Olympiad problems

2007 Balkan MO, 1

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

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 $\angle BAD+\angle ADC = 120$.

Let the numbers $\alpha ,\beta $ satisfy $0<\alpha <\beta <\frac{\pi}{2}$ and let $\gamma $ and $\delta $ be the numbers defined by the conditions: $(\text{i})\ 0<\gamma<\frac{\pi}{2}$, and $\tan\gamma$ is the arithmetic mean of $\tan\alpha$ and $\tan\beta$; $(\text{ii})\ 0<\delta<\frac{\pi}{2}$, and $\frac{1}{\cos\delta}$ is the arithmetic mean of $\frac{1}{\cos\alpha}$ and $\frac{1}{\cos\beta}$. Prove that $\gamma <\delta $.

2019 AMC 12/AHSME, 19

Tags: perimeter , geometry , trigonometry

In $\triangle ABC$ with integer side lengths, \[ \cos A=\frac{11}{16}, \qquad \cos B= \frac{7}{8}, \qquad \text{and} \qquad\cos C=-\frac{1}{4}. \] What is the least possible perimeter for $\triangle ABC$? $\textbf{(A) } 9 \qquad \textbf{(B) } 12 \qquad \textbf{(C) } 23 \qquad \textbf{(D) } 27 \qquad \textbf{(E) } 44$

1962 IMO, 6

Tags: circles , geometry , circumcircle , trigonometry , isosceles

Consider an isosceles triangle. let $R$ be the radius of its circumscribed circle and $r$ be the radius of its inscribed circle. Prove that the distance $d$ between the centers of these two circle is \[ d=\sqrt{R(R-2r)} \]

2014 Math Prize For Girls Problems, 20

Tags: absolute value , ceiling function , trigonometry

How many complex numbers $z$ such that $\left| z \right| < 30$ satisfy the equation \[ e^z = \frac{z - 1}{z + 1} \, ? \]

1985 Tournament Of Towns, (085) 1

Tags: geometry , geometric inequality , trigonometry , inequalities

$a, b$ and $c$ are sides of a triangle, and $\gamma$ is its angle opposite $c$. Prove that $c \ge (a + b) \sin \frac{\gamma}{2}$ (V. Prasolov )

1983 AMC 12/AHSME, 5

Tags: trigonometry

Triangle $ABC$ has a right angle at $C$. If $\sin A = \frac{2}{3}$, then $\tan B$ is $ \textbf{(A)}\ \frac{3}{5}\qquad\textbf{(B)}\ \frac{\sqrt 5}{3}\qquad\textbf{(C)}\ \frac{2}{\sqrt 5}\qquad\textbf{(D)}\ \frac{\sqrt 5}{2}\qquad\textbf{(E)}\ \frac{5}{3} $

II Soros Olympiad 1995 - 96 (Russia), 10.2

Tags: trigonometry , algebra

Without using a calculator, find out what is greater: $\sin 28^o$ or $tg21^o$?

2004 Romania National Olympiad, 3

Tags: trigonometry , 3d geometry , pyramid , geometry

Let $ABCD A^{\prime} B^{\prime} C^{\prime} D^{\prime}$ be a truncated regular pyramid in which $BC^{\prime}$ and $DA^{\prime}$ are perpendicular. (a) Prove that $\measuredangle \left( AB^{\prime},DA^{\prime} \right) = 60^{\circ}$; (b) If the projection of $B^{\prime}$ on $(ABC)$ is the center of the incircle of $ABC$, then prove that $d \left( CB^{\prime},AD^{\prime} \right) = \frac12 BC^{\prime}$. [i]Mircea Fianu[/i]

1960 IMO, 6

Tags: 3d geometry , inradius , sphere , trigonometry , geometry

Consider a cone of revolution with an inscribed sphere tangent to the base of the cone. A cylinder is circumscribed about this sphere so that one of its bases lies in the base of the cone. let $V_1$ be the volume of the cone and $V_2$ be the volume of the cylinder. a) Prove that $V_1 \neq V_2$; b) Find the smallest number $k$ for which $V_1=kV_2$; for this case, construct the angle subtended by a diamter of the base of the cone at the vertex of the cone.

2004 China Girls Math Olympiad, 6

Tags: trigonometry , geometry

Given an acute triangle $ABC$ with $O$ as its circumcenter. Line $AO$ intersects $BC$ at $D$. Points $E$, $F$ are on $AB$, $AC$ respectively such that $A$, $E$, $D$, $F$ are concyclic. Prove that the length of the projection of line segment $EF$ on side $BC$ does not depend on the positions of $E$ and $F$.

2005 Tuymaada Olympiad, 4

Tags: trigonometry , triangle inequality , tetrahedron , 3d geometry , geometry , inequalities

In a triangle $ABC$, let $A_{1}$, $B_{1}$, $C_{1}$ be the points where the excircles touch the sides $BC$, $CA$ and $AB$ respectively. Prove that $A A_{1}$, $B B_{1}$ and $C C_{1}$ are the sidelenghts of a triangle. [i]Proposed by L. Emelyanov[/i]

2001 IMO Shortlist, 3

Tags: inequalities , trigonometry , algebra

Let $x_1,x_2,\ldots,x_n$ be arbitrary real numbers. Prove the inequality \[ \frac{x_1}{1+x_1^2} + \frac{x_2}{1+x_1^2 + x_2^2} + \cdots + \frac{x_n}{1 + x_1^2 + \cdots + x_n^2} < \sqrt{n}. \]

1957 Polish MO Finals, 2

Tags: geometry , trigonometry

Prove that between the sides $ a $, $ b $, $ c $ and the opposite angles $ A $, $ B $, $ C $ of a triangle there is a relationship $$ a^2 \cos^2 A = b^2 \cos^2 B + c^2 \cos^2 C + 2bc \cos B \cos C \cos 2A.$$

2008 Harvard-MIT Mathematics Tournament, 8

Tags: trigonometry

Compute $ \arctan\left(\tan65^\circ \minus{} 2\tan40^\circ\right)$. (Express your answer in degrees.)

2007 Today's Calculation Of Integral, 236

Tags: trigonometry , integration , calculus computations , calculus

Let $a$ be a positive constant. Evaluate the following definite integrals $A,\ B$. \[A=\int_0^{\pi} e^{-ax}\sin ^ 2 x\ dx,\ B=\int_0^{\pi} e^{-ax}\cos ^ 2 x\ dx\]. [i]1998 Shinsyu University entrance exam/Textile Science[/i]

2010 Indonesia TST, 4

Tags: geometry , geometric transformation , similar triangles , angle bisector , trigonometry , reflection , circumcircle

Let $ ABC$ be a non-obtuse triangle with $ CH$ and $ CM$ are the altitude and median, respectively. The angle bisector of $ \angle BAC$ intersects $ CH$ and $ CM$ at $ P$ and $ Q$, respectively. Assume that \[ \angle ABP\equal{}\angle PBQ\equal{}\angle QBC,\] (a) prove that $ ABC$ is a right-angled triangle, and (b) calculate $ \dfrac{BP}{CH}$. [i]Soewono, Bandung[/i]

1940 Putnam, B1

Tags: physics , trigonometry

A projectile, thrown with initial velocity $v_0$ in a direction making angle $\alpha$ with the horizontal, is acted on by no force except gravity. Find the lenght of its path until it strikes a horizontal plane through the starting point. Show that the flight is longest when $$\sin \alpha \log(\sec \alpha+ \tan \alpha)=1.$$

2008 Harvard-MIT Mathematics Tournament, 4

Tags: complex number , trigonometry

Suppose that $ a, b, c, d$ are real numbers satisfying $ a \geq b \geq c \geq d \geq 0$, $ a^2 \plus{} d^2 \equal{} 1$, $ b^2 \plus{} c^2 \equal{} 1$, and $ ac \plus{} bd \equal{} 1/3$. Find the value of $ ab \minus{} cd$.

2013 Uzbekistan National Olympiad, 2

Tags: inequality , inequalities , trigonometry , algebra

Let $x$ and $y$ are real numbers such that $x^2y^2+2yx^2+1=0.$ If $S=\frac{2}{x^2}+1+\frac{1}{x}+y(y+2+\frac{1}{x})$, find (a)max$S$ and (b) min$S$.

1989 Federal Competition For Advanced Students, P2, 4

Tags: trigonometry , geometry

We are given a circle $ k$ and nonparallel tangents $ t_1,t_2$ at points $ P_1,P_2$ on $ k$, respectively. Lines $ t_1$ and $ t_2$ meet at $ A_0$. For a point $ A_3$ on the smaller arc $ P_1 P_2,$ the tangent $ t_3$ to $ k$ at $ P_3$ meets $ t_1$ at $ A_1$ and $ t_2$ at $ A_2$. How must $ P_3$ be chosen so that the triangle $ A_0 A_1 A_2$ has maximum area?

2007 Iran MO (3rd Round), 8

Tags: limit , floor function , logarithm , factorial , trigonometry , ceiling function

In this question you must make all numbers of a clock, each with using 2, exactly 3 times and Mathematical symbols. You are not allowed to use English alphabets and words like $ \sin$ or $ \lim$ or $ a,b$ and no other digits. [img]http://i2.tinypic.com/5x73dza.png[/img]

1998 Spain Mathematical Olympiad, 1

Tags: rotation , trigonometry , geometry

A unit square $ABCD$ with centre $O$ is rotated about $O$ by an angle $\alpha$. Compute the common area of the two squares.

2013 AMC 12/AHSME, 24

Tags: trig identities , inequalities , triangle inequality , law of sines , trigonometry , probability , geometry

Three distinct segments are chosen at random among the segments whose end-points are the vertices of a regular 12-gon. What is the probability that the lengths of these three segments are the three side lengths of a triangle with positive area? $ \textbf{(A)} \ \frac{553}{715} \qquad \textbf{(B)} \ \frac{443}{572} \qquad \textbf{(C)} \ \frac{111}{143} \qquad \textbf{(D)} \ \frac{81}{104} \qquad \textbf{(E)} \ \frac{223}{286}$

2008 Hong kong National Olympiad, 3

Tags: parallelogram , circumcircle , geometry , trigonometry , ratio , inradius , incenter

$ \Delta ABC$ is a triangle such that $ AB \neq AC$. The incircle of $ \Delta ABC$ touches $ BC, CA, AB$ at $ D, E, F$ respectively. $ H$ is a point on the segment $ EF$ such that $ DH \bot EF$. Suppose $ AH \bot BC$, prove that $ H$ is the orthocentre of $ \Delta ABC$. Remark: the original question has missed the condition $ AB \neq AC$