For $0<a,b,c,d<\frac{\pi}{2}$ is true that $$\cos 2a+\cos 2b+ \cos 2c+ \cos 2d= 4 (\sin a \sin b \sin c \sin d -\cos a \cos b \cos c \cos d)$$ Find all possible values of $a+b+c+d$

$ ABC$ is a triangle with $ \angle BAC \equal{} 10{}^\circ$, $ \angle ABC \equal{} 150{}^\circ$. Let $ X$ be a point on $ \left[AC\right]$ such that $ \left|AX\right| \equal{} \left|BC\right|$. Find $ \angle BXC$. $\textbf{(A)}\ 15^\circ \qquad\textbf{(B)}\ 20^\circ \qquad\textbf{(C)}\ 25^\circ \qquad\textbf{(D)}\ 30^\circ \qquad\textbf{(E)}\ 35^\circ$

Prove the inequality \[(n + 1)\cos\frac{\pi}{n + 1}- n\cos\frac{\pi}{n}> 1\] for all natural numbers $n \ge 2.$

Let $ABC$ be a scalene triangle and $k$ be its circumcircle. Let $t_A,t_B,t_C$ be the tangents to $k$ at $A, B, C,$ respectively. Prove that points $AB\cap t_C$, $CA\cap t_B$, and $BC\cap t_A$ exist, and that they are collinear.

Let $ABC$ be a triangle in a plane $\pi$. Find the set of all points $P$ (distinct from $A,B,C$ ) in the plane $\pi$ such that the circumcircles of triangles $ABP$, $BCP$, $CAP$ have the same radii.

$ D$ is a point on the edge $ BC$ of triangle $ ABC$ such that $ AD\equal{}\frac{BD^2}{AB\plus{}AD}\equal{}\frac{CD^2}{AC\plus{}AD}$. $ E$ is a point such that $ D$ is on $ [AE]$ and $ CD\equal{}\frac{DE^2}{CD\plus{}CE}$. Prove that $ AE\equal{}AB\plus{}AC$.

Let $F$ be the focus of parabola $y^2=2px(p>0)$, with directrix $l$ and two points $A,B$ on it. Knowing that $\angle AFB=\frac{\pi}{3}$, find the maximal value of $\frac{|MN|}{|AB|}$, where $M$ is the midpoint of $AB$ and $N$ is the projection of $M$ to $l$.

In triangle $ABC$ let $A'$, $B'$, $C'$ respectively be the midpoints of the sides $BC$, $CA$, $AB$. Furthermore let $L$, $M$, $N$ be the projections of the orthocenter on the three sides $BC$, $CA$, $AB$, and let $k$ denote the nine-point circle. The lines $AA'$, $BB'$, $CC'$ intersect $k$ in the points $D$, $E$, $F$. The tangent lines on $k$ in $D$, $E$, $F$ intersect the lines $MN$, $LN$ and $LM$ in the points $P$, $Q$, $R$. Prove that $P$, $Q$ and $R$ are collinear.

If $T$ and $T_1$ are two triangles with angles $x, y, z$ and $x_1, y_1, z_1$, respectively, prove the inequality \[\frac{\cos x_1}{\sin x}+\frac{\cos y_1}{\sin y}+\frac{\cos z_1}{\sin z} \leq \cot x+\cot y+\cot z.\]

Given that points $ D,E$ lie on the sidelines $ AB,BC$ of triangle $ ABC$, respectively, point $ P$ is in interior of triangle $ ABC$ such that $ PE \equal{} PC$ and $ \bigtriangleup DEP\sim \bigtriangleup PCA.$ Prove that $ BP$ is tangent of the circumcircle of triangle $ PAD.$

The sides of rectangle $ABCD$ have lengths 10 and 11. An equilateral triangle is drawn so that no point of the triangle lies outside $ABCD.$ The maximum possible area of such a triangle can be written in the form $p\sqrt{q}-r,$ where $p, q,$ and $r$ are positive integers, and $q$ is not divisible by the square of any prime number. Find $p+q+r.$

Suppose $a$ is a complex number such that \[a^2+a+\frac{1}{a}+\frac{1}{a^2}+1=0\] If $m$ is a positive integer, find the value of \[a^{2m}+a^m+\frac{1}{a^m}+\frac{1}{a^{2m}}\]

A simple pendulum experiment is constructed from a point mass $m$ attached to a pivot by a massless rod of length $L$ in a constant gravitational field. The rod is released from an angle $\theta_0 < \frac{\pi}{2}$ at rest and the period of motion is found to be $T_0$. Ignore air resistance and friction. At what angle $\theta_g$ during the swing is the tension in the rod the greatest? $\textbf{(A) } \text{The tension is greatest at } \theta_g = \theta_0.\\ \textbf{(B) } \text{The tension is greatest at }\theta_g = 0.\\ \textbf{(C) } \text{The tension is greatest at an angle } 0 < \theta_g < \theta_0.\\ \textbf{(D) } \text{The tension is constant.}\\ \textbf{(E) } \text{None of the above is true for all values of } \theta_0 \text{ with } 0 < \theta_{0} < \frac{\pi}{2}$

Let $ABCD$ be a convex circumscribed quadrilateral such that $\angle ABC+\angle ADC<180^{\circ}$ and $\angle ABD+\angle ACB=\angle ACD+\angle ADB$. Prove that one of the diagonals of quadrilateral $ABCD$ passes through the other diagonals midpoint.

Let $ ABCD$ be an isosceles trapezoid with $ \overline{AD}\parallel{}\overline{BC}$ whose angle at the longer base $ \overline{AD}$ is $ \dfrac{\pi}{3}$. The diagonals have length $ 10\sqrt {21}$, and point $ E$ is at distances $ 10\sqrt {7}$ and $ 30\sqrt {7}$ from vertices $ A$ and $ D$, respectively. Let $ F$ be the foot of the altitude from $ C$ to $ \overline{AD}$. The distance $ EF$ can be expressed in the form $ m\sqrt {n}$, where $ m$ and $ n$ are positive integers and $ n$ is not divisible by the square of any prime. Find $ m \plus{} n$.

A small chunk of ice falls from rest down a frictionless parabolic ice sheet shown in the figure. At the point labeled $\mathbf{A}$ in the diagram, the ice sheet becomes a steady, rough incline of angle $30^\circ$ with respect to the horizontal and friction coefficient $\mu_k$. This incline is of length $\frac{3}{2}h$ and ends at a cliff. The chunk of ice comes to a rest precisely at the end of the incline. What is the coefficient of friction $\mu_k$? [asy] size(200); pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); draw((0,0)--(sqrt(3),0)--(0,1)); draw(anglemark((0,1),(sqrt(3),0),(0,0))); label("$30^\circ$",(1.5,0.03),NW); label("A", (0,1),NE); dot((0,1)); label("rough incline",(0.4,0.4)); draw((0.4,0.5)--(0.5,0.6),EndArrow); dot((-0.2,4/3)); label("parabolic ice sheet",(0.6,4/3)); draw((0.05,1.3)--(-0.05,1.2),EndArrow); label("ice chunk",(-0.5,1.6)); draw((-0.3,1.5)--(-0.25,1.4),EndArrow); draw((-0.2,4/3)--(-0.19, 1.30083)--(-0.18,1.27)--(-0.17,1.240833)--(-0.16,1.21333)--(-0.15,1.1875)--(-0.14,1.16333)--(-0.13,1.140833)--(-0.12,1.12)--(-0.11,1.100833)--(-0.10,1.08333)--(-0.09,1.0675)--(-0.08,1.05333)--(-0.07,1.040833)--(-0.06,1.03)--(-0.05,1.020833)--(-0.04,1.01333)--(-0.03,1.0075)--(-0.02,1.00333)--(-0.01,1.000833)--(0,1)); draw((-0.6,0)--(-0.6,4/3),dashed,EndArrow,BeginArrow); label("$h$",(-0.6,2/3),W); draw((0.2,1.2)--(sqrt(3)+0.2,0.2),dashed,EndArrow,BeginArrow); label("$\frac{3}{2}h$",(sqrt(3)/2+0.2,0.7),NE); [/asy] $ \textbf{(A)}\ 0.866\qquad\textbf{(B)}\ 0.770\qquad\textbf{(C)}\ 0.667\qquad\textbf{(D)}\ 0.385\qquad\textbf{(E)}\ 0.333 $

Find all functions $f: \mathbb{R} \to \mathbb{R}$ such that \[f(x)f(y) = (x+y+1)^2 \cdot f \left( \frac{xy-1}{x+y+1} \right)\] $\forall x,y \in \mathbb{R}$ with $x+y+1 \neq 0$ and $f(x) > 1$ $\forall x > 0.$

Let $S$ be the sum of all $x$ in the interval $[0,2\pi)$ that satisfy \[\tan^2 x - 2\tan x\sin x=0.\] Compute $\lfloor10S\rfloor$.

Let $z = \cos \frac{2\pi}{2011} + i\sin \frac{2\pi}{2011}$, and let \[ P(x) = x^{2008} + 3x^{2007} + 6x^{2006} + \cdots + \frac{2008 \cdot 2009}{2} x + \frac{2009 \cdot 2010}{2} \] for all complex numbers $x$. Evaluate $P(z)P(z^2)P(z^3) \cdots P(z^{2010})$.

Let $n\in\mathbb{N}^*$. Solve the equation $\sum_{k=0}^n C_n^k\cos2kx=\cos nx$ in $\mathbb{R}$.

Let $x$ be an angle between $0^o$ and $90^o$ so that $$\frac{\sin^4 x}{9}+\frac{\cos^4 x}{16 }=\frac{1}{25} .$$ Then what is $\tan x$?

Is $\sum_{n=1}^{+\infty}\frac{\cos n}{n}(1 + \frac{1}{\sqrt{2}} + \cdots + \frac{1}{\sqrt{n}})$ convergent? why?

Let $f(t)$ be the cubic polynomial for $t$ such that $\cos 3x=f(\cos x)$ holds for all real number $x$. Evaluate \[\int_0^1 \{f(t)\}^2 \sqrt{1-t^2}dt\]

[asy]size(8cm); real w = 2.718; // width of block real W = 13.37; // width of the floor real h = 1.414; // height of block real H = 7; // height of block + string real t = 60; // measure of theta pair apex = (w/2, H); // point where the strings meet path block = (0,0)--(w,0)--(w,h)--(0,h)--cycle; // construct the block draw(shift(-W/2,0)*block); // draws white block path arrow = (w,h/2)--(w+W/8,h/2); // path of the arrow draw(shift(-W/2,0)*arrow, EndArrow); // draw the arrow picture pendulum; // making a pendulum... draw(pendulum, block); // block fill(pendulum, block, grey); // shades block draw(pendulum, (w/2,h)--apex); // adds in string add(pendulum); // adds in block + string add(rotate(t, apex) * pendulum); // adds in rotated block + string dot("$\theta$", apex, dir(-90+t/2)*3.14); // marks the apex and labels it with theta draw((apex-(w,0))--(apex+(w,0))); // ceiling draw((-W/2-w/2,0)--(w+W/2,0)); // floor[/asy] A block of mass $m=\text{4.2 kg}$ slides through a frictionless table with speed $v$ and collides with a block of identical mass $m$, initially at rest, that hangs on a pendulum as shown above. The collision is perfectly elastic and the pendulum block swings up to an angle $\theta=12^\circ$, as labeled in the diagram. It takes a time $ t = \text {1.0 s} $ for the block to swing up to this peak. Find $10v$, in $\text{m/s}$ and round to the nearest integer. Do not approximate $ \theta \approx 0 $; however, assume $\theta$ is small enough as to use the small-angle approximation for the period of the pendulum. [i](Ahaan Rungta, 6 points)[/i]

Evaluate the following integrals. (1) $\int_{0}^{\frac{\pi}{4}}\frac{dx}{1+\sin x}.$ (2) $\int_{\frac{4}{3}}^{2}\frac{dx}{x^{2}\sqrt{x-1}}.$