Let $ABCDEF$ be a regular hexagon with side length $a$. At point $A$, the perpendicular $AS$, with length $2a\sqrt{3}$, is erected on the hexagon's plane. The points $M, N, P, Q,$ and $R$ are the projections of point $A$ on the lines $SB, SC, SD, SE,$ and $SF$, respectively. [list=a] [*]Prove that the points $M, N, P, Q, R$ lie on the same plane. [*]Find the measure of the angle between the planes $(MNP)$ and $(ABC)$.[/list]

The sign shown below consists of two uniform legs attached by a frictionless hinge. The coefficient of friction between the ground and the legs is $\mu$. Which of the following gives the maximum value of $\theta$ such that the sign will not collapse? $\textbf{(A) } \sin \theta = 2 \mu \\ \textbf{(B) } \sin \theta /2 = \mu / 2\\ \textbf{(C) } \tan \theta / 2 = \mu\\ \textbf{(D) } \tan \theta = 2 \mu \\ \textbf{(E) } \tan \theta / 2 = 2 \mu$

Prove the inequality $$\sum_{k=1}^{n}\frac{\sin (k+1)x}{\sin kx}< 2\frac{\cos x}{\sin^2x}$$ where $0 < nx < \pi/2$, $n \in N$.

A finite non-empty set of integers is called $3$-[i]good[/i] if the sum of its elements is divisible by $3$. Find the number of $3$-good subsets of $\{0,1,2,\ldots,9\}$.

Let $ a,\ b$ be real numbers satisfying $ \int_0^1 (ax\plus{}b)^2dx\equal{}1$. Determine the values of $ a,\ b$ for which $ \int_0^1 3x(ax\plus{}b)\ dx$ is maximized.

Calculate the following indefinite integrals. [1] $\int (x^2+4x-3)^2(x+2)dx$ [2] $\int \frac{\ln x}{x(\ln x+1)}dx$ [3] $\int \frac{\sin \ (\pi \log _2 x)}{x}dx$ [4] $\int \frac{dx}{\sin x\cos ^ 2 x}$ [5] $\int \sqrt{1-3x}\ dx$

In a triangle $ABC$, points $X$ and $Y$ are on $BC$ and $CA$ respectively such that $CX=CY$,$AX$ is not perpendicular to $BC$ and $BY$ is not perpendicular to $CA$.Let $\Gamma$ be the circle with $C$ as centre and $CX$ as its radius.Find the angles of triangle $ABC$ given that the orthocentres of triangles $AXB$ and $AYB$ lie on $\Gamma$.

The incircle of triangle $ABC$ touches $BC$ at $D$ and $AC$ at $E$. Let $K$ be the point on $CB$ with $CK=BD$, and $L$ be the point on $CA$ with $AE=CL$. Lines $AK$ and $BL$ meet at $P$. If $Q$ is the midpoint of $BC$, $I$ the incenter, and $G$ the centroid of $\triangle ABC$, show that: $(a)$ $IQ$ and $AK$ are parallel, $(b)$ the triangles $AIG$ and $QPG$ have equal area.

Find $ \lim_{n\to\infty} \frac{\pi}{n}\left\{\frac{1}{\sin \frac{\pi (n\plus{}1)}{4n}}\plus{}\frac{1}{\sin \frac{\pi (n\plus{}2)}{4n}}\plus{}\cdots \plus{}\frac{1}{\sin \frac{\pi (n\plus{}n)}{4n}}\right\}$

In triangle $ABC,$ $\sin \angle A=\frac{4}{5}$ and $\angle A<90^\circ$ Let $D$ be a point outside triangle $ABC$ such that $\angle BAD=\angle DAC$ and $\angle BDC = 90^{\circ}.$ Suppose that $AD=1$ and that $\frac{BD} {CD} = \frac{3}{2}.$ If $AB+AC$ can be expressed in the form $\frac{a\sqrt{b}}{c}$ where $a,b,c$ are pairwise relatively prime integers, find $a+b+c$. [i]Author: Ray Li[/i]

Let $ ABCDE$ be a regular pentagon of side length $ 1$. Let $ F$ be the midpoint of $ AB$ and let $ G$ and $ H$ be the points on sides $ CD$ and $ DE$ respectively $ \angle GFD \equal{} \angle HFD \equal{} 30^{\circ}$. Show that the triangle $ GFH$ is equilateral. A square of side $ a$ is inscribed in $ \triangle GFH$ with one side of the square along $ GH$. Prove that: $ FG \equal{} t \equal{} \frac {2 \cos 18^{\circ} \cos^2 36^{\circ}}{\cos 6^{\circ}}$ and $ a \equal{} \frac {t \sqrt {3}}{2 \plus{} \sqrt {3}}$.

For all $a,b,c\in \bb{R}^+ $ such that $a+b+c=1$ and $ ( \frac{1}{(a+b)^2}+\frac{1}{(b+c)^2}+\frac{1}{(c+a)^2} )(a-bc)(b-ac)(c-ab)\le M \cdot abc$. Find min $M$

Let $ABC$ be a triangle and let $M,N,P$ be points on the sides $BC$, $CA$ and $AB$ respectively such that \[ \frac{AP}{PB} = \frac{BM}{MC} = \frac{CN}{AN}. \] Prove that triangle if $MNP$ is equilateral then triangle $ABC$ is equilateral.

An acute angle $XAY$ and a point $P$ inside the angle are given. Construct (using a ruler and a compass) a line that passes through $P$ and intersects the rays $AX$ and $AY$ at $B$ and $C$ such that the area of the triangle $ABC$ equals $AP^2$. [i]Greece[/i]

Prove that there is positive integers $N$ such that the equation $$arctan(N)=\sum_{i=1}^{2020} a_i arctan(i),$$ does not hold for any integers $a_{i}.$

Let $ABC$ be a triangle with circumcenter $O$, incenter $I$, $\angle B=45^\circ$, and $OI\parallel BC$. Find $\cos\angle C$.

Faces $ABC$ and $BCD$ of tetrahedron $ABCD$ meet at an angle of $30^\circ$. The area of face $ABC$ is $120$, the area of face $BCD$ is $80$, and $BC=10$. Find the volume of the tetrahedron.

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]

Solve in the real numbers the equation $ \arctan\sqrt{3^{1-2x}} +\arctan {3^x} =\frac{7\pi }{12} . $ [i]Ovidiu Țâțan[/i]

A sector with acute central angle $ \theta$ is cut from a circle of radius 6. The radius of the circle circumscribed about the sector is $ \textbf{(A)}\ 3\cos\theta \qquad \textbf{(B)}\ 3\sec\theta \qquad \textbf{(C)}\ 3 \cos \frac12 \theta \qquad \textbf{(D)}\ 3 \sec \frac12 \theta \qquad \textbf{(E)}\ 3$

The sequence $(s_n)$, where $s_n= \sum_{k=1}^n \sin k$ for $n = 1, 2,\dots$ is bounded. Find an upper and lower bound.

Point $O$ is the center of the circle into which quadrilateral $ABCD$ is inscribed. If angles $AOC$ and $BAD$ are both equal to $110$ degrees and angle $ABC$ is greater than angle $ADC$, prove that $AB+AD>CD$. Fresh translation.

Evaluate $ \int_{0}^{1}e^{\sqrt{e^{x}}}\ dx\plus{}2\int_{e}^{e^{\sqrt{e}}}\ln (\ln x)\ dx$.

Square $ ABCD$ has side length $ s$, a circle centered at $ E$ has radius $ r$, and $ r$ and $ s$ are both rational. The circle passes through $ D$, and $ D$ lies on $ \overline{BE}$. Point $ F$ lies on the circle, on the same side of $ \overline{BE}$ as $ A$. Segment $ AF$ is tangent to the circle, and $ AF \equal{} \sqrt {9 \plus{} 5\sqrt {2}}$. What is $ r/s$? [asy]unitsize(6mm); defaultpen(linewidth(.8pt)+fontsize(10pt)); dotfactor=3; pair B=(0,0), C=(3,0), D=(3,3), A=(0,3); pair Ep=(3+5*sqrt(2)/6,3+5*sqrt(2)/6); pair F=intersectionpoints(Circle(A,sqrt(9+5*sqrt(2))),Circle(Ep,5/3))[0]; pair[] dots={A,B,C,D,Ep,F}; draw(A--F); draw(Circle(Ep,5/3)); draw(A--B--C--D--cycle); dot(dots); label("$A$",A,NW); label("$B$",B,SW); label("$C$",C,SE); label("$D$",D,SW); label("$E$",Ep,E); label("$F$",F,NW);[/asy]$ \textbf{(A) } \frac {1}{2}\qquad \textbf{(B) } \frac {5}{9}\qquad \textbf{(C) } \frac {3}{5}\qquad \textbf{(D) } \frac {5}{3}\qquad \textbf{(E) } \frac {9}{5}$

For a fixed $\theta \in \lbrack 0, \frac{\pi}{2} \rbrack$, find the smallest $a \in \mathbb{R}^{+}$ which satisfies the following conditions: [b]I. [/b] $\frac{\sqrt a}{\cos \theta} + \frac{\sqrt a}{\sin \theta} > 1$. [b]II.[/b] There exists $x \in \lbrack 1 - \frac{\sqrt a}{\sin \theta}, \frac{\sqrt a}{\cos \theta} \rbrack$ such that $\lbrack (1 - x)\sin \theta - \sqrt{a - x^2 \cos^{2} \theta} \rbrack^{2} + \lbrack x\cos \theta - \sqrt{a - (1 - x)^2 \sin^{2} \theta} \rbrack^{2} \leq a$.