Let $D,E,F$ be points on the sides $BC,CA,AB$ respectively of a triangle $ABC$ such that $BD=CE=AF$ and $\angle BDF=\angle CED=\angle AFE.$ Show that $\triangle ABC$ is equilateral.

Let $ABCD$ be a convex quadrilateral and points $E$ and $F$ on sides $AB,CD$ such that \[\tfrac{AB}{AE}=\tfrac{CD}{DF}=n\] If $S$ is the area of $AEFD$ show that ${S\leq\frac{AB\cdot CD+n(n-1)AD^2+n^2DA\cdot BC}{2n^2}}$

Let $ABC$ be a triangle with $AB=AC$. Also, let $D\in[BC]$ be a point such that $BC>BD>DC>0$, and let $\mathcal{C}_1,\mathcal{C}_2$ be the circumcircles of the triangles $ABD$ and $ADC$ respectively. Let $BB'$ and $CC'$ be diameters in the two circles, and let $M$ be the midpoint of $B'C'$. Prove that the area of the triangle $MBC$ is constant (i.e. it does not depend on the choice of the point $D$). [i]Greece[/i]

Let $a_{n}$ be the area surrounded by the curves $y=e^{-x}$ and the part of $y=e^{-x}|\cos x|,\ (n-1)\pi \leq x\leq n\pi \ (n=1,\ 2,\ 3,\ \cdots).$ Evaluate $\lim_{n\to\infty}(a_{1}+a_{2}+\cdots+a_{n}).$

Let $ ABC$ be an equilateral triangle, take line $ t $ such that $ t\parallel BC $ and $ t $ passes through $ A $. Let point $ D $ be on side $ AC $ , the bisector of angle $ ABD $ intersects line $ t $ in point $ E $. Prove that $ BD = CD + AE $.

Prove that if $ n $ is a natural number and the angle $ \alpha $ is not a multiple of $ \frac{180^{\circ}}{2^n} $, then $$\frac{1}{\sin 2\alpha} + \frac{1}{\sin 4\alpha} + \frac{1}{\sin 8\alpha} + ... + = ctg \alpha - ctg 2^n \alpha.$$

Let $ f : [0, \plus{}\infty) \to \mathbb R$ be a differentiable function. Suppose that $ \left|f(x)\right| \le 5$ and $ f(x)f'(x) \ge \sin x$ for all $ x \ge 0$. Prove that there exists $ \lim_{x\to\plus{}\infty}f(x)$.

Let $D$ be a point inside the equilateral triangle $\triangle ABC$ such that $|AD|=\sqrt{2}, |BD|=3, |CD|=\sqrt{5}$. $m(\widehat{ADB}) = ?$ $\textbf{(A)}\ 120^{\circ} \qquad\textbf{(B)}\ 105^{\circ} \qquad\textbf{(C)}\ 100^{\circ} \qquad\textbf{(D)}\ 95^{\circ} \qquad\textbf{(E)}\ 90^{\circ}$

In acute-angled triangle $ABC$, let $H$ be the foot of perpendicular from $A$ to $BC$ and also suppose that $J$ and $I$ are excenters oposite to the side $AH$ in triangles $ABH$ and $ACH$. If $P$ is the point that incircle touches $BC$, prove that $I,J,P,H$ are concyclic.

For a given pair of values $x$ and $y$ satisfying $x = \sin \alpha , y = \sin \beta$ , there can be four different values of $z = \sin( \alpha +\beta )$. (a) Set up a relation between $x, y$ and $z$ not involving trigonometric functions or radicals. (b) Find those pairs of values $(x, y)$ for which $z = \sin (\alpha +\beta)$ takes on fewer than four distinct values.

Triangle $ABC$ is isosceles with $AC = BC$ and $\angle ACB = 106^\circ$. Point $M$ is in the interior of the triangle so that $\angle MAC = 7^\circ$ and $\angle MCA = 23^\circ$. Find the number of degrees in $\angle CMB$.

Prove that : $\left|\sum_{i=0}^n \left(1-\pi \sin \frac{i\pi}{4n}\cos \frac{i\pi}{4n}\right)\right|<1.$

On the plane find the locus of points whose coordinates satisfy $sin(x + y) = 0$.

Consider a sequence of circles $K_1,K_2,K_3,K_4, \ldots$ of radii $r_1, r_2, r_3, r_4, \ldots$ , respectively, situated inside a triangle $ABC$. The circle $K_1$ is tangent to $AB$ and $AC$; $K_2$ is tangent to $K_1$, $BA$, and $BC$; $K_3$ is tangent to $K_2$, $CA$, and $CB$; $K_4$ is tangent to $K_3$, $AB$, and $AC$; etc. (a) Prove the relation \[r_1 \cot \frac 12 A+ 2 \sqrt{r_1r_2} + r_2 \cot \frac 12 B = r \left(\cot \frac 12 A + \cot \frac 12 B \right) \] where $r$ is the radius of the incircle of the triangle $ABC$. Deduce the existence of a $t_1$ such that \[r_1=r \cot \frac 12 B \cot \frac 12 C \sin^2 t_1\] (b) Prove that the sequence of circles $K_1,K_2, \ldots $ is periodic.

Find the area of rhombus $ABCD$ given that the radii of the circles circumscribed around triangles $ABD$ and $ACD$ are $12.5$ and $25$, respectively.

Let $ABC$ be a triangle such that it's circumcircle radius is equal to the radius of outer inscribed circle with respect to $A$. Suppose that the outer inscribed circle with respect to $A$ touches $BC,AC,AB$ at $M,N,L$. Prove that $O$ (Center of circumcircle) is the orthocenter of $MNL$.

Solve with full solution: \[\left\{\begin{matrix}\sqrt{(\sin x)^2+\frac{1}{(\sin x)^2}}+\sqrt{(\cos y)^2+\frac{1}{(\cos y)^2}}=\sqrt\frac{20y}{x+y} \\\sqrt{(\sin y)^2+\frac{1}{(\sin y)^2}}+\sqrt{(\cos x)^2+\frac{1}{(\cos x)^2}}=\sqrt\frac{20x}{x+y}\end{matrix}\right. \]

Let $ABCDEF$ be a convex hexagon and it`s diagonals have one common point $M$. It is known that the circumcenters of triangles $MAB,MBC,MCD,MDE,MEF,MFA$ lie on a circle. Show that the quadrilaterals $ABDE,BCEF,CDFA$ have equal areas.

An infinitely long slab of glass is rotated. A light ray is pointed at the slab such that the ray is kept horizontal. If $\theta$ is the angle the slab makes with the vertical axis, then $\theta$ is changing as per the function \[ \theta(t) = t^2, \]where $\theta$ is in radians. Let the $\emph{glassious ray}$ be the ray that represents the path of the refracted light in the glass, as shown in the figure. Let $\alpha$ be the angle the glassious ray makes with the horizontal. When $\theta=30^\circ$, what is the rate of change of $\alpha$, with respect to time? Express your answer in radians per second (rad/s) to 3 significant figures. Assume the index of refraction of glass to be $1.50$. Note: the second figure shows the incoming ray and the glassious ray in cyan. [asy] fill((6,-2-sqrt(3))--(3, -2)--(3+4/sqrt(3),2)--(6+4/sqrt(3),2-sqrt(3))--cycle, gray(0.7)); draw((3+2/sqrt(3),0)--(10,0),linetype("4 4")+grey); draw((0,0)--(3+2/sqrt(3),0)); draw((3+2/sqrt(3), 0)--(7+2/sqrt(3), -1)); arrow((6.5+2/sqrt(3), -7/8), dir(180-14.04), 9); draw((3,-2)--(3,2), linetype("4 4")); draw((6,-2-sqrt(3))--(3,-2)--(3+2/sqrt(3),0)--(3+4/sqrt(3), 2)--(6+4/sqrt(3), 2-sqrt(3))); draw(anglemark((3+2/sqrt(3),0),(3,-2),(3,0),15)); label("$\theta$", (3.2, -1.6), N, fontsize(8)); label("$\alpha$", (6, -0.2), fontsize(8)); [/asy] [asy] fill((6,-2-sqrt(3))--(3, -2)--(3+4/sqrt(3),2)--(6+4/sqrt(3),2-sqrt(3))--cycle, gray(0.7)); draw((3+2/sqrt(3),0)--(10,0),linetype("4 4")+grey); draw((0,0)--(3+2/sqrt(3),0), cyan); draw((3+2/sqrt(3), 0)--(7+2/sqrt(3), -1), cyan); arrow((6.5+2/sqrt(3), -7/8), dir(180-14.04), 9, cyan); draw((3,-2)--(3,2), linetype("4 4")); draw((6,-2-sqrt(3))--(3,-2)--(3+2/sqrt(3),0)--(3+4/sqrt(3), 2)--(6+4/sqrt(3), 2-sqrt(3))); draw(anglemark((3+2/sqrt(3),0),(3,-2),(3,0),15)); label("$\theta$", (3.2, -1.6), N, fontsize(8)); label("$\alpha$", (6, -0.2), fontsize(8)); [/asy] [i]Problem proposed by Ahaan Rungta[/i]

Prove that if the angles $ A $, $ B $, $ C $ of a triangle satisfy the equation $$\cos 3A + \cos 3B + \cos 3C = 1,$$ then one of these angles equals $120^\circ $.

Let be two fixed points $ B,C. $ Find the locus of the spatial points $ A $ such that $ ABC $ is a nondegenerate triangle and the expression $$ R^2 (A)\cdot\sin \left( 2\angle ABC\right)\cdot\sin \left( 2\angle BCA\right) $$ has the greatest value possible, where $ R(A) $ denotes the radius of the excirlce of $ ABC. $

A right-triangular wooden block of mass $M$ is at rest on a table, as shown in figure. Two smaller wooden cubes, both with mass $m$, initially rest on the two sides of the larger block. As all contact surfaces are frictionless, the smaller cubes start sliding down the larger block while the block remains at rest. What is the normal force from the system to the table? $\textbf{(A) } 2mg\\ \textbf{(B) } 2mg + Mg\\ \textbf{(C) } mg + Mg\\ \textbf{(D) } Mg + mg( \sin \alpha + \sin \beta)\\ \textbf{(E) } Mg + mg( \cos \alpha + \cos \beta)$

Evaluate $50(\cos 39^{\circ}\cos21^{\circ}+\cos129^{\circ}\cos69^{\circ})$

Let $n$ be the $200$th smallest positive real solution to the equation $x- \frac{\pi}{2} =\ tan x$. Find the greatest integer that does not exceed $\frac{n}{2}$.

Show that $\tan \left( \frac{\pi}{m} \right)$ is irrational for all positive integers $m \ge 5$.