Let $t$ be a real number, and let $$a_n=2\cos \left(\frac{t}{2^n}\right)-1,\quad n=1,2,3,\dots$$ Let $b_n$ be the product $a_1a_2a_3\cdots a_n$. Find a formula for $b_n$ that does not involve a product of $n$ terms, and deduce that $$\lim_{n\to \infty}b_n=\frac{2\cos t+1}{3}$$

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

Let $ ABC$ be an acute triangle, let $ M,N$ be the midpoints of minor arcs $ \widehat{CA},\widehat{AB}$ of the circumcircle of triangle $ ABC,$ point $ D$ is the midpoint of segment $ MN,$ point $ G$ lies on minor arc $ \widehat{BC}.$ Denote by $ I,I_{1},I_{2}$ the incenters of triangle $ ABC,ABG,ACG$ respectively.Let $ P$ be the second intersection of the circumcircle of triangle $ GI_{1}I_{2}$ with the circumcircle of triangle $ ABC.$ Prove that three points $ D,I,P$ are collinear.

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

If $n \ge 2$ is an integer, prove the equality $$\prod_{k=1}^n \tan \frac{\pi}{3}\left(1+\frac{3^k}{3^n-1}\right)=\prod_{k=1}^n \cot \frac{\pi}{3}\left(1-\frac{3^k}{3^n-1}\right)$$

Let $ABC$ be a triangle with $AB \neq AC$ and circumcenter $O$. The bisector of $\angle BAC$ intersects $BC$ at $D$. Let $E$ be the reflection of $D$ with respect to the midpoint of $BC$. The lines through $D$ and $E$ perpendicular to $BC$ intersect the lines $AO$ and $AD$ at $X$ and $Y$ respectively. Prove that the quadrilateral $BXCY$ is cyclic.

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 ${S\leq\frac{AB\cdot CD+n(n-1)AD^2+n^2DA\cdot BC}{2n^2}}$

Triangle $ABC$ is isosceles, with $AB=AC$ and altitude $AM=11.$ Suppose that there is a point $D$ on $\overline{AM}$ with $AD=10$ and $\angle BDC=3\angle BAC.$ Then the perimeter of $\triangle ABC$ may be written in the form $a+\sqrt{b},$ where $a$ and $b$ are integers. Find $a+b.$ [asy] import graph; size(7cm); real lsf=0.5; pen dps=linewidth(0.7)+fontsize(10); defaultpen(dps); pen ds=black; real xmin=-1.55,xmax=7.95,ymin=-4.41,ymax=5.3; draw((1,3)--(0,0)); draw((0,0)--(2,0)); draw((2,0)--(1,3)); draw((1,3)--(1,0)); draw((1,0.7)--(0,0)); draw((1,0.7)--(2,0)); label("$11$",(0.75,1.63),SE*lsf); dot((1,3),ds); label("$A$",(0.96,3.14),NE*lsf); dot((0,0),ds); label("$B$",(-0.15,-0.18),NE*lsf); dot((2,0),ds); label("$C$",(2.06,-0.18),NE*lsf); dot((1,0),ds); label("$M$",(0.97,-0.27),NE*lsf); dot((1,0.7),ds); label("$D$",(1.05,0.77),NE*lsf); clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle); [/asy]

Sides $AB,~ BC, ~CD$ and $DA$, respectively, of convex quadrilateral $ABCD$ are extended past $B,~ C ,~ D$ and $A$ to points $B',~C',~ D'$ and $A'$. Also, $AB = BB' = 6,~ BC = CC' = 7, ~CD = DD' = 8$ and $DA = AA' = 9$; and the area of $ABCD$ is 10. The area of $A 'B 'C'D'$ is $\textbf{(A) }20\qquad\textbf{(B) }40\qquad\textbf{(C) }45\qquad\textbf{(D) }50\qquad \textbf{(E) }60$

Two acute angles $a$ and $b$ satisfy condition $$\sin^2a+\sin^2b = \sin(a+b)$$ Prove that $a + b = \pi /2$.

In acute triangle $ ABC,$ points $ P,Q$ lie on its sidelines $ AB,AC,$ respectively. The circumcircle of triangle $ ABC$ intersects of triangle $ APQ$ at $ X$ (different from $ A$). Let $ Y$ be the reflection of $ X$ in line $ PQ.$ Given $ PX>PB.$ Prove that $ S_{\bigtriangleup XPQ}>S_{\bigtriangleup YBC}.$ Where $ S_{\bigtriangleup XYZ}$ denotes the area of triangle $ XYZ.$

In the triangle $ABC$, the angle  $\alpha = \angle BAC$ and the side $a = BC$ are given. Assume that $a = \sqrt{rR}$, where $r$ is the inradius and $R$ the circumradius. Compute all possible lengths of sides $AB$ and $AC.$

If the area of $\triangle ABC$ is $64$ square units and the geometric mean (mean proportional) between sides $AB$ and $AC$ is $12$ inches, then $\sin A$ is equal to $\textbf{(A) }\dfrac{\sqrt{3}}{2}\qquad\textbf{(B) }\frac{3}{5}\qquad\textbf{(C) }\frac{4}{5}\qquad\textbf{(D) }\frac{8}{9}\qquad \textbf{(E) }\frac{15}{17}$

Let $f_n(x,\ y)=\frac{n}{r\cos \pi r+n^2r^3}\ (r=\sqrt{x^2+y^2})$, $I_n=\int\int_{r\leq 1} f_n(x,\ y)\ dxdy\ (n\geq 2).$ Find $\lim_{n\to\infty} I_n.$ [i]2009 Tokyo Institute of Technology, Master Course in Mathematics[/i]

Prove that for all $n \in \mathbb N$ the following is true: \[2^n \prod_{k=1}^n \sin \frac{k \pi}{2n+1} = \sqrt{2n+1}\]

Let $X$ be a point inside convex quadrilateral $ABCD$ with $\angle AXB+\angle CXD=180^{\circ}$. If $AX=14$, $BX=11$, $CX=5$, $DX=10$, and $AB=CD$, find the sum of the areas of $\triangle AXB$ and $\triangle CXD$. [i]Proposed by Michael Kural[/i]

Consider the isosceles triangle $ABC$ with $AB = AC$, and $M$ the midpoint of $BC$. Find the locus of the points $P$ interior to the triangle, for which $\angle BPM+\angle CPA = \pi$.

$(POL 2)$ Given two segments $AB$ and $CD$ not in the same plane, find the locus of points $M$ such that $MA^2 +MB^2 = MC^2 +MD^2.$

Let $\triangle ABC$ be an acute triangle and $AD$ the bisector of the angle $\angle BAC$ with $D\in(BC)$. Let $E$ and $F$ denote feet of perpendiculars from $D$ to $AB$ and $AC$ respectively. If $BF\cap CE=K$ and $\odot AKE\cap BF=L$ prove that $DL\perp BF$.

Calculate the following integarls. [1] $\int x\cos ^ 2 x dx$ [2] $\int \frac{x-1}{(3x-1)^2}dx$ [3] $\int \frac{x^3}{(2-x^2)^4}dx$ [4] $\int \left({\frac{1}{4\sqrt{x}}+\frac{1}{2x}}\right)dx$ [5] $\int (\ln x)^2 dx$

For real number $a,$ find the minimum value of $\int_{0}^{\frac{\pi}{2}}\left|\frac{\sin 2x}{1+\sin^{2}x}-a\cos x\right| dx.$

Prove that there exists a positive integer $n$ such that $\int_0^1 x\sin\ (x^2-x+1)dx\geq \frac {n}{n+1}\sin \frac{n+2}{n+3}.$

The incenter of the triangle $ ABC$ is $ K.$ The midpoint of $ AB$ is $ C_1$ and that of $ AC$ is $ B_1.$ The lines $ C_1K$ and $ AC$ meet at $ B_2,$ the lines $ B_1K$ and $ AB$ at $ C_2.$ If the areas of the triangles $ AB_2C_2$ and $ ABC$ are equal, what is the measure of angle $ \angle CAB?$

If $\sum_{i=1}^n\cos^{-1}(\alpha_i)=0$, then find $\sum_{i=1}^n\alpha_i$. $\textbf{(A)}~\frac n2$ $\textbf{(B)}~n$ $\textbf{(C)}~n\pi$ $\textbf{(D)}~\frac{n\pi}2$

Let $a_0 = 1$ and define the sequence $\{a_n\}$ by \[a_{n+1} = \frac{\sqrt{3}a_n - 1}{a_n + \sqrt{3}}.\] If $a_{2017}$ can be expressed in the form $a+b\sqrt{c}$ in simplest radical form, compute $a+b+c$. [i]2016 CCA Math Bonanza Lightning #3.2[/i]