The adjoining figure shows two intersecting chords in a circle, with $B$ on minor arc $AD$. Suppose that the radius of the circle is 5, that $BC = 6$, and that $AD$ is bisected by $BC$. Suppose further that $AD$ is the only chord starting at $A$ which is bisected by $BC$. It follows that the sine of the minor arc $AB$ is a rational number. If this fraction is expressed as a fraction $m/n$ in lowest terms, what is the product $mn$? [asy] size(200); defaultpen(linewidth(0.7)+fontsize(10)); pair A=dir(200), D=dir(95), M=midpoint(A--D), C=dir(30), BB=C+2*dir(C--M), B=intersectionpoint(M--BB, Circle(origin, 1)); draw(Circle(origin, 1)^^A--D^^B--C); real r=0.05; pair M1=midpoint(M--D), M2=midpoint(M--A); draw((M1+0.1*dir(90)*dir(A--D))--(M1+0.1*dir(-90)*dir(A--D))); draw((M2+0.1*dir(90)*dir(A--D))--(M2+0.1*dir(-90)*dir(A--D))); pair point=origin; label("$A$", A, dir(point--A)); label("$B$", B, dir(point--B)); label("$C$", C, dir(point--C)); label("$D$", D, dir(point--D));[/asy]

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]

Solve the equation $\cos^2{x}+\cos^2{2x}+\cos^2{3x}=1$

Evaluate $ \int_{0}^{\pi}\sqrt{(\cos x\plus{}\cos 2x\plus{}\cos 3x)^{2}\plus{}(\sin x\plus{}\sin 2x\plus{}\sin 3x)^{2}}\ dx$.

Let $ AD$ be height of triangle $ \triangle ABC$ and $ R$ circumradius. Denote by $ E$ and $ F$ feet of perpendiculars from point $ D$ to sides $ AB$ and $ AC$. If $ AD\equal{}R\sqrt{2}$, prove that circumcenter of triangle $ \triangle ABC$ lies on line $ EF$.

Let $x,\ y$ be real numbers. For a function $f(t)=x\sin t+y\cos t$, draw the domain of the points $(x,\ y)$ for which the following inequality holds. \[\left|\int_{-\pi}^{\pi} f(t)\cos t\ dt\right|\leq \int_{-\pi}^{\pi} \{f(t)\}^2dt.\]

Triangle $ABC$ is inscribed in circle $\omega$. Circle $\gamma$ is tangent to $AB$ and $AC$ at points $P$ and $Q$ respectively. Also circle $\gamma$ is tangent to circle $\omega$ at point $S$. Let the intesection of $AS$ and $PQ$ be $T$. Prove that $\angle{BTP}=\angle{CTQ}$.

Find all $t\in \mathbb R$, such that $\int_{0}^{\frac{\pi}{2}}\mid \sin x+t\cos x\mid dx=1$ .

Compute $\displaystyle\lim_{n\to\infty}\dfrac{\sum_{k=1}^n|\cos(k)|}{n}$.

Let $\triangle ABC$ be an acute triangle with circumcenter $O$ such that $AB<AC$, let $Q$ be the intersection of the external bisector of $\angle A$ with $BC$, and let $P$ be a point in the interior of $\triangle ABC$ such that $\triangle BPA$ is similar to $\triangle APC$. Show that $\angle QPA + \angle OQB = 90^{\circ}$. [i]Alex Zhu.[/i]

A circle passes through the points $A,C$ of triangle $ABC$ intersects with the sides $AB,BC$ at points $D,E$ respectively. Let $ \frac{BD}{CE}=\frac{3}{2}$, $BE=4$, $AD=5$ and $AC=2\sqrt{7} $. Find the angle $ \angle BDC$.

Find the maximal positive integer $n$, so that for any real number $x$ we have $\sin^{n}{x}+\cos^{n}{x} \geq \frac{1}{n}$.

Find the sum of the values of $x$ such that $\cos^3 3x+ \cos^3 5x = 8 \cos^3 4x \cos^3 x$, where $x$ is measured in degrees and $100< x< 200$.

$ABC$ is acute-angled. $D$ s a variable point on the side BC. $O_1$ is the circumcenter of $ABD$, $O_2$ is the circumcenter of $ACD$, and $O$ is the circumcenter of $AO_1O_2$. Find the locus of $O$.

Let $\omega,\omega_1,\omega_2$ be three mutually tangent circles such that $\omega_1,\omega_2$ are externally tangent at $P$, $\omega_1,\omega$ are internally tangent at $A$, and $\omega,\omega_2$ are internally tangent at $B$. Let $O,O_1,O_2$ be the centers of $\omega,\omega_1,\omega_2$, respectively. Given that $X$ is the foot of the perpendicular from $P$ to $AB$, prove that $\angle{O_1XP}=\angle{O_2XP}$. [i]David Yang.[/i]

Find the smallest positive $x$ for which holds the inequality $$\sin x \le \sin (x+1)\le \sin (x+2)\le sin (x+3)\le \sin (x+4) .$$

Assuming that the roots of $x^3 +px^2 +qx +r=0$ are all real and positive, find the relation between $p,q,r$ which is a necessary and sufficient condition that the roots are the cosines of the angles of a triangle.

Find all functions $f:[-1,1] \rightarrow \mathbb{R},$ which satisfy $$f(\sin{x})+f(\cos{x})=2020$$ for any real number $x.$

Consider the function \[f(x)= \sin \biggl( \frac{\pi}{2} \lfloor x \rfloor \biggr).\] Find the period of $f$ and sketch diagram of $f$ in one period. Also prove that $\lim_{x \to 1} f(x)$ does not exist.

Given that $O$ is a regular octahedron, that $C$ is the cube whose vertices are the centers of the faces of $O$, and that the ratio of the volume of $O$ to that of $C$ is $\frac{m}{n}$, where $m$ and $n$ are relatively prime integers, find $m+n$.

Let $ABC$ be an acute triangle. Let $\omega$ be a circle whose centre $L$ lies on the side $BC$. Suppose that $\omega$ is tangent to $AB$ at $B'$ and $AC$ at $C'$. Suppose also that the circumcentre $O$ of triangle $ABC$ lies on the shorter arc $B'C'$ of $\omega$. Prove that the circumcircle of $ABC$ and $\omega$ meet at two points. [i]Proposed by Härmel Nestra, Estonia[/i]

The cyclic octagon $ABCDEFGH$ has sides $a,a,a,a,b,b,b,b$ respectively. Find the radius of the circle that circumscribes $ABCDEFGH.$

Let $ABC$ be an acute-angled triangle of area 1. Show that the triangle whose vertices are the feet of the perpendiculars from the centroid $G$ to $AB$, $BC$, $CA$ has area between $\frac 4{27}$ and $\frac 14$.

The point $P_1, P_2,\cdots ,P_{2018} $ is placed inside or on the boundary of a given regular pentagon. Find all placement methods are made so that $$S=\sum_{1\leq i<j\leq 2018}|P_iP_j| ^2$$takes the maximum value.

In a triangle ABC, D is midpoint of BC . If $\angle ADB = 45 ^{\circ}$ and $\angle ACD = 30^{\circ}$, determine $\angle BAD.$