Olympiad problems

2003 Vietnam National Olympiad, 1

Let $f: \mathbb{R}\to\mathbb{R}$ is a function such that $f( \cot x ) = \cos 2x+\sin 2x$ for all $0 < x < \pi$. Define $g(x) = f(x) f(1-x)$ for $-1 \leq x \leq 1$. Find the maximum and minimum values of $g$ on the closed interval $[-1, 1].$

1967 IMO Longlists, 55

Tags: trigonometry , trigonometric inequality , inequality , algebra

Find all $x$ for which, for all $n,$ \[\sum^n_{k=1} \sin {k x} \leq \frac{\sqrt{3}}{2}.\]

2008 India Regional Mathematical Olympiad, 1

Tags: semicircle , geometry , straight lines , trigonometry , intersection

On a semicircle with diameter $AB$ and centre $S$, points $C$ and $D$ are given such that point $C$ belongs to arc $AD$. Suppose $\angle CSD = 120^\circ$. Let $E$ be the point of intersection of the straight lines $AC$ and $BD$ and $F$ the point of intersection of the straight lines $AD$ and $BC$. Prove that $EF=\sqrt{3}AB$.

2009 Today's Calculation Of Integral, 462

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

Evaluate $ \int_0^1 \frac{(1\minus{}x\plus{}x^2)\cos \ln (x\plus{}\sqrt{1\plus{}x^2})\minus{}\sqrt{1\plus{}x^2}\sin \ln (x\plus{}\sqrt{1\plus{}x^2})}{(1\plus{}x^2)^{\frac{3}{2}}}\ dx$.

1969 IMO Shortlist, 38

Tags: algebra , series summation , trigonometry , trigonometric identities

$(HUN 5)$ Let $r$ and $m (r \le m)$ be natural numbers and $Ak =\frac{2k-1}{2m}\pi$. Evaluate $\frac{1}{m^2}\displaystyle\sum_{k=1}^{m}\displaystyle\sum_{l=1}^{m}\sin(rA_k)\sin(rA_l)\cos(rA_k-rA_l)$

2014 India IMO Training Camp, 1

Tags: geometry , circumcircle , trigonometry , geometric transformation , reflection , dilation , radical axis

In a triangle $ABC$, with $AB\neq AC$ and $A\neq 60^{0},120^{0}$, $D$ is a point on line $AC$ different from $C$. Suppose that the circumcentres and orthocentres of triangles $ABC$ and $ABD$ lie on a circle. Prove that $\angle ABD=\angle ACB$.

2010 Today's Calculation Of Integral, 551

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

In the coordinate plane, find the area of the region bounded by the curve $ C: y\equal{}\frac{x\plus{}1}{x^2\plus{}1}$ and the line $ L: y\equal{}1$.

1995 AIME Problems, 7

Tags: trigonometry , quadratic , number theory , relatively prime , algebra , quadratic formula

Given that $(1+\sin t)(1+\cos t)=5/4$ and \[ (1-\sin t)(1-\cos t)=\frac mn-\sqrt{k}, \] where $k, m,$ and $n$ are positive integers with $m$ and $n$ relatively prime, find $k+m+n.$

2009 Olympic Revenge, 2

Tags: integration , trigonometry , real analysis

Prove that $\int_{0}^{\frac{\pi}{2}} arctg (1 - \sin^2x\cos^2x)dx = \frac{\pi^2}{4} - \pi arctg\sqrt{\frac{\sqrt{2}-1}{2}}$

2010 Today's Calculation Of Integral, 556

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

Prove the following inequality. \[ \sqrt[3]{\int_0^{\frac {\pi}{4}} \frac {x}{\cos ^ 2 x\cos ^ 2 (\tan x)\cos ^ 2(\tan (\tan x))\cos ^ 2(\tan (\tan (\tan x)))}dx}<\frac{4}{\pi}\] Last Edited. Sorry, I have changed the problem. kunny

2009 Ukraine Team Selection Test, 10

Tags: geometry , circumcircle , homothety , trigonometry , quadrilateral , inversion

Let $ ABCD$ be a convex quadrilateral and let $ P$ and $ Q$ be points in $ ABCD$ such that $ PQDA$ and $ QPBC$ are cyclic quadrilaterals. Suppose that there exists a point $ E$ on the line segment $ PQ$ such that $ \angle PAE \equal{} \angle QDE$ and $ \angle PBE \equal{} \angle QCE$. Show that the quadrilateral $ ABCD$ is cyclic. [i]Proposed by John Cuya, Peru[/i]

2011 AIME Problems, 8

Tags: algebra , polynomial , analytic geometry , geometry , geometric transformation , reflection , trigonometry

Let $z_1,z_2,z_3,\dots,z_{12}$ be the 12 zeroes of the polynomial $z^{12}-2^{36}$. For each $j$, let $w_j$ be one of $z_j$ or $i z_j$. Then the maximum possible value of the real part of $\displaystyle\sum_{j=1}^{12} w_j$ can be written as $m+\sqrt{n}$ where $m$ and $n$ are positive integers. Find $m+n$.

2007 Balkan MO, 1

Tags: trigonometry , geometry , geometric transformation , reflection , trapezoid , circumcircle , trig identities

Let $ABCD$ a convex quadrilateral with $AB=BC=CD$, with $AC$ not equal to $BD$ and $E$ be the intersection point of it's diagonals. Prove that $AE=DE$ if and only if $\angle BAD+\angle ADC = 120$.

2005 Today's Calculation Of Integral, 15

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

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

2005 Today's Calculation Of Integral, 41

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

Evaluate \[\int_0^a \sqrt{2ax-x^2}\ dx \ (a>0)\]

1992 Balkan MO, 3

Tags: inequalities , circumcircle , trigonometry , geometry

Let $D$, $E$, $F$ be points on the sides $BC$, $CA$, $AB$ respectively of a triangle $ABC$ (distinct from the vertices). If the quadrilateral $AFDE$ is cyclic, prove that \[ \frac{ 4 \mathcal A[DEF] }{\mathcal A[ABC] } \leq \left( \frac{EF}{AD} \right)^2 . \] [i]Greece[/i]

1995 Turkey MO (2nd round), 4

Tags: ratio , trigonometry , geometry

In a triangle $ABC$ with $AB\neq AC$, the internal and external bisectors of angle $A$ meet the line $BC$ at $D$ and $E$ respectively. If the feet of the perpendiculars from a point $F$ on the circle with diameter $DE$ to $BC,CA,AB$ are $K,L,M$, respectively, show that $KL=KM$.

2014 Online Math Open Problems, 2

Tags: geometry , trigonometry

Consider two circles of radius one, and let $O$ and $O'$ denote their centers. Point $M$ is selected on either circle. If $OO' = 2014$, what is the largest possible area of triangle $OMO'$? [i]Proposed by Evan Chen[/i]

1997 Turkey Team Selection Test, 1

Tags: trigonometry , geometry

A convex $ABCDE$ is inscribed in a unit circle, $AE$ being its diameter. If $AB = a$, $BC = b$, $CD = c$, $DE = d$ and $ab = cd =\frac{1}{4}$, compute $AC + CE$ in terms of $a, b, c, d.$

2013 Brazil Team Selection Test, 1

Tags: geometry , trigonometry , circumcircle , parallelogram , analytic geometry , equal area , ratio

Let $ABC$ be an acute triangle with altitudes $AD$, $BE$, and $CF$, and let $O$ be the center of its circumcircle. Show that the segments $OA$, $OF$, $OB$, $OD$, $OC$, $OE$ dissect the triangle $ABC$ into three pairs of triangles that have equal areas.

2015 International Zhautykov Olympiad, 2

Tags: symmetry , trigonometry , geometry

Inside the triangle $ ABC $ a point $ M $ is given. The line $ BM $ meets the side $ AC $ at $ N $. The point $ K $ is symmetrical to $ M $ with respect to $ AC $. The line $ BK $ meets $ AC $ at $ P $. If $ \angle AMP = \angle CMN $, prove that $ \angle ABP=\angle CBN $.

2005 National High School Mathematics League, 3

Tags: trigonometry , geometry

$\triangle ABC$ is inscribed to unit circle. Bisector of $\angle A,\angle B,\angle C$ intersect the circle at $A_1,B_1,C_1$ respectively. The value of $\frac{\displaystyle AA_1\cdot\cos\frac{A}{2}+BB_1\cdot\cos\frac{B}{2}+CC_1\cdot\cos\frac{C}{2}}{\sin A+\sin B+\sin C}$ is $\text{(A)}2\qquad\text{(B)}4\qquad\text{(C)}6\qquad\text{(D)}8$

2012 China Second Round Olympiad, 2

Tags: trigonometry , geometry

In $\triangle ABC$, the corresponding sides of angle $A,B,C$ are $a,b,c$ respectively. If $a\cos B-b\cos A=\frac{3}{5}c$, find the value of $\frac{\tan A}{\tan B}$.

2011 Today's Calculation Of Integral, 692

Tags: calculus , integration , trigonometry , calculus computations

Evaluate $\int_0^{\frac{\pi}{12}} \frac{\tan ^ 2 x-3}{3\tan ^ 2 x-1}dx$. created by kunny

1988 IMO Longlists, 4

Tags: geometry , circumcircle , trigonometry , inradius , geometric inequality

The triangle $ ABC$ is inscribed in a circle. The interior bisectors of the angles $ A,B$ and $ C$ meet the circle again at $ A', B'$ and $ C'$ respectively. Prove that the area of triangle $ A'B'C'$ is greater than or equal to the area of triangle $ ABC.$