Investigate the behaviour as $t\to+\infty$ of solutions of the systems \[\begin{cases} \dot{x}=y\\ \dot{y}=2\sin y-y-x\end{cases}\] \[\begin{cases} \dot{x}=y\\ \dot{y}=2x-x^{3}-x^{2}-\epsilon y\end{cases}\] where $\epsilon\ll 1$.

Let $ABC$ be an acute-angled triangle. Let $AD,BE,CF$ be internal bisectors with $D, E, F$ on $BC, CA, AB$ respectively. Prove that \[\frac{EF}{BC}+\frac{FD}{CA}+\frac{DE}{AB}\geq 1+\frac{r}{R}\]

For any quadrilateral with the side lengths $a,$ $b,$ $c,$ $d$ and the area $S,$ prove the inequality$S\leq \frac{a+c}{2}\cdot \frac{b+d}{2}.$

A quadrilateral $ABCD$ without parallel sides is circumscribed around a circle with centre $O$. Prove that $O$ is a point of intersection of middle lines of quadrilateral $ABCD$ (i.e. barycentre of points $A,\,B,\,C,\,D$) iff $OA\cdot OC=OB\cdot OD$.

Evaluate $\int_0^{\frac{\pi}{2}} \ln (1+\sqrt[3]{\sin \theta})\cos \theta\ d\theta.$

In acute triangle $ABC$, $AB > AC$. Let $M$ be the midpoint of side $BC$. The exterior angle bisector of $\widehat{BAC}$ meet ray $BC$ at $P$. Point $K$ and $F$ lie on line $PA$ such that $MF \perp BC$ and $MK \perp PA$. Prove that $BC^2 = 4 PF \cdot AK$. [asy] defaultpen(fontsize(10)); size(7cm); pair A = (4.6,4), B = (0,0), C = (5,0), M = midpoint(B--C), I = incenter(A,B,C), P = extension(A, A+dir(I--A)*dir(-90), B,C), K = foot(M,A,P), F = extension(M, (M.x, M.x+1), A,P); draw(K--M--F--P--B--A--C); pair point = I; pair[] p={A,B,C,M,P,F,K}; string s = "A,B,C,M,P,F,K"; int size = p.length; real[] d; real[] mult; for(int i = 0; i<size; ++i) { d[i] = 0; mult[i] = 1;} string[] k= split(s,","); for(int i = 0;i<p.length;++i) { label("$"+k[i]+"$",p[i],mult[i]*dir(point--p[i])*dir(d[i])); }[/asy]

In quadrilateral $ABCD, \angle B$ is a right angle, diagonal $\overline{AC}$ is perpendicular to $\overline{CD},$ $AB=18, BC=21,$ and $CD=14.$ Find the perimeter of $ABCD$.

For every positive integer $n$, define $S_n$ to be the sum \[ S_n = \sum_{k = 1}^{2010} \left( \cos \frac{k! \, \pi}{2010} \right)^n . \] As $n$ approaches infinity, what value does $S_n$ approach?

Prove that if $\alpha$ and $\beta$ are acute angles and $\alpha$ < $\beta$ , then $\frac{tan \alpha}{\alpha} < \frac{tan \beta}{\beta} $

Let $ AB$ be a diameter of a circle $ S$, and let $ L$ be the tangent at $ A$. Furthermore, let $ c$ be a fixed, positive real, and consider all pairs of points $ X$ and $ Y$ lying on $ L$, on opposite sides of $ A$, such that $ |AX|\cdot |AY| \equal{} c$. The lines $ BX$ and $ BY$ intersect $ S$ at points $ P$ and $ Q$, respectively. Show that all the lines $ PQ$ pass through a common point.

Compute the sum $$\sum_{k=0}^{1273}\frac{1}{1 + tan^{2548}\left(\frac{k\pi}{2548}\right)}$$

Let $ C$ be the parameterized curve for a given positive number $ r$ and $ 0\leq t\leq \pi$, $ C: \left\{\begin{array}{ll} x \equal{} 2r(t \minus{} \sin t\cos t) & \quad \\ y \equal{} 2r\sin ^ 2 t & \quad \end{array} \right.$ When the point $ P$ moves on the curve $ C$, (1) Find the magnitude of acceleralation of the point $ P$ at time $ t$. (2) Find the length of the locus by which the point $ P$ sweeps for $ 0\leq t\leq \pi$. (3) Find the volume of the solid by rotation of the region bounded by the curve $ C$ and the $ x$-axis about the $ x$-axis. Edited.

Consider all segments dividing the area of a triangle $ABC$ in two equal parts. Find the length of the shortest segment among them, if the side lengths $a,$ $b,$ $c$ of triangle $ABC$ are given. How many of these shortest segments exist ?

Consider a trihedral angle $Sxyz$ with $\angle xSz = \alpha, \angle xSy = \beta$ and $\angle ySz =\gamma$. Let $A,B,C$ denote the dihedral angles at edges $y, z, x$ respectively. $(a)$ Prove that $\frac{\sin\alpha}{\sin A}=\frac{\sin\beta}{\sin B}=\frac{\sin\gamma}{\sin C}$ $(b)$ Show that $\alpha + \beta = 180^{\circ}$ if and only if $\angle A + \angle B = 180^{\circ}.$ $(c)$ Assume that $\alpha=\beta =\gamma = 90^{\circ}$. Let $O \in Sz$ be a fixed point such that $SO = a$ and let $M,N$ be variable points on $x, y$ respectively. Prove that $\angle SOM +\angle SON +\angle MON$ is constant and find the locus of the incenter of $OSMN$.

A regular dodecagon ($ 12$ sides) is inscribed in a circle with radius $ r$ inches. The area of the dodecagon, in square inches, is: $ \textbf{(A)}\ 3r^2 \qquad \textbf{(B)}\ 2r^2 \qquad \textbf{(C)}\ \frac{3r^2 \sqrt{3}}{4} \qquad \textbf{(D)}\ r^2 \sqrt{3} \qquad \textbf{(E)}\ 3r^2 \sqrt{3}$

Two distinct points $A$ and $B$ are chosen at random from 15 points equally spaced around a circle centered at $O$ such that each pair of points $A$ and $B$ has the same probability of being chosen. The probability that the perpendicular bisectors of $OA$ and $OB$ intersect strictly inside the circle can be expressed in the form $\frac{m}{n}$, where $m,n$ are relatively prime positive integers. Find $m+n$. [i]Ray Li.[/i]

Let $A=(0,0)$ and $B=(b,2)$ be points on the coordinate plane. Let $ABCDEF$ be a convex equilateral hexagon such that $\angle FAB=120^\circ,$ $\overline{AB}\parallel \overline{DE},$ $\overline{BC}\parallel \overline{EF,}$ $\overline{CD}\parallel \overline{FA},$ and the y-coordinates of its vertices are distinct elements of the set $\{0,2,4,6,8,10\}.$ The area of the hexagon can be written in the form $m\sqrt{n},$ where $m$ and $n$ are positive integers and n is not divisible by the square of any prime. Find $m+n.$

What is the sum of all possible values of $\cos\left(2\theta\right)$ if $\cos\left(2\theta\right)=2\cos\left(\theta\right)$ for a real number $\theta$? [i]2019 CCA Math Bonanza Team Round #3[/i]

In a triangle $ABC$, the excircle $\omega_a$ opposite $A$ touches $AB$ at $P$ and $AC$ at $Q$, while the excircle $\omega_b$ opposite $B$ touches $BA$ at $M$ and $BC$ at $N$. Let $K$ be the projection of $C$ onto $MN$ and let $L$ be the projection of $C$ onto $PQ$. Show that the quadrilateral $MKLP$ is cyclic. ([i]Bulgaria[/i])

Let $\theta$ be a real number such that $1 + \sin 2\theta -\left(\frac12 \sin 2\theta\right)^2= 0$. Compute the maximum value of $(1 + \sin \theta )(1 + \cos \theta)$.

A ball of mass $m$ is launched into the air. Ignore air resistance, but assume that there is a wind that exerts a constant force $F_0$ in the -$x$ direction. In terms of $F_0$ and the acceleration due to gravity $g$, at what angle above the positive $x$-axis must the ball be launched in order to come back to the point from which it was launched? $ \textbf{(A)}\ \tan^{-1}(F_0/mg)$ $\textbf{(B)}\ \tan^{-1}(mg/F_0)$ $\textbf{(C)}\ \sin^{-1}(F_0/mg)$ $\textbf{(D)}\ \text{the angle depends on the launch speed}$ $\textbf{(E)}\ \text{no such angle is possible}$

Acute-angled $~$ $\triangle{ABC}$ $~$ is given. A line $~$ $l$ $~$ parallel to side $~$ $AB$ $~$ passing through vertex $~$ $C$ $~$ is drawn. Let the angle bisectors of $~$ $\angle{BAC}$ $~$ and $~$ $\angle{ABC}$ $~$ intersect the sides $~$ $BC$ and $~$ $AC$ at points $~$ $D$ $~$ and $~$ $F$, and line $~$ $l$ $~$ at points $~$ $E$ $~$ and $~$ $G$ $~$ respectively. Prove that if $~$ $\overline{DE}=\overline{GF}$ $~$ then $~$ $\overline{AC}=\overline{BC}\, .$

Given a circle $(O)$ and two fixed points $B,C$ on $(O),$ and an arbitrary point $A$ on $(O)$ such that the triangle $ABC$ is acute. $M$ lies on ray $AB,$ $N$ lies on ray $AC$ such that $MA=MC$ and $NA=NB.$ Let $P$ be the intersection of $(AMN)$ and $(ABC),$ $P\ne A.$ $MN$ intersects $BC$ at $Q.$ a) Prove that $A,P,Q$ are collinear. b) $D$ is the midpoint of $BC.$ Let $K$ be the intersection of $(M,MA)$ and $(N,NA),$ $K\ne A.$ $d$ is the line passing through $A$ and perpendicular to $AK.$ $E$ is the intersection of $d$ and $BC.$ $(ADE)$ intersects $(O)$ at $F,$ $F\ne A.$ Prove that $AF$ passes through a fixed point.

Find the largest possible volume for a tetrahedron which lies inside a hemisphere of radius $1$.

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.$