Olympiad problems

2003 China Team Selection Test, 1

Let $ ABCD$ be a quadrilateral which has an incircle centered at $ O$. Prove that \[ OA\cdot OC\plus{}OB\cdot OD\equal{}\sqrt{AB\cdot BC\cdot CD\cdot DA}\]

2009 Today's Calculation Of Integral, 499

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

Evaluate \[ \int_0^{\pi} (\sqrt[2009]{\cos x}\plus{}\sqrt[2009]{\sin x}\plus{}\sqrt[2009]{\tan x})\ dx.\]

2012 Today's Calculation Of Integral, 794

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

Define a function $f(x)=\int_0^{\frac{\pi}{2}} \frac{\cos |t-x|}{1+\sin |t-x|}dt$ for $0\leq x\leq \pi$. Find the maximum and minimum value of $f(x)$ in $0\leq x\leq \pi$.

1977 IMO Longlists, 6

Tags: trigonometry , function , inequalities , induction , algebra

Let $x_1, x_2, \ldots , x_n \ (n \geq 1)$ be real numbers such that $0 \leq x_j \leq \pi, \ j = 1, 2,\ldots, n.$ Prove that if $\sum_{j=1}^n (\cos x_j +1) $ is an odd integer, then $\sum_{j=1}^n \sin x_j \geq 1.$

2007 Balkan MO Shortlist, A3

Tags: inequalities , trigonometry , three variable inequality , fourier

For $n\in\mathbb{N}$, $n\geq 2$, $a_{i}, b_{i}\in\mathbb{R}$, $1\leq i\leq n$, such that \[\sum_{i=1}^{n}a_{i}^{2}=\sum_{i=1}^{n}b_{i}^{2}=1, \sum_{i=1}^{n}a_{i}b_{i}=0. \] Prove that \[\left(\sum_{i=1}^{n}a_{i}\right)^{2}+\left(\sum_{i=1}^{n}b_{i}\right)^{2}\leq n. \] [i]Cezar Lupu & Tudorel Lupu[/i]

1998 Romania Team Selection Test, 2

Tags: geometry , trigonometry , 3d geometry , prism , inequalities

A parallelepiped has surface area 216 and volume 216. Show that it is a cube.

2014 NIMO Problems, 5

Tags: circumcircle , ratio , trigonometry , complex number , pythagorean theorem , geometry

Triangle $ABC$ has sidelengths $AB = 14, BC = 15,$ and $CA = 13$. We draw a circle with diameter $AB$ such that it passes $BC$ again at $D$ and passes $CA$ again at $E$. If the circumradius of $\triangle CDE$ can be expressed as $\tfrac{m}{n}$ where $m, n$ are coprime positive integers, determine $100m+n$. [i]Proposed by Lewis Chen[/i]

1989 China Team Selection Test, 2

Tags: trigonometry , function , angle bisector , geometry

$AD$ is the altitude on side $BC$ of triangle $ABC$. If $BC+AD-AB-AC = 0$, find the range of $\angle BAC$. [i]Alternative formulation.[/i] Let $AD$ be the altitude of triangle $ABC$ to the side $BC$. If $BC+AD=AB+AC$, then find the range of $\angle{A}$.

2009 Putnam, A3

Tags: linear algebra , matrix , trigonometry , limit , induction , function

Let $ d_n$ be the determinant of the $ n\times n$ matrix whose entries, from left to right and then from top to bottom, are $ \cos 1,\cos 2,\dots,\cos n^2.$ (For example, $ d_3 \equal{} \begin{vmatrix}\cos 1 & \cos2 & \cos3 \\ \cos4 & \cos5 & \cos 6 \\ \cos7 & \cos8 & \cos 9\end{vmatrix}.$ The argument of $ \cos$ is always in radians, not degrees.) Evaluate $ \lim_{n\to\infty}d_n.$

2006 IMC, 3

Tags: trigonometry , logarithm

Compare $\tan(\sin x)$ with $\sin(\tan x)$, for $x\in \left]0,\frac{\pi}{2}\right[$.

2005 South East Mathematical Olympiad, 8

Tags: trigonometry , calculus , derivative , function , inequalities

Let $0 < \alpha, \beta, \gamma < \frac{\pi}{2}$ and $\sin^{3} \alpha + \sin^{3} \beta + \sin^3 \gamma = 1$. Prove that \[ \tan^{2} \alpha + \tan^{2} \beta + \tan^{2} \gamma \geq \frac{3 \sqrt{3}}{2} . \]

2011 District Olympiad, 2

Tags: complex number , absolute value , geometry , rectangle , trigonometry

[b]a)[/b] Show that if four distinct complex numbers have the same absolute value and their sum vanishes, then they represent a rectangle. [b]b)[/b] Let $ x,y,z,t $ be four real numbers, and $ k $ be an integer. Prove the following implication: $$ \sum_{j\in\{ x,y,z,t\}} \sin j = 0 = \sum_{j\in\{ x,y,z,t\}} \cos j\implies \sum_{j\in\{ x,y,z,t\}} \sin (1+2n)j. $$

1981 Vietnam National Olympiad, 1

Tags: trigonometry , perimeter , circumcircle , inradius , geometry

Prove that a triangle $ABC$ is right-angled if and only if \[\sin A + \sin B + \sin C = \cos A + \cos B + \cos C + 1\]

2005 Today's Calculation Of Integral, 1

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

Calculate the following indefinite integral. [1] $\int \frac{e^{2x}}{(e^x+1)^2}dx$ [2] $\int \sin x\cos 3x dx$ [3] $\int \sin 2x\sin 3x dx$ [4] $\int \frac{dx}{4x^2-12x+9}$ [5] $\int \cos ^4 x dx$

2010 Korea National Olympiad, 1

Tags: trigonometry , inequalities

$ x, y, z $ are positive real numbers such that $ x+y+z=1 $. Prove that \[ \sqrt{ \frac{x}{1-x} } + \sqrt{ \frac{y}{1-y} } + \sqrt{ \frac{z}{1-z} } > 2 \]

2011 AMC 12/AHSME, 20

Tags: geometry , circumcircle , trigonometry

Triangle $ABC$ has $AB=13$, $BC=14$, and $AC=15$. The points $D, E,$ and $F$ are the midpoints of $\overline{AB}$, $\overline{BC}$, and $\overline{AC}$ respectively. Let $ X \ne E$ be the intersection of the circumcircles of $\triangle BDE$ and $\triangle CEF$. What is $XA+XB+XC$? $ \textbf{(A)}\ 24 \qquad \textbf{(B)}\ 14\sqrt{3} \qquad \textbf{(C)}\ \frac{195}{8} \qquad \textbf{(D)}\ \frac{129\sqrt{7}}{14} \qquad \textbf{(E)}\ \frac{69\sqrt{2}}{4} $

2009 IMO Shortlist, 6

Tags: geometry , circumcircle , trigonometry , symmetry

Let the sides $AD$ and $BC$ of the quadrilateral $ABCD$ (such that $AB$ is not parallel to $CD$) intersect at point $P$. Points $O_1$ and $O_2$ are circumcenters and points $H_1$ and $H_2$ are orthocenters of triangles $ABP$ and $CDP$, respectively. Denote the midpoints of segments $O_1H_1$ and $O_2H_2$ by $E_1$ and $E_2$, respectively. Prove that the perpendicular from $E_1$ on $CD$, the perpendicular from $E_2$ on $AB$ and the lines $H_1H_2$ are concurrent. [i]Proposed by Eugene Bilopitov, Ukraine[/i]

1963 Miklós Schweitzer, 8

Tags: trigonometry , function , integration , search , real analysis

Let the Fourier series \[ \frac{a_0}{2}+ \sum _{k\geq 1}(a_k\cos kx+b_k \sin kx)\] of a function $ f(x)$ be absolutely convergent, and let \[ a^2_k+b^2_k \geq a_{k+1}^2+b_{k+1}^2 \;(k=1,2,...)\ .\] Show that \[ \frac1h \int_0^{2\pi} (f(x+h)-f(x-h))^2dx \;(h>0)\] is uniformly bounded in $ h$. [K. Tandori]

2014 All-Russian Olympiad, 1

Tags: trigonometry , algebra

Does there exist positive $a\in\mathbb{R}$, such that \[|\cos x|+|\cos ax| >\sin x +\sin ax \] for all $x\in\mathbb{R}$? [i]N. Agakhanov[/i]

2003 National High School Mathematics League, 4

Tags: trigonometry

If $x\in\left[-\frac{5\pi}{12},-\frac{\pi}{3}\right]$, then the maximum value of $y=\tan\left(x+\frac{2\pi}{3}\right)-\tan\left(x+\frac{\pi}{6}\right)+\cos\left(x+\frac{\pi}{6}\right)$ is $\text{(A)}\frac{12}{5}\sqrt2\qquad\text{(B)}\frac{11}{6}\sqrt2\qquad\text{(C)}\frac{11}{6}\sqrt3\qquad\text{(D)}\frac{12}{5}\sqrt3$

2004 Thailand Mathematical Olympiad, 1

Tags: trigonometry

Given that $\cos 4A =\frac13$ and $-\frac{\pi}{4} \le A \le \frac{\pi}{4}$ , find the value of $\cos^8 A - \sin^8 A$.

2011 Math Prize For Girls Problems, 11

Tags: linear algebra , matrix , algebra , polynomial , trigonometry , mewing

The sequence $a_0$, $a_1$, $a_2$, $\ldots\,$ satisfies the recurrence equation \[ a_n = 2 a_{n-1} - 2 a_{n - 2} + a_{n - 3} \] for every integer $n \ge 3$. If $a_{20} = 1$, $a_{25} = 10$, and $a_{30} = 100$, what is the value of $a_{1331}$?

1981 IMO Shortlist, 15

Tags: trigonometry , geometry , incenter , geometric inequality , minimization

Consider a variable point $P$ inside a given triangle $ABC$. Let $D$, $E$, $F$ be the feet of the perpendiculars from the point $P$ to the lines $BC$, $CA$, $AB$, respectively. Find all points $P$ which minimize the sum \[ {BC\over PD}+{CA\over PE}+{AB\over PF}. \]

Today's calculation of integrals, 767

Tags: calculus , integration , trigonometry , calculus computations

For $0\leq t\leq 1$, define $f(t)=\int_0^{2\pi} |\sin x-t|dx.$ Evaluate $\int_0^1 f(t)dt.$

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