Let $ABC$ be an acute triangle and $O$ be its circumcenter. Let $D$ be the midpoint of $[AB]$. The circumcircle of $\triangle ADO$ meets $[AC]$ at $A$ and $E$. If $|AE|=7$, $|DE|=8$, and $m(\widehat{AOD}) = 45^\circ$, what is the area of $\triangle ABC$? $ \textbf{(A)}\ 56\sqrt 3 \qquad\textbf{(B)}\ 56 \sqrt 2 \qquad\textbf{(C)}\ 50 \sqrt 2 \qquad\textbf{(D)}\ 84 \qquad\textbf{(E)}\ \text{None of the preceding} $

While there do not exist pairwise distinct real numbers $a,b,c$ satisfying $a^2+b^2+c^2 = ab+bc+ca$, there do exist complex numbers with that property. Let $a,b,c$ be complex numbers such that $a^2+b^2+c^2 = ab+bc+ca$ and $|a+b+c| = 21$. Given that $|a-b| = 2\sqrt{3}$, $|a| = 3\sqrt{3}$, compute $|b|^2+|c|^2$. [hide="Clarifications"] [list] [*] The problem should read $|a+b+c| = 21$. An earlier version of the test read $|a+b+c| = 7$; that value is incorrect. [*] $|b|^2+|c|^2$ should be a positive integer, not a fraction; an earlier version of the test read ``... for relatively prime positive integers $m$ and $n$. Find $m+n$.''[/list][/hide] [i]Ray Li[/i]

Let $(x_n)_{n \in \mathbb{N}}$ be the sequence defined as $x_n = \sin(2 \pi n! e)$ for all $n \in \mathbb{N}$. Compute $\lim_{n \to \infty} x_n$.

Suppose that $ ABC$ is an acute triangle such that $ \tan A$, $ \tan B$, $ \tan C$ are the three roots of the equation $ x^3 \plus{} px^2 \plus{} qx \plus{} p \equal{} 0$, where $ q\neq 1$. Show that $ p \le \minus{} 3\sqrt 3$ and $ q > 1$.

Let $E$ be a point on side $[AD]$ of rhombus $ABCD$. Lines $AB$ and $CE$ meet at $F$, lines $BE$ and $DF$ meet at $G$. If $m(\widehat{DAB}) = 60^\circ $, what is$m(\widehat{DGB})$? $ \textbf{a)}\ 45^\circ \qquad\textbf{b)}\ 50^\circ \qquad\textbf{c)}\ 60^\circ \qquad\textbf{d)}\ 65^\circ \qquad\textbf{e)}\ 75^\circ $

It is given the function $f:\mathbb{R} \to \mathbb{R}$ such that it holds $f(\sin x)=\sin (2011x)$. Find the value of $f(\cos x)$.

In a triangle $ ABC,$ let $ D$ and $ E$ be the intersections of the bisectors of $ \angle ABC$ and $ \angle ACB$ with the sides $ AC,AB,$ respectively. Determine the angles $ \angle A,\angle B, \angle C$ if $ \angle BDE \equal{} 24 ^{\circ},$ $ \angle CED \equal{} 18 ^{\circ}.$

What is the greatest value of $\sin x \cos y + \sin y \cos z + \sin z \cos x$, where $x,y,z$ are real numbers? $ \textbf{(A)}\ \sqrt 2 \qquad\textbf{(B)}\ \dfrac 32 \qquad\textbf{(C)}\ \dfrac {\sqrt 3}2 \qquad\textbf{(D)}\ 2 \qquad\textbf{(E)}\ 3 $

For real numbers $a,b,c,d$ not all equal to $0$ , define a real function $f(x) = a +b\cos{2x} + c\sin{5x} +d \cos{8x}$. Suppose $f(t) = 4a$ for some real $t$. prove that there exist a real number $s$ s.t. $f(s)<0$

Let $ ABC$ be an acute-angled triangle. $ C_{1}$ and $ C_{2}$ are two circles of diameters $ AB$ and $ AC$, respectively. $ C_{2}$ and $ AB$ intersect again at $ F$, and $ C_{1}$ and $ AC$ intersect again at $ E$. Also, $ BE$ meets $ C_{2}$ at $ P$ and $ CF$ meets $ C_{1}$ at $ Q$. Prove that $ AP=AQ$.

Determine all real numbers $\alpha$ with the property that all numbers in the sequence $\cos\alpha,\cos2\alpha,\cos2^2\alpha,\ldots,\cos2^n\alpha,\ldots$ are negative.

A function $ f$ is defined by $ f(z) \equal{} i\bar z$, where $ i \equal{}\sqrt{\minus{}\!1}$ and $ \bar z$ is the complex conjugate of $ z$. How many values of $ z$ satisfy both $ |z| \equal{} 5$ and $ f (z) \equal{} z$? $ \textbf{(A)}\ 0 \qquad \textbf{(B)}\ 1 \qquad \textbf{(C)}\ 2 \qquad \textbf{(D)}\ 4 \qquad \textbf{(E)}\ 8$

Let $A_1A_2A_3\dots A_{20}$ be a $20$-sided polygon $P$ in the plane, where all of the side lengths of $P$ are equal, the interior angle at $A_i$ measures $108$ degrees for all odd $i$, and the interior angle $A_i$ measures $216$ degrees for all even $i$. Prove that the lines $A_2A_8$, $A_4A_{10}$, $A_5A_{13}$, $A_6A_{16}$, and $A_7A_{19}$ all intersect at the same point. [asy] import graph; size(10cm); pair temp= (-1,0); pair A01 = (0,0); pair A02 = rotate(306,A01)*temp; pair A03 = rotate(144,A02)*A01; pair A04 = rotate(252,A03)*A02; pair A05 = rotate(144,A04)*A03; pair A06 = rotate(252,A05)*A04; pair A07 = rotate(144,A06)*A05; pair A08 = rotate(252,A07)*A06; pair A09 = rotate(144,A08)*A07; pair A10 = rotate(252,A09)*A08; pair A11 = rotate(144,A10)*A09; pair A12 = rotate(252,A11)*A10; pair A13 = rotate(144,A12)*A11; pair A14 = rotate(252,A13)*A12; pair A15 = rotate(144,A14)*A13; pair A16 = rotate(252,A15)*A14; pair A17 = rotate(144,A16)*A15; pair A18 = rotate(252,A17)*A16; pair A19 = rotate(144,A18)*A17; pair A20 = rotate(252,A19)*A18; dot(A01); dot(A02); dot(A03); dot(A04); dot(A05); dot(A06); dot(A07); dot(A08); dot(A09); dot(A10); dot(A11); dot(A12); dot(A13); dot(A14); dot(A15); dot(A16); dot(A17); dot(A18); dot(A19); dot(A20); draw(A01--A02--A03--A04--A05--A06--A07--A08--A09--A10--A11--A12--A13--A14--A15--A16--A17--A18--A19--A20--cycle); label("$A_{1}$",A01,E); label("$A_{2}$",A02,W); label("$A_{3}$",A03,NE); label("$A_{4}$",A04,SW); label("$A_{5}$",A05,N); label("$A_{6}$",A06,S); label("$A_{7}$",A07,N); label("$A_{8}$",A08,SE); label("$A_{9}$",A09,NW); label("$A_{10}$",A10,E); label("$A_{11}$",A11,W); label("$A_{12}$",A12,E); label("$A_{13}$",A13,SW); label("$A_{14}$",A14,NE); label("$A_{15}$",A15,S); label("$A_{16}$",A16,N); label("$A_{17}$",A17,S); label("$A_{18}$",A18,NW); label("$A_{19}$",A19,SE); label("$A_{20}$",A20,W);[/asy]

(1) Evaluate $ \int_1^{3\sqrt{3}} \left(\frac{1}{\sqrt[3]{x^2}}\minus{}\frac{1}{1\plus{}\sqrt[3]{x^2}}\right)\ dx.$ (2) Find the positive real numbers $ a,\ b$ such that for $ t>1,$ $ \lim_{t\rightarrow \infty} \left(\int_1^t \frac{1}{1\plus{}\sqrt[3]{x^2}}\ dx\minus{}at^b\right)$ converges.

Find the minimum value of $f(x)=\int_0^{\frac{\pi}{4}} |\tan t-x|dt.$

Let \[X = \cos\frac{2\pi}7 + \cos\frac{4\pi}7 + \cos\frac{6\pi}7 + \cdots + \cos\frac{2006\pi}7 + \cos\frac{2008\pi}7.\] Compute $\Big|\lfloor 2008 X\rfloor\Big|$.

In acute triangle $ABC$ angle $B$ is greater than$C$. Let $M$ is midpoint of $BC$. $D$ and $E$ are the feet of the altitude from $C$ and $B$ respectively. $K$ and $L$ are midpoint of $ME$ and $MD$ respectively. If $KL$ intersect the line through $A$ parallel to $BC$ in $T$, prove that $TA=TM$.

Given any convex polygon, show that there are three consecutive vertices such that the polygon lies inside the circle through them.

Given is a number $a$ with 0 $\le \alpha \le \pi$. A sequence $c_0,c_1, c_2,...$ is defined as $$c_0=\cos \alpha$$ $$C_{n+1}=\sqrt{\frac{1+c_n}{2}} \,\, for \,\,\, n=0,1,2,...$$ Calculate $\lim_{n\to \infty}2^{2n+1}(1-c_n)$

Let $ ABC$ be a non-isosceles triangle, in which $ X,Y,$ and $ Z$ are the tangency points of the incircle of center $ I$ with sides $ BC,CA$ and $ AB$ respectively. Denoting by $ O$ the circumcircle of $ \triangle{ABC}$, line $ OI$ meets $ BC$ at a point $ D.$ The perpendicular dropped from $ X$ to $ YZ$ intersects $ AD$ at $ E$. Prove that $ YZ$ is the perpendicular bisector of $ [EX]$.

In rectangle $ ABCD$, $ AD \equal{} 1$, $ P$ is on $ \overline{AB}$, and $ \overline{DB}$ and $ \overline{DP}$ trisect $ \angle ADC$. What is the perimeter of $ \triangle BDP$? [asy]unitsize(2cm); defaultpen(linewidth(.8pt)); dotfactor=4; pair D=(0,0), C=(sqrt(3),0), B=(sqrt(3),1), A=(0,1), P=(sqrt(3)/3,1); pair[] dotted={A,B,C,D,P}; draw(A--B--C--D--cycle); draw(B--D--P); dot(dotted); label("$A$",A,NW); label("$B$",B,NE); label("$C$",C,SE); label("$D$",D,SW); label("$P$",P,N);[/asy]$ \textbf{(A)}\ 3 \plus{} \frac {\sqrt3}{3} \qquad\textbf{(B)}\ 2 \plus{} \frac {4\sqrt3}{3}\qquad\textbf{(C)}\ 2 \plus{} 2\sqrt2\qquad\textbf{(D)}\ \frac {3 \plus{} 3\sqrt5}{2} \qquad\textbf{(E)}\ 2 \plus{} \frac {5\sqrt3}{3}$

Evaluate \[ \log_{10}(\tan 1^{\circ})+ \log_{10}(\tan 2^{\circ})+ \log_{10}(\tan 3^{\circ})+ \cdots + \log_{10}(\tan 88^{\circ})+\log_{10}(\tan 89^{\circ}). \] $ \textbf{(A)}\ 0 \qquad\textbf{(B)}\ \frac{1}{2}\log_{10}(\frac{\sqrt{3}}{2}) \qquad\textbf{(C)}\ \frac{1}{2}\log_{10}2 \qquad\textbf{(D)}\ 1 \qquad\textbf{(E)}\ \text{none of these} $

Prove that if $A,B,$ and $C$ are angles of a triangle measured in radians then $A \cos B +\sin A \cos C >0.$

Find $ \lim_{n\to\infty} \int_0^{\frac{\pi}{2}} x|\cos (2n\plus{}1)x|\ dx$.

The exinscribed circle of a triangle $ABC$ corresponding to its vertex $A$ touches the sidelines $AB$ and $AC$ in the points $M$ and $P$, respectively, and touches its side $BC$ in the point $N$. Show that if the midpoint of the segment $MP$ lies on the circumcircle of triangle $ABC$, then the points $O$, $N$, $I$ are collinear, where $I$ is the incenter and $O$ is the circumcenter of triangle $ABC$.