P; } // NOTE: I've tampered with the angles to make the diagram not-to-scale. The correct numbers should be 72 instead of 76, and 45 instead of 55. pair A=(0,1), B=(0,0), C=(1,0), D=(1,1), F=intersectionpoints(A--A+2*dir(-76),B--C)[0], H=intersectionpoints(A--A+2*dir(-76+55),D--C)[0], E=F+(0,1), G=H-(1,0), P=intersectionpoints(E--F,G--H)[0]; draw(A--B--C--D--cycle); draw(F--A--H); draw(E--F); draw(G--H); label("$A$",D2(A),NW); label("$B$",D2(B),SW); label("$C$",D2(C),SE); label("$D$",D2(D),NE); label("$E$",D2(E),plain.N); label("$F$",D2(F),S); label("$G$",D2(G),W); label("$H$",D2(H),plain.E); label("$P$",D2(P),SE); [/asy]

Find the smallest positive $x$ for which holds the inequality $$\sin x \le \sin (x+1)\le \sin (x+2)\le sin (x+3)\le \sin (x+4) .$$

Prove that $$x\cos x \le \frac{\pi^2}{16}$$ for $0 \le x \le \frac{\pi}{2}$

There are $2n$ complex numbers that satisfy both $z^{28}-z^{8}-1=0$ and $|z|=1$. These numbers have the form $z_{m}=\cos\theta_{m}+i\sin\theta_{m}$, where $0\leq\theta_{1}<\theta_{2}< \dots <\theta_{2n}<360$ and angles are measured in degrees. Find the value of $\theta_{2}+\theta_{4}+\dots+\theta_{2n}$.

The circumcircle of triangle $ABC$ has centre $O$. $P$ is the midpoint of $\widehat{BAC}$ and $QP$ is the diameter. Let $I$ be the incentre of $\triangle ABC$ and let $D$ be the intersection of $PI$ and $BC$. The circumcircle of $\triangle AID$ and the extension of $PA$ meet at $F$. The point $E$ lies on the line segment $PD$ such that $DE=DQ$. Let $R,r$ be the radius of the inscribed circle and circumcircle of $\triangle ABC$, respectively. Show that if $\angle AEF=\angle APE$, then $\sin^2\angle BAC=\dfrac{2r}R$

Let $ABC$ be a triangle. Prove that there exists a point $D$ on the side $AB$ of the triangle $ABC$, such that $CD$ is the geometric mean of $AD$ and $DB$, iff the triangle $ABC$ satisfies the inequality $\sin A\sin B\le\sin^2\frac{C}{2}$. [hide="Comment"][i]Alternative formulation, from IMO ShortList 1974, Finland 2:[/i] We consider a triangle $ABC$. Prove that: $\sin(A) \sin(B) \leq \sin^2 \left( \frac{C}{2} \right)$ is a necessary and sufficient condition for the existence of a point $D$ on the segment $AB$ so that $CD$ is the geometrical mean of $AD$ and $BD$.[/hide]

Evaluate $\int_{-a}^a \left(x+\frac{1}{\sin x+\frac{1}{e^x-e^{-x}}}\right)dx\ (a>0)$. created by kunny

Let $a$ and $b$ be relatively prime positive integers such that $a/b$ is the maximum possible value of \[\sin^2x_1+\sin^2x_2+\sin^2x_3+\cdots+\sin^2x_{2007},\] where, for $1\leq i\leq 2007$, $x_i$ is a nonnegative real number, and \[x_1+x_2+x_3+\cdots+x_{2007}=\pi.\] Find the value of $a+b$.

In a triangle $ABC$ with $ \angle A = 36^o$ and $AB = AC$, the bisector of the angle at $C$ meets the oposite side at $D$. Compute the angles of $\triangle BCD$. Express the length of side $BC$ in terms of the length $b$ of side $AC$ without using trigonometric functions.

Find $\lim_{n\to\infty} n\int_0^{\frac{\pi}{2}} \frac{1}{(1+\cos x)^n}dx\ (n=1,\ 2,\ \cdots).$

Find $\lim_{n\to\infty}\sum_{k=0}^{\infty} \int_{2k\pi}^{(2k+1)\pi} xe^{-x}\sin x\ dx.$

Let $n$ and $k$ be positive integers such that $1 \leq n \leq N+1$, $1 \leq k \leq N+1$. Show that: \[ \min_{n \neq k} |\sin n - \sin k| < \frac{2}{N}. \]

Define $f(x)=\max \{\sin x, \cos x\} .$ Find at how many points in $(-2 \pi, 2 \pi), f(x)$ is not differentiable? [list=1] [*] 0 [*] 2 [*] 4 [*] $\infty$ [/list]

An equilateral triangle is given. A point lies on the incircle of this triangle. If the smallest two distances from the point to the sides of the triangle is $1$ and $4$, the sidelength of this equilateral triangle can be expressed as $\tfrac{a\sqrt b}c$ where $(a,c)=1$ and $b$ is not divisible by the square of an integer greater than $1$. Find $a+b+c$.

Acute-angled triangle $ABC$ is inscribed into circle $\Omega$. Lines tangent to $\Omega$ at $B$ and $C$ intersect at $P$. Points $D$ and $E$ are on $AB$ and $AC$ such that $PD$ and $PE$ are perpendicular to $AB$ and $AC$ respectively. Prove that the orthocentre of triangle $ADE$ is the midpoint of $BC$.

Let $ABC$ be an isosceles triangle with apex at $C.$ Let $D$ and $E$ be two points on the sides $AC$ and $BC$ such that the angle bisectors $\angle DEB$ and $\angle ADE$ meet at $F,$ which lies on segment $AB.$ Prove that $F$ is the midpoint of $AB.$

Points $E$, $F$, $G$, and $H$ lie on sides $AB$, $BC$, $CD$, and $DA$ of a convex quadrilateral $ABCD$ such that $\frac{AE}{EB} \cdot \frac{BF}{FC} \cdot \frac{CG}{GD} \cdot \frac{DH}{HA}=1$. Points $A$, $B$, $C$, and $D$ lie on sides $H_1E_1$, $E_1F_1$, $F_1G_1$, and $G_1H_1$ of a convex quadrilateral $E_1F_1G_1H_1$ such that $E_1F_1 \parallel EF$, $F_1G_1 \parallel FG$, $G_1H_1 \parallel GH$, and $H_1E_1 \parallel HE$. Given that $\frac{E_1A}{AH_1}=a$, express $\frac{F_1C}{CG_1}$ in terms of $a$. 

Find the function $f(x)$ such that $f(x)=\cos (2mx)+\int_{0}^{\pi}f(t)|\cos t|\ dt$ for positive inetger $m.$

Let $M$ be the intersection of the diagonals $AC$ and $BD$ of cyclic quadrilateral $ABCD$. If $|AB|=5$, $|CD|=3$, and $m(\widehat{AMB}) = 60^\circ$, what is the circumradius of the quadrilateral? $ \textbf{(A)}\ 5\sqrt 3 \qquad\textbf{(B)}\ \dfrac {7\sqrt 3}{3} \qquad\textbf{(C)}\ 6 \qquad\textbf{(D)}\ 4 \qquad\textbf{(E)}\ \sqrt{34} $

Let $f(x)$ be a differentiable function such that $f'(x)+f(x)=4xe^{-x}\sin 2x,\ \ f(0)=0.$ Find $\lim_{n\to\infty}\sum_{k=1}^{n}f(k\pi).$

In $\vartriangle ABC$ with edges $a, b$ and $c$, suppose $b + c = 6$ and the area $S$ is $a^2 - (b -c)^2$. Find the value of $\cos A$ and the largest possible value of $S$.

Let $ f(x)$ be a nonnegative, integrable function on $ (0,2\pi)$ whose Fourier series is $ f(x)\equal{}a_0\plus{}\sum_{k\equal{}1}^{\infty} a_k \cos (n_k x)$, where none of the positive integers $ n_k$ divides another. Prove that $ |a_k| \leq a_0$. [i]G. Halasz[/i]

Concentric circles $\Omega_1$ and $\Omega_2$ with radii $1$ and $100$, respectively, are drawn with center $O$. Points $A$ and $B$ are chosen independently at random on the circumferences of $\Omega_1$ and $\Omega_2$, respectively. Denote by $\ell$ the tangent line to $\Omega_1$ passing through $A$, and denote by $P$ the reflection of $B$ across $\ell$. Compute the expected value of $OP^2$. [i]Proposed by Lewis Chen[/i]