Olympiad problems

2007 AIME Problems, 9

Tags: number theory , rectangle , trigonometry , geometric transformation , homothety , geometry

In right triangle $ABC$ with right angle $C$, $CA=30$ and $CB=16$. Its legs $\overline{CA}$ and $\overline{CB}$ are extended beyond $A$ and $B$. Points $O_{1}$ and $O_{2}$ lie in the exterior of the triangle and are the centers of two circles with equal radii. The circle with center $O_{1}$ is tangent to the hypotenuse and to the extension of leg CA, the circle with center $O_{2}$ is tangent to the hypotenuse and to the extension of leg CB, and the circles are externally tangent to each other. The length of the radius of either circle can be expressed as $p/q$, where $p$ and $q$ are relatively prime positive integers. Find $p+q$.

1971 IMO Longlists, 7

Tags: trigonometry , geometry , circumcircle

In a triangle $ABC$, let $H$ be its orthocenter, $O$ its circumcenter, and $R$ its circumradius. Prove that: [b](a)[/b] $|OH| = R \sqrt{1-8 \cos \alpha \cdot \cos \beta \cdot \cos \gamma}$ where $\alpha, \beta, \gamma$ are angles of the triangle $ABC;$ [b](b)[/b] $O \equiv H$ if and only if $ABC$ is equilateral.

2012 Today's Calculation Of Integral, 811

Tags: trigonometry , calculus , calculus computations , integration

Let $a$ be real number. Evaluate $\int_a^{a+\pi} |x|\cos x\ dx.$

1972 Vietnam National Olympiad, 1

Tags: trigonometry , polynomial , algebra , functional equation

Let $\alpha$ be an arbitrary angle and let $x = cos\alpha, y = cosn\alpha$ ($n \in Z$). i) Prove that to each value $x \in [-1, 1]$ corresponds one and only one value of $y$. Thus we can write $y$ as a function of $x, y = T_n(x)$. Compute $T_1(x), T_2(x)$ and prove that $T_{n+1}(x) = 2xT_n(x) - T_{n-1}(x)$. From this it follows that $T_n(x)$ is a polynomial of degree $n$. ii) Prove that the polynomial $T_n(x$) has $n$ distinct roots in $[-1, 1]$.

1997 Taiwan National Olympiad, 8

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

Let $O$ be the circumcenter and $R$ be the circumradius of an acute triangle $ABC$. Let $AO$ meet the circumcircle of $OBC$ again at $D$, $BO$ meet the circumcircle of $OCA$ again at $E$, and $CO$ meet the circumcircle of $OAB$ again at $F$. Show that $OD.OE.OF\geq 8R^{3}$.

2010 Today's Calculation Of Integral, 646

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

Evaluate \[\int_0^{\pi} a^x\cos bx\ dx,\ \int_0^{\pi} a^x\sin bx\ dx\ (a>0,\ a\neq 1,\ b\in{\mathbb{N^{+}}})\] Own

2007 Peru IMO TST, 2

Tags: analytic geometry , geometry , trigonometry , circumcircle , ratio , rectangle , geometric transformation

Let $ABC$ be a triangle such that $CA \neq CB$, the points $A_{1}$ and $B_{1}$ are tangency points for the ex-circles relative to sides $CB$ and $CA$, respectively, and $I$ the incircle. The line $CI$ intersects the cincumcircle of the triangle $ABC$ in the point $P$. The line that trough $P$ that is perpendicular to $CP$, intersects the line $AB$ in $Q$. Prove that the lines $QI$ and $A_{1}B_{1}$ are parallels.

2014 AIME Problems, 9

Tags: algebra , vieta , quadratic , polynomial , trigonometry

Let $x_1<x_2<x_3$ be three real roots of equation $\sqrt{2014}x^3-4029x^2+2=0$. Find $x_2(x_1+x_3)$.

2008 AIME Problems, 8

Tags: modular arithmetic , trigonometry

Let $ a\equal{}\pi/2008$. Find the smallest positive integer $ n$ such that \[ 2[\cos(a)\sin(a)\plus{}\cos(4a)\sin(2a)\plus{}\cos(9a)\sin(3a)\plus{}\cdots\plus{}\cos(n^2a)\sin(na)]\] is an integer.

1966 IMO Longlists, 10

Tags: equation , trigonometry , algebra , trigonometric equation

How many real solutions are there to the equation $x = 1964 \sin x - 189$ ?

1949 Miklós Schweitzer, 2

Tags: trigonometry , limit , real analysis , integration

Compute $ \lim_{n\rightarrow \infty} \int_{0}^{\pi} \frac {\sin{x}}{1 \plus{} \cos^2 nx}dx$ .

2009 Iran Team Selection Test, 1

Tags: trigonometry , inradius , geometry , incenter

Let $ ABC$ be a triangle and $ A'$ , $ B'$ and $ C'$ lie on $ BC$ , $ CA$ and $ AB$ respectively such that the incenter of $ A'B'C'$ and $ ABC$ are coincide and the inradius of $ A'B'C'$ is half of inradius of $ ABC$ . Prove that $ ABC$ is equilateral .

1998 Baltic Way, 11

Tags: trigonometry , inequalities , geometry

If $a,b,c$ be the lengths of the sides of a triangle. Let $R$ denote its circumradius. Prove that \[ R\ge \frac{a^2+b^2}{2\sqrt{2a^2+2b^2-c^2}}\] When does equality hold?

1999 AMC 12/AHSME, 27

Tags: trigonometry

In triangle $ ABC$, $ 3\sin A \plus{} 4\cos B \equal{} 6$ and $ 4\sin B \plus{} 3\cos A \equal{} 1$. Then $ \angle C$ in degrees is $ \textbf{(A)}\ 30\qquad \textbf{(B)}\ 60\qquad \textbf{(C)}\ 90\qquad \textbf{(D)}\ 120\qquad \textbf{(E)}\ 150$

1991 Czech And Slovak Olympiad IIIA, 1

Tags: inequalities , trigonometry

Prove that for any real numbers $p,q,r,\phi$,: $$\cos^2\phi+q \sin \phi \cos \phi +r\sin^2 \phi \ge \frac12 (p+r-\sqrt{(p-r)^2+q^2})$$

2023 Bulgarian Spring Mathematical Competition, 11.1

Tags: trigonometry , algebra

Find all real $a$ such that the equation $3^{\cos (2x)+1}-(a-5)3^{\cos^2(2x)}=7$ has a real root. [hide=Remark] This was the statement given at the contest, but there was actually a typo and the intended equation was $3^{\cos (2x)+1}-(a-5)3^{\cos^2(x)}=7$, which is much easier.

2004 National Olympiad First Round, 29

Tags: trigonometry , circumcircle , law of cosines , cyclic quadrilateral , trapezoid , geometry

Let $M$ be the intersection of the diagonals $AC$ and $BD$ of cyclic quadrilateral $ABCD$. If $|AB|=5$, $|CD|=3$, and $m(\widehat{AMB}) = 60^\circ$, what is the circumradius of the quadrilateral? $ \textbf{(A)}\ 5\sqrt 3 \qquad\textbf{(B)}\ \dfrac {7\sqrt 3}{3} \qquad\textbf{(C)}\ 6 \qquad\textbf{(D)}\ 4 \qquad\textbf{(E)}\ \sqrt{34} $

2008 Putnam, A5

Tags: rotation , trigonometry , analytic geometry , algebra , polynomial

Let $ n\ge 3$ be an integer. Let $ f(x)$ and $ g(x)$ be polynomials with real coefficients such that the points $ (f(1),g(1)),(f(2),g(2)),\dots,(f(n),g(n))$ in $ \mathbb{R}^2$ are the vertices of a regular $ n$-gon in counterclockwise order. Prove that at least one of $ f(x)$ and $ g(x)$ has degree greater than or equal to $ n\minus{}1.$

1994 AIME Problems, 10

Tags: trigonometry , ratio , inequalities , calculus , integration

In triangle $ABC,$ angle $C$ is a right angle and the altitude from $C$ meets $\overline{AB}$ at $D.$ The lengths of the sides of $\triangle ABC$ are integers, $BD=29^3,$ and $\cos B=m/n,$ where $m$ and $n$ are relatively prime positive integers. Find $m+n.$

2005 Today's Calculation Of Integral, 19

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

Calculate the following indefinite integrals. [1] $\int \tan ^ 3 x dx$ [2] $\int a^{mx+n}dx\ (a>0,a\neq 1, mn\neq 0)$ [3] $\int \cos ^ 5 x dx$ [4] $\int \sin ^ 2 x\cos ^ 3 x dx$ [5]$ \int \frac{dx}{\sin x}$

1986 IMO Longlists, 51

Tags: geometry , rectangle , inequalities , trigonometry , cyclic quadrilateral

Let $a, b, c, d$ be the lengths of the sides of a quadrilateral circumscribed about a circle and let $S$ be its area. Prove that $S \leq \sqrt{abcd}$ and find conditions for equality.

1984 AIME Problems, 9

Tags: geometry , 3d geometry , tetrahedron , trigonometry

In tetrahedron $ABCD$, edge $AB$ has length 3 cm. The area of face $ABC$ is 15 $\text{cm}^2$ and the area of face $ABD$ is 12 $\text{cm}^2$. These two faces meet each other at a $30^\circ$ angle. Find the volume of the tetrahedron in $\text{cm}^3$.

2014 Singapore Senior Math Olympiad, 21

Tags: trigonometry , algebra , quadratic , vieta

Let $n$ be an integer, and let $\triangle ABC$ be a right-angles triangle with right angle at $C$. It is given that $\sin A$ and $\sin B$ are the roots of the quadratic equation \[(5n+8)x^2-(7n-20)x+120=0.\] Find the value of $n$

2009 Today's Calculation Of Integral, 413

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

Find the maximum and minimum value of $ F(x) \equal{} \frac {1}{2}x \plus{} \int_0^x (t \minus{} x)\sin t\ dt$ for $ 0\leq x\leq \pi$.

2013 Sharygin Geometry Olympiad, 8

Tags: geometry , trigonometry , incenter

Two fixed circles are given on the plane, one of them lies inside the other one. From a point $C$ moving arbitrarily on the external circle, draw two chords $CA, CB$ of the larger circle such that they tangent to the smalaler one. Find the locus of the incenter of triangle $ABC$.