Olympiad problems

1996 South africa National Olympiad, 3

The sides of triangle $ABC$ has integer lengths. Given that $AC=6$ and $\angle BAC=120^\circ$, determine the lengths of the other two sides.

2014 Baltic Way, 1

Tags: trigonometry , modular arithmetic , algebra

Show that \[\cos(56^{\circ}) \cdot \cos(2 \cdot 56^{\circ}) \cdot \cos(2^2\cdot 56^{\circ})\cdot . . . \cdot \cos(2^{23}\cdot 56^{\circ}) = \frac{1}{2^{24}} .\]

1987 IMO, 2

Tags: circumcircle , trigonometry , area , geometry

In an acute-angled triangle $ABC$ the interior bisector of angle $A$ meets $BC$ at $L$ and meets the circumcircle of $ABC$ again at $N$. From $L$ perpendiculars are drawn to $AB$ and $AC$, with feet $K$ and $M$ respectively. Prove that the quadrilateral $AKNM$ and the triangle $ABC$ have equal areas.

2000 Harvard-MIT Mathematics Tournament, 18

Tags: trigonometry , limit

What is the value of $ \sum_{n=1}^\infty (\tan^{-1}\sqrt{n}-\tan^{-1}\sqrt{n+1})$?

2009 Today's Calculation Of Integral, 429

Tags: calculus , integration , trigonometry , calculus computations

Find the length of the curve expressed by the polar equation: $ r\equal{}1\plus{}\cos \theta \ (0\leq \theta \leq \pi)$.

2003 National Olympiad First Round, 25

Tags: geometry , circumcircle , trigonometry , perpendicular bisector

Let $ABC$ be an acute triangle and $O$ be its circumcenter. Let $D$ be the midpoint of $[AB]$. The circumcircle of $\triangle ADO$ meets $[AC]$ at $A$ and $E$. If $|AE|=7$, $|DE|=8$, and $m(\widehat{AOD}) = 45^\circ$, what is the area of $\triangle ABC$? $ \textbf{(A)}\ 56\sqrt 3 \qquad\textbf{(B)}\ 56 \sqrt 2 \qquad\textbf{(C)}\ 50 \sqrt 2 \qquad\textbf{(D)}\ 84 \qquad\textbf{(E)}\ \text{None of the preceding} $

2002 Romania Team Selection Test, 1

Tags: symmetry , trigonometry , geometry

Let $ABCDE$ be a cyclic pentagon inscribed in a circle of centre $O$ which has angles $\angle B=120^{\circ},\angle C=120^{\circ},$ $\angle D=130^{\circ},\angle E=100^{\circ}$. Show that the diagonals $BD$ and $CE$ meet at a point belonging to the diameter $AO$. [i]Dinu Șerbănescu[/i]

IV Soros Olympiad 1997 - 98 (Russia), 10.4

Tags: trigonometry , algebra

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.

1981 All Soviet Union Mathematical Olympiad, 311

Tags: inequalities , trigonometry , algebra

It is known about real $a$ and $b$ that the inequality $$a \cos x + b \cos (3x) > 1$$ has no real solutions. Prove that $|b|\le 1$.

2010 South africa National Olympiad, 2

Tags: trigonometry , geometry

Consider a triangle $ABC$ with $BC = 3$. Choose a point $D$ on $BC$ such that $BD = 2$. Find the value of \[AB^2 + 2AC^2 - 3AD^2.\]

2005 Today's Calculation Of Integral, 39

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

Find the minimum value of the following function $f(x) $ defined at $0<x<\frac{\pi}{2}$. \[f(x)=\int_0^x \frac{d\theta}{\cos \theta}+\int_x^{\frac{\pi}{2}} \frac{d\theta}{\sin \theta}\]

2006 Moldova Team Selection Test, 1

Tags: circumcircle , trigonometry , inequalities , incenter , geometry

Let the point $P$ in the interior of the triangle $ABC$. $(AP, (BP, (CP$ intersect the circumcircle of $ABC$ at $A_{1}, B_{1}, C_{1}$. Prove that the maximal value of the sum of the areas $A_{1}BC$, $B_{1}AC$, $C_{1}AB$ is $p(R-r)$, where $p, r, R$ are the usual notations for the triangle $ABC$.

2018 Purple Comet Problems, 25

Tags: trigonometry

If a and b are in the interval $\left(0, \frac{\pi}{2}\right)$ such that $13(\sin a + \sin b) + 43(\cos a + \cos b) = 2\sqrt{2018}$, then $\tan a + \tan b = \frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m + n$.

1999 India National Olympiad, 4

Tags: trigonometry , complex number , geometry

Let $\Gamma$ and $\Gamma'$ be two concentric circles. Let $ABC$ and $A'B'C'$ be any two equilateral triangles inscribed in $\Gamma$ and $\Gamma'$ respectively. If $P$ and $P'$ are any two points on $\Gamma$ and $\Gamma'$ respectively, show that \[ P'A^2 + P'B^2 + P'C^2 = A'P^2 + B'P^2 + C'P^2. \]

2013 District Olympiad, 3

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

Problem 3. Let $f:\left[ 0,\frac{\pi }{2} \right]\to \left[ 0,\infty \right)$ an increasing function .Prove that: (a) $\int_{0}^{\frac{\pi }{2}}{\left( f\left( x \right)-f\left( \frac{\pi }{4} \right) \right)}\left( \sin x-\cos x \right)dx\ge 0.$ (b) Exist $a\in \left[ \frac{\pi }{4},\frac{\pi }{2} \right]$ such that $\int_{0}^{a}{f\left( x \right)\sin x\ dx=}\int_{0}^{a}{f\left( x \right)\cos x\ dx}.$

2012 IMO Shortlist, G4

Tags: circumcircle , trigonometry , triangle , reflection , geometry

Let $ABC$ be a triangle with $AB \neq AC$ and circumcenter $O$. The bisector of $\angle BAC$ intersects $BC$ at $D$. Let $E$ be the reflection of $D$ with respect to the midpoint of $BC$. The lines through $D$ and $E$ perpendicular to $BC$ intersect the lines $AO$ and $AD$ at $X$ and $Y$ respectively. Prove that the quadrilateral $BXCY$ is cyclic.

1999 Ukraine Team Selection Test, 4

Tags: inequalities , trigonometry , algebra

If $n \in N$ and $0 < x <\frac{\pi}{2n}$, prove the inequality $\frac{\sin 2x}{\sin x}+\frac{\sin 3x}{\sin 2x} +...+\frac{\sin (n+1)x}{\sin nx} < 2\frac{\cos x}{\sin^2 x}$. .

2002 Dutch Mathematical Olympiad, 5

Tags: geometry , trigonometry , angle

In triangle $ABC$, angle $A$ is twice as large as angle $B$. $AB = 3$ and $AC = 2$. Calculate $BC$.

1998 South africa National Olympiad, 3

Tags: trigonometry , geometry

$A,\ B,\ C,\ D,\ E$ and $F$ lie (in that order) on the circumference of a circle. The chords $AD,\ BE$ and $CF$ are concurrent. $P,\ Q$ and $R$ are the midpoints of $AD,\ BE$ and $CF$ respectively. Two further chords $AG \parallel BE$ and $AH \parallel CF$ are drawn. Show that $PQR$ is similar to $DGH$.

2020 BMT Fall, 23

Tags: trigonometry , algebra

Let $0 < \theta < 2\pi$ be a real number for which $\cos (\theta) + \cos (2\theta) + \cos (3\theta) + ...+ \cos (2020\theta) = 0$ and $\theta =\frac{\pi}{n}$ for some positive integer $n$. Compute the sum of the possible values of $n \le 2020$.

2013 Saint Petersburg Mathematical Olympiad, 1

Tags: trigonometry , floor function , calculus , integration , algebra

Find the minimum positive noninteger root of $ \sin x=\sin \lfloor x \rfloor $. F. Petrov

2005 Thailand Mathematical Olympiad, 21

Tags: trigonometry , algebra , min , inequalities

Compute the minimum value of $cos(a-b) + cos(b-c) + cos(c-a)$ as $a,b,c$ ranges over the real numbers.

2020 LIMIT Category 2, 10

Tags: limit , trigonometry , geometry

In a triangle $\triangle XYZ$, $\tan(x)\tan(z)=2$, $\tan(y)\tan(z)=18$. Then what is $\tan^2(z)$?

2013 F = Ma, 20

Tags: trigonometry

A simple pendulum experiment is constructed from a point mass $m$ attached to a pivot by a massless rod of length $L$ in a constant gravitational field. The rod is released from an angle $\theta_0 < \frac{\pi}{2}$ at rest and the period of motion is found to be $T_0$. Ignore air resistance and friction. What is the maximum value of the tension in the rod? $\textbf{(A) } mg\\ \textbf{(B) } 2mg\\ \textbf{(C) } mL\theta_0/T_0^2\\ \textbf{(D) } mg \sin \theta_0\\ \textbf{(E) } mg(3 - 2 \cos \theta_0)$

2011 Iran MO (3rd Round), 3

Tags: circumcircle , ratio , trigonometry , projective geometry , geometry

In triangle $ABC$, $X$ and $Y$ are the tangency points of incircle (with center $I$) with sides $AB$ and $AC$ respectively. A tangent line to the circumcircle of triangle $ABC$ (with center $O$) at point $A$, intersects the extension of $BC$ at $D$. If $D,X$ and $Y$ are collinear then prove that $D,I$ and $O$ are also collinear. [i]proposed by Amirhossein Zabeti[/i]