This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

Tags were heavily modified to better represent problems.

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Found problems: 3349

2013 AIME Problems, 8
Tags: trapezoid , algebra , circumcircle , quadratic , trigonometry , geometry
A hexagon that is inscribed in a circle has side lengths $22$, $22$, $20$, $22$, $22$, and $20$ in that order. The radius of the circle can be written as $p+\sqrt{q}$, where $p$ and $q$ are positive integers. Find $p+q$.

2012 Tuymaada Olympiad, 3
Tags: geometry , trigonometry , inequalities
A circle is contained in a quadrilateral with successive sides of lengths $3,6,5$ and $8$. Prove that the length of its radius is less than $3$. [i]Proposed by K. Kokhas[/i]

1996 AIME Problems, 4
Tags: geometry , floor function , trigonometry , 3d geometry , dilation , ratio , geometric transformation
A wooden cube, whose edges are one centimeter long, rests on a horizontal surface. Illuminated by a point source of light that is $x$ centimeters directly above an upper vertex, the cube casts a shadow on the horizontal surface. The area of the shadow, which does not inclued the area beneath the cube is 48 square centimeters. Find the greatest integer that does not exceed $1000x.$

1991 India National Olympiad, 3
Tags: trigonometry
Given a triangle $ABC$ let \begin{eqnarray*} x &=& \tan\left(\dfrac{B-C}{2}\right) \tan \left(\dfrac{A}{2}\right) \\ y &=& \tan\left(\dfrac{C-A}{2}\right) \tan \left(\dfrac{B}{2}\right) \\ z &=& \tan\left(\dfrac{A-B}{2}\right) \tan \left(\dfrac{C}{2}\right). \end{eqnarray*} Prove that $x+ y + z + xyz = 0$.

2007 Junior Balkan Team Selection Tests - Romania, 3
Tags: conic , ellipse , symmetry , geometry , trigonometry , inequalities , perpendicular bisector
Let $ABC$ an isosceles triangle, $P$ a point belonging to its interior. Denote $M$, $N$ the intersection points of the circle $\mathcal{C}(A, AP)$ with the sides $AB$ and $AC$, respectively. Find the position of $P$ if $MN+BP+CP$ is minimum.

2009 Today's Calculation Of Integral, 454
Tags: trigonometry , calculus , calculus computations , integration
Let $ a$ be positive constant number. Evaluate $ \int_{ \minus{} a}^a \frac {x^2\cos x \plus{} e^{x}}{e^{x} \plus{} 1}\ dx.$

1953 AMC 12/AHSME, 34
Tags: geometry , trig identities , trigonometry , circumcircle , law of sines
If one side of a triangle is $ 12$ inches and the opposite angle is $ 30$ degrees, then the diameter of the circumscribed circle is: $ \textbf{(A)}\ 18\text{ inches} \qquad\textbf{(B)}\ 30\text{ inches} \qquad\textbf{(C)}\ 24\text{ inches} \qquad\textbf{(D)}\ 20\text{ inches}\\ \textbf{(E)}\ \text{none of these}$

Today's calculation of integrals, 767
Tags: integration , calculus , calculus computations , trigonometry
For $0\leq t\leq 1$, define $f(t)=\int_0^{2\pi} |\sin x-t|dx.$ Evaluate $\int_0^1 f(t)dt.$

2007 Balkan MO Shortlist, A3
Tags: trigonometry , three variable inequality , fourier , inequalities
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]

2010 China Team Selection Test, 1
Tags: incenter , geometry , circumcircle , radical axis , power of a point , trigonometry
Let $\omega$ be a semicircle and $AB$ its diameter. $\omega_1$ and $\omega_2$ are two different circles, both tangent to $\omega$ and to $AB$, and $\omega_1$ is also tangent to $\omega_2$. Let $P,Q$ be the tangent points of $\omega_1$ and $\omega_2$ to $AB$ respectively, and $P$ is between $A$ and $Q$. Let $C$ be the tangent point of $\omega_1$ and $\omega$. Find $\tan\angle ACQ$.

2011 Belarus Team Selection Test, 1
Tags: functional equation , trigonometry , functional , algebra
Find all real $a$ such that there exists a function $f: R \to R$ satisfying the equation $f(\sin x )+ a f(\cos x) = \cos 2x$ for all real $x$. I.Voronovich

1995 Turkey MO (2nd round), 4
Tags: geometry , trigonometry , ratio
In a triangle $ABC$ with $AB\neq AC$, the internal and external bisectors of angle $A$ meet the line $BC$ at $D$ and $E$ respectively. If the feet of the perpendiculars from a point $F$ on the circle with diameter $DE$ to $BC,CA,AB$ are $K,L,M$, respectively, show that $KL=KM$.

2010 Romanian Master of Mathematics, 3
Tags: geometric transformation , trig identities , geometry , homothety , projective geometry , ratio , trigonometry
Let $A_1A_2A_3A_4$ be a quadrilateral with no pair of parallel sides. For each $i=1, 2, 3, 4$, define $\omega_1$ to be the circle touching the quadrilateral externally, and which is tangent to the lines $A_{i-1}A_i, A_iA_{i+1}$ and $A_{i+1}A_{i+2}$ (indices are considered modulo $4$ so $A_0=A_4, A_5=A_1$ and $A_6=A_2$). Let $T_i$ be the point of tangency of $\omega_i$ with the side $A_iA_{i+1}$. Prove that the lines $A_1A_2, A_3A_4$ and $T_2T_4$ are concurrent if and only if the lines $A_2A_3, A_4A_1$ and $T_1T_3$ are concurrent. [i]Pavel Kozhevnikov, Russia[/i]

2002 China Team Selection Test, 1
Tags: inequalities , geometry , trigonometry
In acute triangle $ ABC$, show that: $ \sin^3{A}\cos^2{(B \minus{} C)} \plus{} \sin^3{B}\cos^2{(C \minus{} A)} \plus{} \sin^3{C}\cos^2{(A \minus{} B)} \leq 3\sin{A} \sin{B} \sin{C}$ and find out when the equality holds.

2012 China Second Round Olympiad, 9
Tags: algebra , inequalities , function , trigonometry
Given a function $f(x)=a\sin x-\frac{1}{2}\cos 2x+a-\frac{3}{a}+\frac{1}{2}$, where $a\in\mathbb{R}, a\ne 0$. [b](1)[/b] If for any $x\in\mathbb{R}$, inequality $f(x)\le 0$ holds, find all possible value of $a$. [b](2)[/b] If $a\ge 2$, and there exists $x\in\mathbb{R}$, such that $f(x)\le 0$. Find all possible value of $a$.

1997 Flanders Math Olympiad, 3
Tags: trigonometry , ratio , geometry , limit , geometric sequence , inequalities
$\Delta oa_1b_1$ is isosceles with $\angle a_1ob_1 = 36^\circ$. Construct $a_2,b_2,a_3,b_3,...$ as below, with $|oa_{i+1}| = |a_ib_i|$ and $\angle a_iob_i = 36^\circ$, Call the summed area of the first $k$ triangles $A_k$. Let $S$ be the area of the isocseles triangle, drawn in - - -, with top angle $108^\circ$ and $|oc|=|od|=|oa_1|$, going through the points $b_2$ and $a_2$ as shown on the picture. (yes, $cd$ is parallel to $a_1b_1$ there) Show $A_k < S$ for every positive integer $k$. [img]http://www.mathlinks.ro/Forum/album_pic.php?pic_id=284[/img]

1988 IMO Longlists, 4
Tags: geometry , geometric inequality , circumcircle , inradius , trigonometry
The triangle $ ABC$ is inscribed in a circle. The interior bisectors of the angles $ A,B$ and $ C$ meet the circle again at $ A', B'$ and $ C'$ respectively. Prove that the area of triangle $ A'B'C'$ is greater than or equal to the area of triangle $ ABC.$

2010 Purple Comet Problems, 27
Tags: vector , inequalities , trigonometry
Let $a$ and $b$ be real numbers satisfying $2(\sin a + \cos a) \sin b = 3 - \cos b$. Find $3 \tan^2a+4\tan^2 b$.

1999 Harvard-MIT Mathematics Tournament, 2
Tags: calculus , geometry , rectangle , trigonometry
A rectangle has sides of length $\sin x$ and $\cos x$ for some $x$. What is the largest possible area of such a rectangle?

1991 AIME Problems, 4
Tags: function , logarithm , trigonometry
How many real numbers $x$ satisfy the equation $\frac{1}{5}\log_2 x = \sin (5\pi x)$?

1993 Vietnam National Olympiad, 1
Tags: calculus , real analysis , algebra , trigonometry
$f : [-\sqrt{1995},\sqrt{1995}] \to\mathbb{R}$ is defined by $f(x) = x(1993+\sqrt{1995-x^{2}})$. Find its maximum and minimum values.

2005 Today's Calculation Of Integral, 61
Tags: trigonometry , calculus , calculus computations , integration
Evaluate \[\sum_{k=0}^{2004} \int_{-1}^1 \frac{\sqrt{1-x^2}}{\sqrt{k+1}-x}dx\]

2010 Korea National Olympiad, 1
Tags: inequalities , trigonometry
$ x, y, z $ are positive real numbers such that $ x+y+z=1 $. Prove that \[ \sqrt{ \frac{x}{1-x} } + \sqrt{ \frac{y}{1-y} } + \sqrt{ \frac{z}{1-z} } > 2 \]

1990 AMC 12/AHSME, 21
Tags: 3d geometry , pythagorean theorem , trigonometry , geometry , pyramid
Consider a pyramid $P-ABCD$ whose base $ABCD$ is a square and whose vertex $P$ is equidistant from $A$, $B$, $C$, and $D$. If $AB=1$ and $\angle APD=2\theta$ then the volume of the pyramid is $\text{(A)} \ \frac{\sin \theta}{6} \qquad \text{(B)} \ \frac{\cot \theta}{6} \qquad \text{(C)} \ \frac1{6\sin \theta} \qquad \text{(D)} \ \frac{1-\sin 2\theta}{6} \qquad \text{(E)} \ \frac{\sqrt{\cos 2\theta}}{6\sin \theta}$

2011 Vietnam National Olympiad, 4
Tags: combinatorics , ceiling function , trigonometry
A convex pentagon $ABCDE$ satisfies that the sidelengths and $AC,AD\leq \sqrt 3.$ Let us choose $2011$ distinct points inside this pentagon. Prove that there exists an unit circle with centre on one edge of the pentagon, and which contains at least $403$ points out of the $2011$ given points. {Edited} {I posted it correctly before but because of a little confusion deleted the sidelength part, sorry.}