Let $ABC$ be a triangle with $AB=13$, $BC=14$, and $AC=15$. Let $M$ be the midpoint of $BC$ and let $\Gamma$ be the circle passing through $A$ and tangent to line $BC$ at $M$. Let $\Gamma$ intersect lines $AB$ and $AC$ at points $D$ and $E$, respectively, and let $N$ be the midpoint of $DE$. Suppose line $MN$ intersects lines $AB$ and $AC$ at points $P$ and $O$, respectively. If the ratio $MN:NO:OP$ can be written in the form $a:b:c$ with $a,b,c$ positive integers satisfying $\gcd(a,b,c)=1$, find $a+b+c$. [i]James Tao[/i]

In a triangle $ABC$, let $D$ and $E$ be the feet of the angle bisectors of angles $A$ and $B$, respectively. A rhombus is inscribed into the quadrilateral $AEDB$ (all vertices of the rhombus lie on different sides of $AEDB$). Let $\varphi$ be the non-obtuse angle of the rhombus. Prove that $\varphi \le \max \{ \angle BAC, \angle ABC \}$.

Show that the number$$4\sin\frac{\pi}{34}\left(\sin\frac{3\pi}{34}+\sin\frac{7\pi}{34}+\sin\frac{11\pi}{34}+\sin\frac{15\pi}{34}\right)$$ is an integer and determine it.

What is the sum of all positive integer values of $n$ that satisfy the equation \[ \cos \Bigl( \frac{\pi}{n} \Bigr) \cos \Bigl( \frac{2\pi}{n} \Bigr) \cos \Bigl( \frac{4\pi}{n} \Bigr) \cos \Bigl( \frac{8\pi}{n} \Bigr) \cos \Bigl( \frac{16\pi}{n} \Bigr) = \frac{1}{32} \, ? \]

Two circles $S_1$ and $S_2$ with centers $O_1$ and $O_2$ respectively intersect at $A$ and $B$. The tangents at $A$ to $S_1$ and $S_2$ meet segments $BO_2$ and $BO_1$ at $K$ and $L$ respectively. Show that $KL \parallel O_1O_2.$

$A$ is an acute angle. Show that $$\left(1 +\frac{1}{sen A}\right)\left(1 +\frac{1}{cos A}\right)> 5$$

Prove that the sum of the face angles at each vertex of a tetrahedron is a straight angle if and only if the faces are congruent triangles.

The altitudes through the vertices $ A,B,C$ of an acute-angled triangle $ ABC$ meet the opposite sides at $ D,E, F,$ respectively. The line through $ D$ parallel to $ EF$ meets the lines $ AC$ and $ AB$ at $ Q$ and $ R,$ respectively. The line $ EF$ meets $ BC$ at $ P.$ Prove that the circumcircle of the triangle $ PQR$ passes through the midpoint of $ BC.$

Let $\alpha,\beta,\gamma$ be positive real numbers such that $\alpha+\beta+\gamma < \pi$, $\alpha+\beta > \gamma$,$ \beta+\gamma > \alpha$, $\gamma + \alpha > \beta.$ Prove that with the segments of lengths $\sin \alpha, \sin \beta, \sin \gamma $ we can construct a triangle and that its area is not greater than \[A=\dfrac 18\left( \sin 2\alpha+\sin 2\beta+ \sin 2\gamma \right).\] [i]Proposed by Soviet Union[/i]

Prove that for all real numbers $\alpha_1,\alpha_2,\ldots,\alpha_n$ we have \[ \sum_{i=1}^n \sum_{j=1}^n ij \cos {(\alpha_i - \alpha_j )} \geq 0. \] [i]Octavian Stanasila[/i]

Let $f(x)=\sin(\sin(x))$. Evaluate \[ \lim_{h \to 0} \dfrac {f(x+h)-f(h)}{x} \] at $x=\pi$.

Find the smallest value of the expression: $$y=\frac{x^2}{8}+x \cos x +\cos 2x$$

Let $n$ be a positive integer having at least two different prime factors. Show that there exists a permutation $a_1, a_2, \dots , a_n$ of the integers $1, 2, \dots , n$ such that \[\sum_{k=1}^{n} k \cdot \cos \frac{2 \pi a_k}{n}=0.\]

Let $ABC$ be an isosceles triangle with apex at $C.$ Let $D$ and $E$ be two points on the sides $AC$ and $BC$ such that the angle bisectors $\angle DEB$ and $\angle ADE$ meet at $F,$ which lies on segment $AB.$ Prove that $F$ is the midpoint of $AB.$

Compute $\displaystyle\int_0^{\pi/3}x\tan^2(x)dx$.

Does there exist an angle $ \alpha\in(0,\pi/2)$ such that $ \sin\alpha$, $ \cos\alpha$, $ \tan\alpha$ and $ \cot\alpha$, taken in some order, are consecutive terms of an arithmetic progression?

If $\sin 2x \sin 3x = \cos 2x \cos 3x$, then one value for $x$ is A. $18^\circ$ B. $30^\circ$ C. $36^\circ$ D. $45^\circ$ E. $60^\circ$

Find all $x$ and $y$ which are rational multiples of $\pi$ with $0<x<y<\frac{\pi}{2}$ and $\tan x+\tan y =2$.

Triangle $ABC$ lies entirely in the first quadrant of the Cartesian plane, and its sides have slopes $63$, $73$, $97$. Suppose the curve $\mathcal V$ with equation $y=(x+3)(x^2+3)$ passes through the vertices of $ABC$. Find the sum of the slopes of the three tangents to $\mathcal V$ at each of $A$, $B$, $C$. [i]Proposed by Akshaj[/i]

Let $n$ be a positive integer. Show that \[\sum^{n}_{k=1}\tan^{2}\frac{k \pi}{2n+1}\] is an odd integer.

Square $AXYZ$ is inscribed in equiangular hexagon $ABCDEF$ with $X$ on $\overline{BC}$, $Y$ on $\overline{DE}$, and $Z$ on $\overline{EF}$. Suppose that $AB=40$, and $EF=41(\sqrt{3}-1)$. What is the side-length of the square? [asy] size(200); defaultpen(linewidth(1)); pair A=origin,B=(2.5,0),C=B+2.5*dir(60), D=C+1.75*dir(120),E=D-(3.19,0),F=E-1.8*dir(60); pair X=waypoint(B--C,0.345),Z=rotate(90,A)*X,Y=rotate(90,Z)*A; draw(A--B--C--D--E--F--cycle); draw(A--X--Y--Z--cycle,linewidth(0.9)+linetype("2 2")); dot("$A$",A,W,linewidth(4)); dot("$B$",B,dir(0),linewidth(4)); dot("$C$",C,dir(0),linewidth(4)); dot("$D$",D,dir(20),linewidth(4)); dot("$E$",E,dir(100),linewidth(4)); dot("$F$",F,W,linewidth(4)); dot("$X$",X,dir(0),linewidth(4)); dot("$Y$",Y,N,linewidth(4)); dot("$Z$",Z,W,linewidth(4)); [/asy] $ \textbf{(A)}\ 29\sqrt{3} \qquad\textbf{(B)}\ \frac{21}{2}\sqrt{2}+\frac{41}{2}\sqrt{3}\qquad\textbf{(C)}\ 20\sqrt{3}+16$ $\textbf{(D)}\ 20\sqrt{2}+13\sqrt{3} \qquad\textbf{(E)}\ 21\sqrt{6}$

Given triangle $ABC$ in the plane such that $\angle CAB = a > \pi/2$. Let $PQ$ be a segment whose midpoint is the point $A$. Prove that $(BP+CQ) \tan a/2 \geq BC$.

Show that for any vectors $a, b$ in Euclidean space, \[|a \times b|^3 \leq \frac{3 \sqrt 3}{8} |a|^2 |b|^2 |a-b|^2\] Remark. Here $\times$ denotes the vector product.

Let $a$ and $b$ be real numbers for which the following is true: $acscx + b cot x \ge 1$, for all $0 <x < \pi$ Find the least value of $a^2 + b$.

Find the positive integer $N$ for which there exist reals $\alpha, \beta, \gamma, \theta$ which obey \begin{align*} 0.1 &= \sin \gamma \cos \theta \sin \alpha, \\ 0.2 &= \sin \gamma \sin \theta \cos \alpha, \\ 0.3 &= \cos \gamma \cos \theta \sin \beta, \\ 0.4 &= \cos \gamma \sin \theta \cos \beta, \\ 0.5 &\ge \left\lvert N-100 \cos2\theta \right\rvert. \end{align*}[i]Proposed by Evan Chen[/i]