Olympiad problems

2006 Germany Team Selection Test, 1

Find all real solutions $x$ of the equation $\cos\cos\cos\cos x=\sin\sin\sin\sin x$. (Angles are measured in radians.)

2013 F = Ma, 2

Tags: trigonometry , reflection , geometric transformation , geometry

Jordi stands 20 m from a wall and Diego stands 10 m from the same wall. Jordi throws a ball at an angle of 30 above the horizontal, and it collides elastically with the wall. How fast does Jordi need to throw the ball so that Diego will catch it? Consider Jordi and Diego to be the same height, and both are on the same perpendicular line from the wall. $\textbf{(A) } 11 \text{ m/s}\\ \textbf{(B) } 15 \text{ m/s}\\ \textbf{(C) } 19 \text{ m/s}\\ \textbf{(D) } 30 \text{ m/s}\\ \textbf{(E) } 35 \text{ m/s}$

2005 China National Olympiad, 1

Tags: inequalities , trigonometry

Suppose $\theta_{i}\in(-\frac{\pi}{2},\frac{\pi}{2}), i = 1,2,3,4$. Prove that, there exist $x\in \mathbb{R}$, satisfying two inequalities \begin{eqnarray*} \cos^2\theta_1\cos^2\theta_2-(\sin\theta\sin\theta_2-x)^2 &\geq& 0, \\ \cos^2\theta_3\cos^2\theta_4-(\sin\theta_3\sin\theta_4-x)^2 & \geq & 0 \end{eqnarray*} if and only if \[ \sum^4_{i=1}\sin^2\theta_i\leq2(1+\prod^4_{i=1}\sin\theta_i + \prod^4_{i=1}\cos\theta_i). \]

2008 Bosnia Herzegovina Team Selection Test, 1

Tags: derivative , geometry , trigonometry , inequalities , calculus

Prove that in an isosceles triangle $ \triangle ABC$ with $ AC\equal{}BC\equal{}b$ following inequality holds $ b> \pi r$, where $ r$ is inradius.

2000 Harvard-MIT Mathematics Tournament, 5

Tags: trigonometry , algebra

Given $\cos (\alpha + \beta) + sin (\alpha - \beta) = 0$, $\tan \beta =\frac{1}{2000}$, find $\tan \alpha$.

2007 Swedish Mathematical Competition, 3

Tags: circumradius , trigonometry , geometry

Let $\alpha$, $\beta$, $\gamma$ be the angles of a triangle. If $a$, $b$, $c$ are the side length of the triangle and $R$ is the circumradius, show that \[ \cot \alpha + \cot \beta +\cot \gamma =\frac{R\left(a^2+b^2+c^2\right)}{abc} \]

2008 Bulgaria Team Selection Test, 2

Tags: power of a point , radical axis , geometry , homothety , trigonometry , geometric transformation , circumcircle

In the triangle $ABC$, $AM$ is median, $M \in BC$, $BB_{1}$ and $CC_{1}$ are altitudes, $C_{1} \in AB$, $B_{1} \in AC$. The line through $A$ which is perpendicular to $AM$ cuts the lines $BB_{1}$ and $CC_{1}$ at points $E$ and $F$, respectively. Let $k$ be the circumcircle of $\triangle EFM$. Suppose also that $k_{1}$ and $k_{2}$ are circles touching both $EF$ and the arc $EF$ of $k$ which does not contain $M$. If $P$ and $Q$ are the points at which $k_{1}$ intersects $k_{2}$, prove that $P$, $Q$, and $M$ are collinear.

2014 Contests, 2

Tags: trigonometry , function

Let $f$ be the function defined by $f(x) = 4x(1 - x)$. Let $n$ be a positive integer. Prove that there exist distinct real numbers $x_1$, $x_2$, $\ldots\,$, $x_n$ such that $x_{i + 1} = f(x_i)$ for each integer $i$ with $1 \le i \le n - 1$, and such that $x_1 = f(x_n)$.

1970 Spain Mathematical Olympiad, 7

Tags: trigonometry

Calculate the values of the cosines of the angles $x$ that satisfy the next equation: $$\sin^2 x - 2 \cos^2 x +\frac12 \sin 2x = 0.$$

2000 National High School Mathematics League, 2

Tags: trigonometry

If $\sin\alpha>0,\cos\alpha<0,\sin\frac{\alpha}{3}>\cos\frac{\alpha}{3}$, then the range value of $\frac{\alpha}{3}$ is $\text{(A)}\left(2k\pi+\frac{\pi}{6},2k\pi+\frac{\pi}{3}\right),k\in\mathbb{Z}$ $\text{(B)}\left(\frac{2k\pi}{3}+\frac{\pi}{6},\frac{2k\pi}{3}+\frac{\pi}{3}\right),k\in\mathbb{Z}$ $\text{(C)}\left(2k\pi+\frac{5\pi}{6},2k\pi+\pi\right),k\in\mathbb{Z}$ $\text{(D)}\left(2k\pi+\frac{\pi}{4},2k\pi+\frac{\pi}{3}\right)\cup\left(2k\pi+\frac{5\pi}{6},2k\pi+\pi\right),k\in\mathbb{Z}$

2010 Today's Calculation Of Integral, 553

Tags: calculus , function , integration , trigonometry , calculus computations

Find the continuous function such that $ f(x)\equal{}\frac{e^{2x}}{2(e\minus{}1)}\int_0^1 e^{\minus{}y}f(y)dy\plus{}\int_0^{\frac 12} f(y)dy\plus{}\int_0^{\frac 12}\sin ^ 2(\pi y)dy$.

2015 Azerbaijan National Olympiad, 5

Tags: trigonometry , geometry

In the convex quadrilateral $ABCD$ angle $\angle{BAD}=90$,$\angle{BAC}=2\cdot\angle{BDC}$ and $\angle{DBA}+\angle{DCB}=180$. Then find the angle $\angle{DBA}$

2009 Today's Calculation Of Integral, 505

Tags: calculus computations , trigonometry , integration , calculus

In the $ xyz$ space with the origin $ O$, given a cuboid $ K: |x|\leq \sqrt {3},\ |y|\leq \sqrt {3},\ 0\leq z\leq 2$ and the plane $ \alpha : z \equal{} 2$. Draw the perpendicular $ PH$ from $ P$ to the plane. Find the volume of the solid formed by all points of $ P$ which are included in $ K$ such that $ \overline{OP}\leq \overline{PH}$.

2001 Croatia National Olympiad, Problem 2

Tags: ratio , geometry , triangle , trigonometry

In a triangle $ABC$ with $AC\ne BC$, $M$ is the midpoint of $AB$ and $\angle A=\alpha$, $\angle B=\beta$, $\angle ACM=\varphi$ and $\angle BSM=\Psi$. Prove that $$\frac{\sin\alpha\sin\beta}{\sin(\alpha-\beta)}=\frac{\sin\varphi\sin\Psi}{\sin(\varphi-\Psi)}.$$

1983 National High School Mathematics League, 9

Tags: trigonometry

In $\triangle ABC,\sin A=\frac{3}{5},\cos B=\frac{5}{13}$, then $\cos C=$________.

2011 Today's Calculation Of Integral, 735

Tags: trigonometry , logarithm , calculus computations , calculus , integration

Evaluate the following definite integrals: (a) $\int_0^{\frac{\sqrt{\pi}}{2}} x\tan (x^2)\ dx$ (b) $\int_0^{\frac 13} xe^{3x}\ dx$ (c) $\int_e^{e^e} \frac{1}{x\ln x}\ dx$ (d) $\int_2^3 \frac{x^2+1}{x(x+1)}\ dx$

2007 Moldova Team Selection Test, 3

Tags: trigonometry , radical axis , trapezoid , projective geometry , ratio , circumcircle , geometry

Let $M, N$ be points inside the angle $\angle BAC$ usch that $\angle MAB\equiv \angle NAC$. If $M_{1}, M_{2}$ and $N_{1}, N_{2}$ are the projections of $M$ and $N$ on $AB, AC$ respectively then prove that $M, N$ and $P$ the intersection of $M_{1}N_{2}$ with $N_{1}M_{2}$ are collinear.

2001 Brazil National Olympiad, 4

Tags: inequalities , algebra , function , trigonometry

A calculator treats angles as radians. It initially displays 1. What is the largest value that can be achieved by pressing the buttons cos or sin a total of 2001 times? (So you might press cos five times, then sin six times and so on with a total of 2001 presses.)

2003 China Second Round Olympiad, 1

Tags: trigonometry , geometry

From point $P$ outside a circle draw two tangents to the circle touching at $A$ and $B$. Draw a secant line intersecting the circle at points $C$ and $D$, with $C$ between $P$ and $D$. Choose point $Q$ on the chord $CD$ such that $\angle DAQ=\angle PBC$. Prove that $\angle DBQ=\angle PAC$.

2010 AMC 10, 19

Tags: perpendicular bisector , trigonometry , pythagorean theorem , quadratic , trig identities , law of cosines , geometry

A circle with center $ O$ has area $ 156\pi$. Triangle $ ABC$ is equilateral, $ \overline{BC}$ is a chord on the circle, $ OA \equal{} 4\sqrt3$, and point $ O$ is outside $ \triangle ABC$. What is the side length of $ \triangle ABC$? $ \textbf{(A)}\ 2\sqrt3 \qquad\textbf{(B)}\ 6 \qquad\textbf{(C)}\ 4\sqrt3 \qquad\textbf{(D)}\ 12 \qquad\textbf{(E)}\ 18$

1993 Taiwan National Olympiad, 6

Tags: trigonometry , algebra , induction , irrational number

Let $m$ be equal to $1$ or $2$ and $n<10799$ be a positive integer. Determine all such $n$ for which $\sum_{k=1}^{n}\frac{1}{\sin{k}\sin{(k+1)}}=m\frac{\sin{n}}{\sin^{2}{1}}$.

2006 Stanford Mathematics Tournament, 5

Tags: trigonometry , quadratic

There exist two positive numbers $ x$ such that $ \sin(\arccos(\tan(\arcsin x)))\equal{}x$. Find the product of the two possible $ x$.

2002 Iran MO (3rd Round), 9

Tags: invariant , geometric transformation , trigonometry , ratio , reflection , geometry , trapezoid

Let $ M$ and $ N$ be points on the side $ BC$ of triangle $ ABC$, with the point $ M$ lying on the segment $ BN$, such that $ BM \equal{} CN$. Let $ P$ and $ Q$ be points on the segments $ AN$ and $ AM$, respectively, such that $ \measuredangle PMC \equal{}\measuredangle MAB$ and $ \measuredangle QNB \equal{}\measuredangle NAC$. Prove that $ \measuredangle QBC \equal{}\measuredangle PCB$.

2007 Today's Calculation Of Integral, 180

Tags: trigonometry , integration , limit , calculus , geometry , calculus computations

Let $a_{n}$ be the area surrounded by the curves $y=e^{-x}$ and the part of $y=e^{-x}|\cos x|,\ (n-1)\pi \leq x\leq n\pi \ (n=1,\ 2,\ 3,\ \cdots).$ Evaluate $\lim_{n\to\infty}(a_{1}+a_{2}+\cdots+a_{n}).$

2014 AIME Problems, 14

Tags: number theory , trig identities , relatively prime , trigonometry , geometry

In $\triangle ABC$, $AB=10$, $\angle A=30^\circ$, and $\angle C=45^\circ$. Let $H,D$, and $M$ be points on line $\overline{BC}$ such that $\overline{AH}\perp\overline{BC}$, $\angle BAD=\angle CAD$, and $BM=CM$. Point $N$ is the midpoint of segment $\overline{HM}$, and point $P$ is on ray $AD$ such that $\overline{PN}\perp\overline{BC}$. Then $AP^2=\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.