Olympiad problems

2013 Bogdan Stan, 3

Let $ a,b,c $ be three real numbers such that $ \cos a+\cos b+\cos c=\sin a+\sin b+\sin c=0. $ Prove that [b]i)[/b] $ \cos 6a+\cos 6b+\cos 6c=3\cos (2a+2b+2c) $ [b]ii)[/b] $ \sin 6a+\sin 6b+\sin 6c=3\sin (2a+2b+2c) $ [i]Vasile Pop[/i]

Indonesia MO Shortlist - geometry, g3.3

Tags: logarithm , geometry , geometric transformation , dilation , trapezoid , trigonometry , parallelogram

Let $ABCD$ be a trapezoid (quadrilateral with one pair of parallel sides) such that $AB < CD$. Suppose that $AC$ and $BD$ meet at $E$ and $AD$ and $BC$ meet at $F$. Construct the parallelograms $AEDK$ and $BECL$. Prove that $EF$ passes through the midpoint of the segment $KL$.

2000 CentroAmerican, 2

Tags: trigonometry

Let $ ABC$ be an acute-angled triangle. $ C_{1}$ and $ C_{2}$ are two circles of diameters $ AB$ and $ AC$, respectively. $ C_{2}$ and $ AB$ intersect again at $ F$, and $ C_{1}$ and $ AC$ intersect again at $ E$. Also, $ BE$ meets $ C_{2}$ at $ P$ and $ CF$ meets $ C_{1}$ at $ Q$. Prove that $ AP=AQ$.

1996 Taiwan National Olympiad, 1

Tags: number theory , trigonometry

Suppose that $a,b,c$ are real numbers in $(0,\frac{\pi}{2})$ such that $a+b+c=\frac{\pi}{4}$ and $\tan{a}=\frac{1}{x},\tan{b}=\frac{1}{y},\tan{c}=\frac{1}{z}$ , where $x,y,z$ are positive integer numbers. Find $x,y,z$.

1997 Taiwan National Olympiad, 5

Tags: trigonometry , geometry , 3d geometry , tetrahedron , parallelogram

Let $ABCD$ is a tetrahedron. Show that a)If $AB=CD,AC=DB,AD=BC$ then triangles $ABC,ABD,ACD,BCD$ are acute. b)If the triangles $ABC,ABD,ACD,BCD$ have the same area , then $AB=CD,AC=DB,AD=BC$.

1952 Miklós Schweitzer, 10

Tags: function , real analysis , trigonometry , integration

Let $ n$ be a positive integer. Prove that, for $ 0<x<\frac{\pi}{n\plus{}1}$, $ \sin{x}\minus{}\frac{\sin{2x}}{2}\plus{}\cdots\plus{}(\minus{}1)^{n\plus{}1}\frac{\sin{nx}}{n}\minus{}\frac{x}{2}$ is positive if $ n$ is odd and negative if $ n$ is even.

1994 Baltic Way, 3

Tags: inequalities , trigonometry

Find the largest value of the expression \[xy+x\sqrt{1-x^2}+y\sqrt{1-y^2}-\sqrt{(1-x^2)(1-y^2)}\]

1999 Baltic Way, 13

Tags: incenter , trigonometry , geometry

The bisectors of the angles $A$ and $B$ of the triangle $ABC$ meet the sides $BC$ and $CA$ at the points $D$ and $E$, respectively. Assuming that $AE+BD=AB$, determine the angle $C$.

2001 All-Russian Olympiad, 2

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

Let the circle $ {\omega}_{1}$ be internally tangent to another circle $ {\omega}_{2}$ at $ N$.Take a point $ K$ on $ {\omega}_{1}$ and draw a tangent $ AB$ which intersects $ {\omega}_{2}$ at $ A$ and $ B$. Let $M$ be the midpoint of the arc $ AB$ which is on the opposite side of $ N$. Prove that, the circumradius of the $ \triangle KBM$ doesnt depend on the choice of $ K$.

1990 APMO, 3

Tags: calculus , geometry , trigonometry

Consider all the triangles $ABC$ which have a fixed base $AB$ and whose altitude from $C$ is a constant $h$. For which of these triangles is the product of its altitudes a maximum?

2009 Today's Calculation Of Integral, 496

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

Evaluate $ \int_{ \minus{} 1}^ {a^2} \frac {1}{x^2 \plus{} a^2}\ dx\ (a > 0).$ You may not use $ \tan ^{ \minus{} 1} x$ or Complex Integral here.

1992 All Soviet Union Mathematical Olympiad, 574

Tags: function , trigonometry , algebra

Let $$f(x) = a \cos(x + 1) + b \cos(x + 2) + c \cos(x + 3)$$, where $a, b, c$ are real. Given that $f(x)$ has at least two zeros in the interval $(0, \pi)$, find all its real zeros.

1998 Swedish Mathematical Competition, 2

Tags: inequalities , geometry , trigonometry

$ABC$ is a triangle. Show that $c \ge (a+b) \sin \frac{C}{2}$

2007 Today's Calculation Of Integral, 235

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

Show that a function $ f(x)\equal{}\int_{\minus{}1}^1 (1\minus{}|\ t\ |)\cos (xt)\ dt$ is continuous at $ x\equal{}0$.

2008 Sharygin Geometry Olympiad, 6

Tags: geometry , circumcircle , inradius , trigonometry

(B.Frenkin) The product of two sides in a triangle is equal to $ 8Rr$, where $ R$ and $ r$ are the circumradius and the inradius of the triangle. Prove that the angle between these sides is less than $ 60^{\circ}$.

I Soros Olympiad 1994-95 (Rus + Ukr), 10.5

Tags: algebra , product , trigonometry

For an arbitrary natural $n$, prove the equality $$\sin \frac{\pi}{2n}\sin \frac{3\pi}{2n}\sin \frac{5\pi}{2n}...\sin \frac{n'\pi}{2n}=2^{\dfrac{1-n}{2}}$$ where $n'$ is the largest odd number not exceeding $n$.

1988 IMO Longlists, 37

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

[b]i.)[/b] Four balls of radius 1 are mutually tangent, three resting on the floor and the fourth resting on the others. A tedrahedron, each of whose edges has length $ s,$ is circumscribed around the balls. Find the value of $ s.$ [b]ii.)[/b] Suppose that $ ABCD$ and $ EFGH$ are opposite faces of a retangular solid, with $ \angle DHC \equal{} 45^{\circ}$ and $ \angle FHB \equal{} 60^{\circ}.$ Find the cosine of $ \angle BHD.$

2008 IMO, 1

Tags: geometry , trigonometry , radical axis , circumcircle

Let $ H$ be the orthocenter of an acute-angled triangle $ ABC$. The circle $ \Gamma_{A}$ centered at the midpoint of $ BC$ and passing through $ H$ intersects the sideline $ BC$ at points $ A_{1}$ and $ A_{2}$. Similarly, define the points $ B_{1}$, $ B_{2}$, $ C_{1}$ and $ C_{2}$. Prove that the six points $ A_{1}$, $ A_{2}$, $ B_{1}$, $ B_{2}$, $ C_{1}$ and $ C_{2}$ are concyclic. [i]Author: Andrey Gavrilyuk, Russia[/i]

1963 IMO, 5

Tags: trigonometry , trig identities , algebra , polynomial , trigonometric equation , trigonometric identities

Prove that $\cos{\frac{\pi}{7}}-\cos{\frac{2\pi}{7}}+\cos{\frac{3\pi}{7}}=\frac{1}{2}$

1988 AIME Problems, 14

Tags: graphing lines , geometry , trigonometry , analytic geometry , slope , geometric transformation , reflection

Let $C$ be the graph of $xy = 1$, and denote by $C^*$ the reflection of $C$ in the line $y = 2x$. Let the equation of $C^*$ be written in the form \[ 12x^2 + bxy + cy^2 + d = 0. \] Find the product $bc$.

2009 Today's Calculation Of Integral, 405

Tags: trigonometry , calculus , integration , calculus computations

Calculate $ \displaystyle \left|\frac {\int_0^{\frac {\pi}{2}} (x\cos x + 1)e^{\sin x}\ dx}{\int_0^{\frac {\pi}{2}} (x\sin x - 1)e^{\cos x}\ dx}\right|$.

1979 IMO Longlists, 33

Tags: trigonometry , inequalities , inequality , geometric inequality , trigonometric identities

Show that $\frac{20}{60} <\sin 20^{\circ} < \frac{21}{60}.$

2010 Today's Calculation Of Integral, 567

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

Let $ a$ be a positive real numbers. In the coordinate plane denote by $ S$ the area of the figure bounded by the curve $ y=\sin x\ (0\leq x\leq \pi)$ and the $x$-axis and denote $T$ by the area of the figure bounded by the curves $y=\sin x\ \left(0\leq x\leq \frac{\pi}{2}\right),\ y=a\cos x\ \left(0\leq x\leq \frac{\pi}{2}\right)$ and the $x$-axis. Find the value of $a$ such that $ S: T=3: 1$.

2005 India IMO Training Camp, 3

Tags: trigonometry , algebra , function , inequalities

For real numbers $a,b,c,d$ not all equal to $0$ , define a real function $f(x) = a +b\cos{2x} + c\sin{5x} +d \cos{8x}$. Suppose $f(t) = 4a$ for some real $t$. prove that there exist a real number $s$ s.t. $f(s)<0$

2010 Today's Calculation Of Integral, 627

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

Evaluate $\int_{\frac{\pi}{4}}^{\frac{\pi}{3}} \frac{(2\sin \theta +1)\cos ^ 3 \theta}{(\sin ^ 2 \theta +1)^2}d\theta .$ [i]Proposed by kunny[/i]