$I$ is the incenter of triangle $ABC$. perpendicular from $I$ to $AI$ meet $AB$ and $AC$ at ${B}'$ and ${C}'$ respectively . Suppose that ${B}''$ and ${C}''$ are points on half-line $BC$ and $CB$ such that $B{B}''=BA$ and $C{C}''=CA$. Suppose that the second intersection of circumcircles of $A{B}'{B}''$ and $A{C}'{C}''$ is $T$. Prove that the circumcenter of $AIT$ is on the $BC$.

Prove that $cos \frac{2\pi}{5} +cos \frac{4\pi}{5} = -\frac{1}{2}$.

Let $n$ be a positive integer, and $a_j$, for $j=1,2,\ldots,n$ are complex numbers. Suppose $I$ is an arbitrary nonempty subset of $\{1,2,\ldots,n\}$, the inequality $\left|-1+ \prod_{j\in I} (1+a_j) \right| \leq \frac 12$ always holds. Prove that $\sum_{j=1}^n |a_j| \leq 3$.

Let $I$ be the incenter of $\triangle ABC$, and $O$ be the excenter corresponding to $B$. If $|BI|=12$, $|IO|=18$, and $|BC|=15$, then what is $|AB|$? $ \textbf{(A)}\ 16 \qquad\textbf{(B)}\ 18 \qquad\textbf{(C)}\ 20 \qquad\textbf{(D)}\ 22 \qquad\textbf{(E)}\ 24 $

Prove that $\sin\sqrt{x}<\sqrt{\sin x}$ for every real $x$ such that $0<x<\frac{\pi}{2}$.

In trapezoid $ ABCD$, $ AB \parallel CD$, $ \angle CAB < 90^\circ$, $ AB \equal{} 5$, $ CD \equal{} 3$, $ AC \equal{} 15$. What are the sum of different integer values of possible $ BD$? $\textbf{(A)}\ 101 \qquad\textbf{(B)}\ 108 \qquad\textbf{(C)}\ 115 \qquad\textbf{(D)}\ 125 \qquad\textbf{(E)}\ \text{None}$

In a circle we enscribe a regular $2001$-gon and inside it a regular $667$-gon with shared vertices. Prove that the surface in the $2001$-gon but not in the $667$-gon is of the form $k.sin^3\left(\frac{\pi}{2001}\right).cos^3\left(\frac{\pi}{2001}\right)$ with $k$ a positive integer. Find $k$.

Prove that for any angles $x,y,z$ holds the equality $$1-\cos^2x-\cos^2y- y-\cos^2z +2 \cos x \cos y \cos z= 4 \sin \frac{x+y+z}{2} \sin \frac{x+y-z}{2} \sin \frac{x-y+z}{2} \sin\frac{-x-y+z}{2}. $$

In the diagram below, angle $ABC$ is a right angle. Point $D$ is on $\overline{BC}$, and $\overline{AD}$ bisects angle $CAB$. Points $E$ and $F$ are on $\overline{AB}$ and $\overline{AC}$, respectively, so that $AE=3$ and $AF=10.$ Given that $EB=9$ and $FC=27$, find the integer closest to the area of quadrilateral $DCFG.$ [asy] size(250); pair A=(0,12), E=(0,8), B=origin, C=(24*sqrt(2),0), D=(6*sqrt(2),0), F=A+10*dir(A--C), G=intersectionpoint(E--F, A--D); draw(A--B--C--A--D^^E--F); pair point=G+1*dir(250); label("$A$", A, dir(point--A)); label("$B$", B, dir(point--B)); label("$C$", C, dir(point--C)); label("$D$", D, dir(point--D)); label("$E$", E, dir(point--E)); label("$F$", F, dir(point--F)); label("$G$", G, dir(point--G)); markscalefactor=0.1; draw(rightanglemark(A,B,C)); label("10", A--F, dir(90)*dir(A--F)); label("27", F--C, dir(90)*dir(F--C)); label("3", (0,10), W); label("9", (0,4), W);[/asy]

Let $A_{1}$, $B_{1}$, $C_{1}$ be the feet of the altitudes of an acute-angled triangle $ABC$ issuing from the vertices $A$, $B$, $C$, respectively. Let $K$ and $M$ be points on the segments $A_{1}C_{1}$ and $B_{1}C_{1}$, respectively, such that $\measuredangle KAM = \measuredangle A_{1}AC$. Prove that the line $AK$ is the angle bisector of the angle $C_{1}KM$.

Prove that the numbers $ A $, $ B $, $ C $ defined by the formulas $$ A = tg \beta tg \gamma + 5,\\ B = tg \gamma tg \alpha + 5,\\ C = tg \alpha tg \beta + 5,$$ where $ \alpha>0 $, $ \beta > 0 $, $ \gamma > 0 $ and $ \alpha + \beta + \gamma = 90^\circ $, satisfy the inequality $$ \sqrt{A} + \sqrt{B} + \sqrt{C} < 4 \sqrt{3}.$$

Show that there are real numbers $A,B$ such that the identity \[\sum_{k=1}^n\tan(k)\tan(k-1)=A\tan(n)+Bn\] holds for every positive integer $n.$

Let $ABC$ be an acute-angled triangle. The tangents to its circumcircle at $A, B, C$ form a triangle $PQR$ with $C \in PQ$ and $B \in PR$. Let $C_{1}$ be the foot of the altitude from $C$ in $\Delta ABC$ . Prove that $CC_{1}$ bisects $\widehat{QC_{1}P}$ .

Point $D$ is on side $CB$ of triangle $ABC$. If \[ \angle{CAD} = \angle{DAB} = 60^\circ,\quad AC = 3\quad\mbox{ and }\quad AB = 6, \] then the length of $AD$ is $\text{(A)} \ 2 \qquad \text{(B)} \ 2.5 \qquad \text{(C)} \ 3 \qquad \text{(D)} \ 3.5 \qquad \text{(E)} \ 4$

Answer the following questions: (1) For complex numbers $\alpha ,\ \beta$, if $\alpha \beta =0$, then prove that $\alpha =0$ or $\beta =0$. (2) For complex number $\alpha$, if $\alpha^2$ is a positive real number, then prove that $\alpha$ is a real number. (3) For complex numbers $\alpha_1,\ \alpha_2,\ \cdots,\ \alpha_{2n+1}\ (n=1,\ 2,\ \cdots)$, assume that $\alpha_1\alpha_2,\ \cdots ,\ \alpha_k\alpha_{k+1},\ \cdots,\ \alpha_{2n}\alpha_{2n+1}$ and $\alpha_{2n+1}\alpha_1$ are all positive real numbers. Prove that $\alpha_1,\ \alpha_2,\ \cdots,\ \alpha_{2n+1}$ are all real numbers.

Evaluate $\int_0^{\frac{\pi}{12}} \frac{\tan ^ 2 x-3}{3\tan ^ 2 x-1}dx$. created by kunny

Given a triangle $ABC$ where $AC > BC$, $D$ is located on the circumcircle of $ABC$ such that $D$ is the midpoint of the arc $AB$ that contains $C$. $E$ is a point on $AC$ such that $DE$ is perpendicular to $AC$. Prove that $AE = EC + CB$.

Prove that every partition of $3$-dimensional space into three disjoint subsets has the following property: One of these subsets contains all possible distances; i.e., for every $a \in \mathbb R^+$, there are points $M$ and $N$ inside that subset such that distance between $M$ and $N$ is exactly $a.$

In $\triangle ABC$, $\angle A = 60$, $AB>AC$, point $O$ is the circumcenter and $H$ is the intersection point of two altitudes $BE$ and $CF$. Points $M$ and $N$ are on the line segments $BH$ and $HF$ respectively, and satisfy $BM=CN$. Determine the value of $\frac{MH+NH}{OH}$.

The sides of triangle $ABC$ has integer lengths. Given that $AC=6$ and $\angle BAC=120^\circ$, determine the lengths of the other two sides.

Compute the value of the sum \begin{align*} \frac{1}{1 + \tan^3 0^\circ} &+ \frac{1}{1 + \tan^3 10^\circ} + \frac{1}{1 + \tan^3 20^\circ} + \frac{1}{1 + \tan^3 30^\circ} + \frac{1}{1 + \tan^3 40^\circ} \\ &+ \frac{1}{1 + \tan^3 50^\circ} + \frac{1}{1 + \tan^3 60^\circ} + \frac{1}{1 + \tan^3 70^\circ} + \frac{1}{1 + \tan^3 80^\circ} \, . \end{align*}

Let $n$ be a positive integer and let $(1+iT)^n=f(T)+ig(T)$ where $i$ is the square root of $-1$, and $f$ and $g$ are polynomials with real coefficients. Show that for any real number $k$ the equation $f(T)+kg(T)=0$ has only real roots.

In $\triangle ABC$ , prove that\[\frac{tan\frac{A}{2}+tan\frac{B}{2}+tan\frac{C}{2}}{\sqrt{3}}\geq\sqrt[6]{tan^2\frac{A}{2}+tan^2\frac{B}{2}+tan^2\frac{C}{2}}.\]

Given a cube with edges of length 1, $e$ the midpoint of $[bc]$, and $m$ midpoint of the face $cdc_1d_1$, as on the figure. Find the area of intersection of the cube with the plane through the points $a,m,e$. [img]http://www.mathlinks.ro/Forum/album_pic.php?pic_id=279[/img]

Let $ABC$ be a triangle inscribed in circle $\Gamma$, centered at $O$ with radius $333.$ Let $M$ be the midpoint of $AB$, $N$ be the midpoint of $AC$, and $D$ be the point where line $AO$ intersects $BC$. Given that lines $MN$ and $BO$ concur on $\Gamma$ and that $BC = 665$, find the length of segment $AD$. [i]Author: Alex Zhu[/i]