Let K be a point on the side BC of the triangle ABC. The incircles of the triangles ABK and ACK touch BC at points M and N, respectively. Show that [tex]BM\cdot CN>KM \cdot KN[/tex].

Let $ x$ be a real number such that the five numbers $ \cos(2 \pi x)$, $ \cos(4 \pi x)$, $ \cos(8 \pi x)$, $ \cos(16 \pi x)$, and $ \cos(32 \pi x)$ are all nonpositive. What is the smallest possible positive value of $ x$?

Let $ABC$ be a triangle each of whose angles is greater than $30^{\circ}$. Suppose a circle centered with $P$ cuts segments $BC$ in $T,Q; CA$ in $K,L$ and $AB$ in $M,N$ such that they are on a circle in counterclockwise direction in that order.Suppose further $PQK,PLM,PNT$ are equilateral. Prove that: $a)$ The radius of the circle is $\frac{2abc}{a^2+b^2+c^2+4\sqrt{3}S}$ where $S$ is area. $b) a\cdot AP=b\cdot BP=c\cdot PC.$

Triangle $ABC$ has a right angle at $C$. If $\sin A = \frac{2}{3}$, then $\tan B$ is $ \textbf{(A)}\ \frac{3}{5}\qquad\textbf{(B)}\ \frac{\sqrt 5}{3}\qquad\textbf{(C)}\ \frac{2}{\sqrt 5}\qquad\textbf{(D)}\ \frac{\sqrt 5}{2}\qquad\textbf{(E)}\ \frac{5}{3} $

Characterize all triangles $ABC$ s.t. \[ AI_a : BI_b : CI_c = BC: CA : AB \] where $I_a$ etc. are the corresponding excentres to the vertices $A, B , C$

Compute $$\frac{\sum_{k=0}^{2022}\sin\frac{k\pi}{3033}}{\sum_{k=0}^{2022}\cos\frac{k\pi}{3033}}.$$

Let ABC be a triangle and let $r, r_a, r_b, r_c$ denote the inradius and ex-radii opposite to the vertices $A, B, C$, respectively. Suppose that $a>r_a, b>r_b, c>r_c$. Prove that [b](a)[/b] $\triangle ABC$ is acute. [b](b)[/b] $a+b+c > r+r_a+r_b+r_c$.

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]

Let $0 < a, b, c < 1$ with $ab + bc + ca = 1$. Prove that \[\frac{a}{1-a^2} + \frac{b}{1-b^2} + \frac{c}{1-c^2} \geq \frac {3 \sqrt{3}}{2}.\] Determine when equality holds.

Angles $\alpha, \beta, \gamma$ satisfy the inequality $\sin \alpha +\sin \beta +\sin \gamma \ge 2$. Prove that $\cos \alpha + \cos \beta +\cos \gamma \le \sqrt5.$

Find the positive integer $ n$ such that \[\arctan\frac{1}{3}\plus{}\arctan\frac{1}{4}\plus{}\arctan\frac{1}{5}\plus{}\arctan\frac{1}{n}\equal{}\frac{\pi}{4}.\]

In triangle $\Delta ADC$ we got $AD=DC$ and $D=100^\circ$. In triangle $\Delta CAB$ we got $CA=AB$ and $A=20^\circ$. Prove that $AB=BC+CD$.

Find maximal value of the function $f(x)=8-3\sin^2 (3x)+6 \sin (6x)$

Let $\alpha, \beta$ be real numbers such that $\sin\alpha\sin\beta=\frac{1}{3}$. Prove that the set of possible values of $\cos \alpha \cos \beta$ is the interval $\left[-\frac{2}{3}, \frac{2}{3}\right]$.

The incircle of a triangle $ABC$ touches the sides $BC$, $CA$, $AB$ at the points $D$, $E$, $F$ respectively. Let the line $AD$ intersect this incircle of triangle $ABC$ at a point $X$ (apart from $D$). Assume that this point $X$ is the midpoint of the segment $AD$, this means, $AX = XD$. Let the line $BX$ meet the incircle of triangle $ABC$ at a point $Y$ (apart from $X$), and let the line $CX$ meet the incircle of triangle $ABC$ at a point $Z$ (apart from $X$). Show that $EY = FZ$.

Points $ K$, $ L$, $ M$, and $ N$ lie in the plane of the square $ ABCD$ so that $ AKB$, $ BLC$, $ CMD$, and $ DNA$ are equilateral triangles. If $ ABCD$ has an area of $ 16$, find the area of $ KLMN$. [asy]unitsize(2cm); defaultpen(fontsize(8)+linewidth(0.8)); pair A=(-0.5,0.5), B=(0.5,0.5), C=(0.5,-0.5), D=(-0.5,-0.5); pair K=(0,1.366), L=(1.366,0), M=(0,-1.366), N=(-1.366,0); draw(A--N--K--A--B--K--L--B--C--L--M--C--D--M--N--D--A); label("$A$",A,SE); label("$B$",B,SW); label("$C$",C,NW); label("$D$",D,NE); label("$K$",K,NNW); label("$L$",L,E); label("$M$",M,S); label("$N$",N,W);[/asy] $ \textbf{(A)}\ 32 \qquad \textbf{(B)}\ 16 \plus{} 16\sqrt {3} \qquad \textbf{(C)}\ 48 \qquad \textbf{(D)}\ 32 \plus{} 16\sqrt {3} \qquad \textbf{(E)}\ 64$

Let $f,g$ are defined in $(a,b)$ such that $f(x),g(x)\in\mathcal{C}^2$ and non-decreasing in an interval $(a,b)$ . Also suppose $f^{\prime \prime}(x)=g(x),g^{\prime \prime}(x)=f(x)$. Also it is given that $f(x)g(x)$ is linear in $(a,b)$. Show that $f\equiv 0 \text{ and } g\equiv 0$ in $(a,b)$.

Let P be an interior point of a regular n-gon $ A_1 A_2 ...A_n$, the lines $ A_i P$ meet the regular n-gon at another point $ B_i$, where $ i\equal{}1,2,...,n$. Prove that sums of all $ PA_i\geq$ sum of all $ PB_i$.

Find all functions $f: \mathbb{R} \to \mathbb{R}$ such that \[f(x)f(y) = (x+y+1)^2 \cdot f \left( \frac{xy-1}{x+y+1} \right)\] $\forall x,y \in \mathbb{R}$ with $x+y+1 \neq 0$ and $f(x) > 1$ $\forall x > 0.$

Evaluate \[\sum_{n=1}^{\infty} \int_{2n\pi}^{2(n+1)\pi} \frac{x\sin x+\cos x}{x^2}\ dx\ (n=1,2,\cdots)\]

Among all the triangles which have a fixed side $l$ and a fixed area $S$, determine for which triangles the product of the altitudes is maximum.

Prove the identity: $$\cos \frac{\pi}{7} \cdot \cos \frac{4\pi}{7} \cdot \cos \frac{5\pi}{7} = \frac{1}{8}$$

Let $x_1,x_2,\ldots,x_n$ be arbitrary real numbers. Prove the inequality \[ \frac{x_1}{1+x_1^2} + \frac{x_2}{1+x_1^2 + x_2^2} + \cdots + \frac{x_n}{1 + x_1^2 + \cdots + x_n^2} < \sqrt{n}. \]

Triangle $ABC$ is inscribed in circle $\omega$ with $AB = 5$, $BC = 7$, and $AC = 3$. The bisector of angle $A$ meets side $BC$ at $D$ and circle $\omega$ at a second point $E$. Let $\gamma$ be the circle with diameter $DE$. Circles $\omega$ and $\gamma$ meet at $E$ and a second point $F$. Then $AF^2 = \frac mn$, where m and n are relatively prime positive integers. Find $m + n$.

In a triangle $ABC$, $T$ is the centroid and $ \angle TAB = \angle ACT$. Find the maximum possible value of $sin \angle CAT +sin \angle CBT$.