In $\triangle PBC$, $\angle PBC=60^{o}$, the tangent at point $P$ to the circumcircle$g$ of $\triangle PBC$ intersects with line $CB$ at $A$. Points $D$ and $E$ lie on the line segment $PA$ and $g$ respectively, satisfying $\angle DBE=90^{o}$, $PD=PE$. $BE$ and $PC$ meet at $F$. It is known that lines $AF,BP,CD$ are concurrent. a) Prove that $BF$ bisect $\angle PBC$ b) Find $\tan \angle PCB$

Let $P$ be a given point inside the triangle $ABC$. Suppose $L,M,N$ are the midpoints of $BC, CA, AB$ respectively and \[PL: PM: PN= BC: CA: AB.\] The extensions of $AP, BP, CP$ meet the circumcircle of $ABC$ at $D,E,F$ respectively. Prove that the circumcentres of $APF, APE, BPF, BPD, CPD, CPE$ are concyclic.

The lengths of the sides of a triangle are consescutive integers, and the largest angle is twice the smallest angle. The cosine of the smallest angle is $\textbf {(A) } \frac 34 \qquad \textbf {(B) } \frac{7}{10} \qquad \textbf {(C) } \frac 23 \qquad \textbf {(D) } \frac{9}{14} \qquad \textbf {(E) } \text{none of these}$

Let $ ABC$ be a triangle, let $ AB > AC$. Its incircle touches side $ BC$ at point $ E$. Point $ D$ is the second intersection of the incircle with segment $ AE$ (different from $ E$). Point $ F$ (different from $ E$) is taken on segment $ AE$ such that $ CE \equal{} CF$. The ray $ CF$ meets $ BD$ at point $ G$. Show that $ CF \equal{} FG$.

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 $AC.$

Let $ABCD$ be a quadrilateral circumscribed about a circle, whose interior and exterior angles are at least 60 degrees. Prove that \[ \frac{1}{3}|AB^3 - AD^3| \le |BC^3 - CD^3| \le 3|AB^3 - AD^3|. \] When does equality hold?

Let $(O)$ be a circle, and let $ABP$ be a line segment such that $A,B$ lie on $(O)$ and $P$ is a point outside $(O)$. Let $C$ be a point on $(O)$ such that $PC$ is tangent to $(O)$ and let $D$ be the point on $(O)$ such that $CD$ is a diameter of $(O)$ and intersects $AB$ inside $(O)$. Suppose that the lines $DB$ and $OP$ intersect at $E$. Prove that $AC$ is perpendicular to $CE$.

In $\triangle ABC$, $M$ is the midpoint of side $BC$, $AN$ bisects $\angle BAC$, $BN\perp AN$ and $\theta$ is the measure of $\angle BAC$. If sides $AB$ and $AC$ have lengths $14$ and $19$, respectively, then length $MN$ equals [asy] size(230); defaultpen(linewidth(0.7)+fontsize(10)); pair B=origin, A=14*dir(36), C=intersectionpoint(B--(9001,0), Circle(A,19)), M=midpoint(B--C), b=A+14*dir(A--C), N=foot(A, B, b); draw(N--B--A--N--M--C--A^^B--M); markscalefactor=0.1; draw(rightanglemark(B,N,A)); pair point=N; label("$A$", A, dir(point--A)); label("$B$", B, dir(point--B)); label("$C$", C, dir(point--C)); label("$M$", M, S); label("$N$", N, dir(30)); label("$19$", (A+C)/2, dir(A--C)*dir(90)); label("$14$", (A+B)/2, dir(A--B)*dir(270)); [/asy] $\displaystyle \text{(A)} \ 2 \qquad \text{(B)} \ \frac{5}{2} \qquad \text{(C)} \ \frac{5}{2} - \sin \theta \qquad \text{(D)} \ \frac{5}{2} - \frac{1}{2} \sin \theta \qquad \text{(E)} \ \frac{5}{2} - \frac{1}{2} \sin \left(\frac{1}{2} \theta\right)$

Let $ABCD$ be a convex quadrilateral such that $BC=AD$. Let $M$ and $N$ be the midpoints of $AB$ and $CD$, respectively. The lines $AD$ and $BC$ meet the line $MN$ at $P$ and $Q$, respectively. Prove that $CQ=DP$.

Let $ABC$ be a triangle whose side lengths are opposite the angle $A,B,C$ are $a,b,c$ respectively. Prove that $$\frac{ab\sin{2C}+bc\sin{ 2A}+ca\sin{2B}}{ab+bc+ca}\leq\frac{\sqrt{3}}{2}$$. [i](nooonuii)[/i]

The incircle of triangle $ \triangle ABC$ touches the sides $ BC$, $ CA$, $ AB$ at $ D, E, F$ respectively. $ X$ is a point inside triangle of $ \triangle ABC$ such that the incircle of triangle $ \triangle XBC$ touches $ BC$ at $ D$, and touches $ CX$ and $ XB$ at $ Y$ and $ Z$ respectively. Show that $ E, F, Z, Y$ are concyclic.

A college student is about to break up with her boyfriend, a mathematics major who is apparently more interested in math than her. Frustrated, she cries, ”You mathematicians have no soul! It’s all numbers and equations! What is the root of your incompetence?!” Her boyfriend assumes she means the square root of himself, or the square root of i. What two answers should he give?

For every nonnegative integer $ n$ and every real number $ x$ prove the inequality: $ |\cos x|\plus{}|\cos 2x|\plus{}\ldots\plus{}|\cos 2^nx|\geq \dfrac{n}{2\sqrt{2}}$

$ABC$ is a triangle. Reflect each vertex in the opposite side to get the triangle $A'B'C'$. Find a necessary and sufficient condition on $ABC$ for $A'B'C'$ to be equilateral.

Let $A,B,C$ be angles of an acute triangle with \begin{align*} \cos^2 A + \cos^2 B + 2 \sin A \sin B \cos C &= \frac{15}{8} \text{ and} \\ \cos^2 B + \cos^2 C + 2 \sin B \sin C \cos A &= \frac{14}{9}. \end{align*} There are positive integers $p$, $q$, $r$, and $s$ for which \[ \cos^2 C + \cos^2 A + 2 \sin C \sin A \cos B = \frac{p-q\sqrt{r}}{s}, \] where $p+q$ and $s$ are relatively prime and $r$ is not divisible by the square of any prime. Find $p+q+r+s$. [i]Note: due to an oversight by the exam-setters, there is no acute triangle satisfying these conditions. You should instead assume $ABC$ is obtuse with $\angle B > 90^{\circ}$.[/i]

Let $f(x)=\cos(a_1+x)+{1\over2}\cos(a_2+x)+{1\over4}\cos(a_3+x)+\ldots+{1\over2^{n-1}}\cos(a_n+x)$, where $a_i$ are real constants and $x$ is a real variable. If $f(x_1)=f(x_2)=0$, prove that $x_1-x_2$ is a multiple of $\pi$.

Prove that in every triangle with sides $a, b, c$ and opposite angles $A, B, C$, is fulfilled (measuring the angles in radians) $$\frac{a A+bB+cC}{a+b+c} \ge \frac{\pi}{3}$$ Hint: Use $a \ge b \ge c \Rightarrow A \ge B \ge C$.

The angle bisector of the acute angle formed at the origin by the graphs of the lines $y=x$ and $y=3x$ has equation $y=kx$. What is $k$? $\textbf{(A)} \: \frac{1+\sqrt{5}}{2} \qquad \textbf{(B)} \: \frac{1+\sqrt{7}}{2} \qquad \textbf{(C)} \: \frac{2+\sqrt{3}}{2} \qquad \textbf{(D)} \: 2\qquad \textbf{(E)} \: \frac{2+\sqrt{5}}{2}$

Let $R$ and $T$ be points respectively on sides $[BC]$ and $[CD]$ of a square $ABCD$ with side length $6$ such that $|CR|+|RT|+|TC|=12$. What is $\tan (\widehat{RAT})$ $ \textbf{(A)}\ 2\sqrt 3 \qquad\textbf{(B)}\ \sqrt 3 \qquad\textbf{(C)}\ \dfrac 13 \qquad\textbf{(D)}\ \dfrac 12 \qquad\textbf{(E)}\ 1 $

$M$ is midpoint of side $BC$ of triangle $ABC$, and $I$ is incenter of triangle $ABC$, and $T$ is midpoint of arc $BC$, that does not contain $A$. Prove that \[\cos B+\cos C=1\Longleftrightarrow MI=MT\]

$ABC$ is a triangle and $\angle A=90^{\circ}$. Let $D$ be the meet point of the interior bisector of $\angle A$ and $BC$. And let $I_a$ be the $A-$excenter of $\triangle ABC$. Prove that: \[\frac{AD}{DI_a}\leq\sqrt{2}-1.\]

If $a,b,c$ be the lengths of the sides of a triangle. Let $R$ denote its circumradius. Prove that \[ R\ge \frac{a^2+b^2}{2\sqrt{2a^2+2b^2-c^2}}\] When does equality hold?

Find the exact value of $\sin 36^o$.

Consider the obtuse-angled triangle $\triangle ABC$ and its side lengths $a,b,c$. Prove that $a^3\cos\angle A +b^3\cos\angle B + c^3\cos\angle C < abc$.

Let $ABC$ be a triangle with $AB = AC$. The incircle touches $BC$, $AC$ and $AB$ at $D$, $E$ and $F$ respectively. Let $P$ be a point on the arc $\overarc{EF}$ that does not contain $D$. Let $Q$ be the second point of intersection of $BP$ and the incircle of $ABC$. The lines $EP$ and $EQ$ meet the line $BC$ at $M$ and $N$, respectively. Prove that the four points $P, F, B, M$ lie on a circle and $\frac{EM}{EN} = \frac{BF}{BP}$.