Let $ABC$ be a triangle with $\angle C=60^\circ$ and $AC<BC$. The point $D$ lies on the side $BC$ and satisfies $BD=AC$. The side $AC$ is extended to the point $E$ where $AC=CE$. Prove that $AB=DE$.

Solve the equation $ \sin^3x \cos 3x \plus{} \cos^3x \sin 3x \equal{} \frac{3}{8}$.

Prove that for any real number $ x$ the following inequality is true: $ \max\{|\sin x|, |\sin(x\plus{}2010)|\}>\dfrac1{\sqrt{17}}$

A spherical shell of mass $M$ and radius $R$ is completely filled with a frictionless fluid, also of mass M. It is released from rest, and then it rolls without slipping down an incline that makes an angle $\theta$ with the horizontal. What will be the acceleration of the shell down the incline just after it is released? Assume the acceleration of free fall is $g$. The moment of inertia of a thin shell of radius $r$ and mass $m$ about the center of mass is $I = \frac{2}{3}mr^2$; the momentof inertia of a solid sphere of radius r and mass m about the center of mass is $I = \frac{2}{5}mr^2$. $\textbf{(A) } g \sin \theta \\ \textbf{(B) } \frac{3}{4} g \sin \theta\\ \textbf{(C) } \frac{1}{2} g \sin \theta\\ \textbf{(D) } \frac{3}{8} g \sin \theta\\ \textbf{(E) } \frac{3}{5} g \sin \theta$

Let $P$, $Q$, and $R$ be the points where the incircle of a triangle $ABC$ touches the sides $AB$, $BC$, and $CA$, respectively. Prove the inequality $\frac{BC} {PQ} + \frac{CA} {QR} + \frac{AB} {RP} \geq 6$.

In triangle $ABC$, $AB>AC$. The bisectors of $\angle{B},\angle{C}$ intersect the sides $AC,AB$ at $P,Q$, respectively. Let $I$ be the incenter of $\Delta ABC$. Suppose that $IP=IQ$. How much isthe value of $\angle A$?

Let $a,b,c$ denote the sides of a triangle and $[ABC]$ the area of the triangle as usual. $(a)$ If $6[ABC] = 2a^2+bc$, determine $A,B,C$. $(b)$ For all triangles, prove that $3a^2+3b^2 - c^2 \ge 4 \sqrt{3} [ABC]$.

Let $P$ be an arbitrary point inside triangle $ABC$, denote by $A_{1}$ (different from $P$) the second intersection of line $AP$ with the circumcircle of triangle $PBC$ and define $B_{1},C_{1}$ similarly. Prove that $\left(1 \plus{} 2\cdot\frac {PA}{PA_{1}}\right)\left(1 \plus{} 2\cdot\frac {PB}{PB_{1}}\right)\left(1 \plus{} 2\cdot\frac {PC}{PC_{1}}\right)\geq 8$.

Does there exist a function $f: \mathbb{R}\rightarrow\mathbb{R}$ such that $f(x+f(y))=f(x)+\sin y$, for all reals $x,y$ ?

Let $f : \mathbb R \to \mathbb R$ be of the form $f(x) = x + \epsilon \sin x,$ where $0 < |\epsilon| \leq 1.$ Define for any $x \in \mathbb R,$ \[x_n=\underbrace{f \ o \ \ldots \ o \ f}_{n \text{ times}} (x).\] Show that for every $x \in \mathbb R$ there exists an integer $k$ such that $\lim_{n\to \infty } x_n = k\pi.$

Find all $x$ for which the inequality holds $$9 \sin x +40 \cos x \ge 41.$$

Draw on the coordinate plane the set of points whose coordinates satisfy the equation $$\sin x \cos^2 y +\sin y \cos^2 x =0$$

Compute $\sin^4\left(7.5^\circ\right)+\sin^4\left(82.5^\circ\right)$. [i]2019 CCA Math Bonanza Lightning Round #2.3[/i]

Find all functions $f(x)$ defined in the range $\left(-\frac\pi2,\frac\pi2\right)$ that are differentiable at $0$ and satisfy $$f(x)=\frac12\left(1+\frac1{\cos x}\right)f\left(\frac x2\right)$$ for every $x$ in the range $\left(-\frac\pi2,\frac\pi2\right)$. [i]L. Davidov[/i]

a) Prove that for $n = 2,3,4,...$ holds: $$\sin a + \sin 2a + ...+ \sin (n-1)a=\frac{\cos a \left(\frac{a}{2}\right) - \cos \left(n-\frac{1}{2}\right) a}{2 \sin \left(\frac{a}{2}\right)}$$ b) A point on the circumference of a wheel, which, remaining in a vertical plane, rolls along a horizontal path, describes, at one revolution of the wheel, a curve having a length equal to four times the diameter of the wheel. Prove this by first considering tilting a regular $n$-gon. [hide=original wording for part b]Een punt van de omtrek van een wiel dat, in een verticaal vlak blijvend, rolt over een horizontaal gedachte weg, beschrijft bij één omwenteling van het wiel een kromme die een lengte heeft die gelijk is aan viermaal de middellijn van het wiel. Bewijs dit door eerst een rondkantelende regelmatige n-hoek te beschouwen.[/hide]

The polynomial $P(x) = x^3 + \sqrt{6} x^2 - \sqrt{2} x - \sqrt{3}$ has three distinct real roots. Compute the sum of all $0 \le \theta < 360$ such that $P(\tan \theta^\circ) = 0$. [i]Proposed by Lewis Chen[/i]

Draw on the plane $(p, q)$ all points with coordinates $(p,q)$, for which the equation $\sin^2x+p\sin x+q=0$ has solutions and all its positive solutions form an arithmetic progression.

Let $\triangle ABC$ be an acute triangle of circumcentre $O$. Let the tangents to the circumcircle of $\triangle ABC$ in points $B$ and $C$ meet at point $P$. The circle of centre $P$ and radius $PB=PC$ meets the internal angle bisector of $\angle BAC$ inside $\triangle ABC$ at point $S$, and $OS \cap BC = D$. The projections of $S$ on $AC$ and $AB$ respectively are $E$ and $F$. Prove that $AD$, $BE$ and $CF$ are concurrent. [i]Author: Cosmin Pohoata[/i]

Evaluate ${ \int_0^{\frac{\pi}{8}}\frac{\cos x}{\cos (x-\frac{\pi}{8}})}\ dx$.

Given that points $ D,E$ lie on the sidelines $ AB,BC$ of triangle $ ABC$, respectively, point $ P$ is in interior of triangle $ ABC$ such that $ PE \equal{} PC$ and $ \bigtriangleup DEP\sim \bigtriangleup PCA.$ Prove that $ BP$ is tangent of the circumcircle of triangle $ PAD.$

A quadrilateral $ABCD$ is circumscribed about a circle $\omega$. The lines $AB$ and $CD$ meet at $O$. A circle $\omega_1$ is tangent to side $BC$ at $K$ and to the extensions of sides $AB$ and $CD$, and a circle $\omega_2$ is tangent to side $AD$ at $L$ and to the extensions of sides $AB$ and $CD$. Suppose that points $O$, $K$, $L$ lie on a line. Prove that the midpoints of $BC$ and $AD$ and the center of $\omega$ also lie on a line.

$f(x)$ is a monic polynomial of degree $2$ with integer coefficients such that $f(x)$ doesn't have any real roots and also $f(0)$ is a square-free integer (and is not $1$ or $-1$). Prove that for every integer $n$ the polynomial $f(x^n)$ is irreducible over $\mathbb Z[x]$. [i]proposed by Mohammadmahdi Yazdi[/i]

Points $D$ and $E$ divides the base $BC$ of an isosceles triangle $ABC$ into three equal parts and $D$ is between $B$ and $E.$ Show that $\angle BAD<\angle DAE.$

A function $f$ has its domain equal to the set of integers $0$, $1$, $\ldots$, $11$, and $f(n)\geq 0$ for all such $n$, and $f$ satisfies [list] [*]$f(0)=0$ [*]$f(6)=1$ [*]If $x\geq 0$, $y\geq 0$, and $x+y\leq 11$, then $f(x+y)=\tfrac{f(x)+f(y)}{1-f(x)f(y)}$.[/list] Find $f(2)^2+f(10)^2$.