Olympiad problems

2011 China Western Mathematical Olympiad, 2

Tags: trigonometry , inequalities

Let $a,b,c > 0$, prove that \[\frac{(a-b)^2}{(c+a)(c+b)} + \frac{(b-c)^2}{(a+b)(a+c)} + \frac{(c-a)^2}{(b+c)(b+a)} \geq \frac{(a-b)^2}{a^2+b^2+c^2}\]

1988 All Soviet Union Mathematical Olympiad, 484

Tags: trigonometry , system of equations , minimum , algebra , sum

What is the smallest $n$ for which there is a solution to $$\begin{cases} \sin x_1 + \sin x_2 + ... + \sin x_n = 0 \\ \sin x_1 + 2 \sin x_2 + ... + n \sin x_n = 100 \end{cases}$$ ?

1969 IMO Shortlist, 6

Tags: binomial coefficient , trigonometry , combinatorics

$(BEL 6)$ Evaluate $\left(\cos\frac{\pi}{4} + i \sin\frac{\pi}{4}\right)^{10}$ in two different ways and prove that $\dbinom{10}{1}-\dbinom{10}{3}+\frac{1}{2}\dbinom{10}{5}=2^4$

2007 Indonesia TST, 1

Tags: trigonometry , circumcircle , geometry

Let $ P$ be a point in triangle $ ABC$, and define $ \alpha,\beta,\gamma$ as follows: \[ \alpha\equal{}\angle BPC\minus{}\angle BAC, \quad \beta\equal{}\angle CPA\minus{}\angle \angle CBA, \quad \gamma\equal{}\angle APB\minus{}\angle ACB.\] Prove that \[ PA\dfrac{\sin \angle BAC}{\sin \alpha}\equal{}PB\dfrac{\sin \angle CBA}{\sin \beta}\equal{}PC\dfrac{\sin \angle ACB}{\sin \gamma}.\]

2006 Pre-Preparation Course Examination, 2

Tags: polynomial , trigonometry , quadratic , algebra

a) Show that you can divide an angle $\theta$ to three equal parts using compass and ruler if and only if the polynomial $4t^3-3t-\cos (\theta)$ is reducible over $\mathbb{Q}(\cos (\theta))$. b) Is it always possible to divide an angle into five equal parts?

2014 Singapore Senior Math Olympiad, 26

Tags: trigonometry , algebra , function , domain

Suppose that $x$ is measured in radians. Find the maximum value of \[\frac{\sin2x+\sin4x+\sin6x}{\cos2x+\cos4x+\cos6x}\] for $0\le x\le \frac{\pi}{16}$

2006 Kyiv Mathematical Festival, 3

Tags: circumcircle , incenter , trigonometry , trig identities , law of sines , geometry

See all the problems from 5-th Kyiv math festival [url=http://www.mathlinks.ro/Forum/viewtopic.php?p=506789#p506789]here[/url] Let $O$ be the circumcenter and $H$ be the intersection point of the altitudes of acute triangle $ABC.$ The straight lines $BH$ and $CH$ intersect the segments $CO$ and $BO$ at points $D$ and $E$ respectively. Prove that if triangles $ODH$ and $OEH$ are isosceles then triangle $ABC$ is isosceles too.

2012 Iran Team Selection Test, 3

Tags: parallelogram , geometric transformation , homothety , trigonometry , geometry

Suppose $ABCD$ is a parallelogram. Consider circles $w_1$ and $w_2$ such that $w_1$ is tangent to segments $AB$ and $AD$ and $w_2$ is tangent to segments $BC$ and $CD$. Suppose that there exists a circle which is tangent to lines $AD$ and $DC$ and externally tangent to $w_1$ and $w_2$. Prove that there exists a circle which is tangent to lines $AB$ and $BC$ and also externally tangent to circles $w_1$ and $w_2$. [i]Proposed by Ali Khezeli[/i]

2009 Moldova Team Selection Test, 3

Tags: geometry , geometric transformation , reflection , circumcircle , trigonometry , symmetry , incenter

[color=darkred]Quadrilateral $ ABCD$ is inscribed in the circle of diameter $ BD$. Point $ A_1$ is reflection of point $ A$ wrt $ BD$ and $ B_1$ is reflection of $ B$ wrt $ AC$. Denote $ \{P\}\equal{}CA_1 \cap BD$ and $ \{Q\}\equal{}DB_1\cap AC$. Prove that $ AC\perp PQ$.[/color]

2010 Germany Team Selection Test, 3

Tags: incenter , circumcircle , trigonometry , inradius , geometry

Let $ABCD$ be a circumscribed quadrilateral. Let $g$ be a line through $A$ which meets the segment $BC$ in $M$ and the line $CD$ in $N$. Denote by $I_1$, $I_2$ and $I_3$ the incenters of $\triangle ABM$, $\triangle MNC$ and $\triangle NDA$, respectively. Prove that the orthocenter of $\triangle I_1I_2I_3$ lies on $g$. [i]Proposed by Nikolay Beluhov, Bulgaria[/i]

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 $k+m+n.$

2005 Today's Calculation Of Integral, 2

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

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

1977 Swedish Mathematical Competition, 4

Tags: algebra , trigonometry

Show that if \[ \frac{\cos x}{\cos y}+\frac{\sin x}{\sin y}=-1 \] then \[ \frac{\cos^3 y}{\cos x}+\frac{\sin^3 y}{\sin x}=1 \]

1966 German National Olympiad, 5

Tags: trigonometry , algebra , trigonometric identities , equation

Prove that \[\tan 7 30^{\prime }=\sqrt{6}+\sqrt{2}-\sqrt{3}-2.\]

2008 Germany Team Selection Test, 2

Tags: inequalities , circumcircle , trigonometry , geometry

For three points $ X,Y,Z$ let $ R_{XYZ}$ be the circumcircle radius of the triangle $ XYZ.$ If $ ABC$ is a triangle with incircle centre $ I$ then we have: \[ \frac{1}{R_{ABI}} \plus{} \frac{1}{R_{BCI}} \plus{} \frac{1}{R_{CAI}} \leq \frac{1}{\bar{AI}} \plus{} \frac{1}{\bar{BI}} \plus{} \frac{1}{\bar{CI}}.\]

2002 France Team Selection Test, 2

Tags: incenter , circumcircle , trigonometry , geometry

Let $ ABC$ be a non-equilateral triangle. Denote by $ I$ the incenter and by $ O$ the circumcenter of the triangle $ ABC$. Prove that $ \angle AIO\leq\frac{\pi}{2}$ holds if and only if $ 2\cdot BC\leq AB\plus{}AC$.

2015 Balkan MO Shortlist, G1

Tags: geometry , circumcircle , trigonometry

In an acute angled triangle $ABC$ , let $BB' $ and $CC'$ be the altitudes. Ray $C'B'$ intersects the circumcircle at $B''$ andl let $\alpha_A$ be the angle $\widehat{ABB''}$. Similarly are defined the angles $\alpha_B$ and $\alpha_C$. Prove that $$\displaystyle\sin \alpha _A \sin \alpha _B \sin \alpha _C\leq \frac{3\sqrt{6}}{32}$$ (Romania)

1967 IMO Shortlist, 5

Tags: trigonometry , algebra , triangle , trigonometric identities

Show that a triangle whose angles $A$, $B$, $C$ satisfy the equality \[ \frac{\sin^2 A + \sin^2 B + \sin^2 C}{\cos^2 A + \cos^2 B + \cos^2 C} = 2 \] is a rectangular triangle.

2006 Iran MO (3rd Round), 3

Tags: inequalities , trigonometry , geometry , incenter , geometric transformation , reflection , circumcircle

In triangle $ABC$, if $L,M,N$ are midpoints of $AB,AC,BC$. And $H$ is orthogonal center of triangle $ABC$, then prove that \[LH^{2}+MH^{2}+NH^{2}\leq\frac14(AB^{2}+AC^{2}+BC^{2})\]

2001 IMO Shortlist, 5

Tags: geometry , circumcircle , trigonometry , triangle

Let $ABC$ be an acute triangle. Let $DAC,EAB$, and $FBC$ be isosceles triangles exterior to $ABC$, with $DA=DC, EA=EB$, and $FB=FC$, such that \[ \angle ADC = 2\angle BAC, \quad \angle BEA= 2 \angle ABC, \quad \angle CFB = 2 \angle ACB. \] Let $D'$ be the intersection of lines $DB$ and $EF$, let $E'$ be the intersection of $EC$ and $DF$, and let $F'$ be the intersection of $FA$ and $DE$. Find, with proof, the value of the sum \[ \frac{DB}{DD'}+\frac{EC}{EE'}+\frac{FA}{FF'}. \]

2003 Alexandru Myller, 4

Tags: trigonometry , geometry

Let $\displaystyle ABCD$ be a a convex quadrilateral and $\displaystyle O$ be a point in its interior. Let $\displaystyle a,b,c,d,e,f$ be the areas of the triangles $\displaystyle OAB,OBC,OCD,ODA,OAC,OBD$. Prove that \[ \displaystyle \left| ac - bd \right| = ef . \]