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

1986 Spain Mathematical Olympiad, 6
Tags: trigonometry , geometry , product
Evaluate $$\prod_{k=1}^{14} cos \big(\frac{k\pi}{15}\big)$$

1959 AMC 12/AHSME, 36
Tags: geometry , triangle geometry , trigonometry
The base of a triangle is $80$, and one side of the base angle is $60^\circ$. The sum of the lengths of the other two sides is $90$. The shortest side is: $ \textbf{(A)}\ 45 \qquad\textbf{(B)}\ 40\qquad\textbf{(C)}\ 36\qquad\textbf{(D)}\ 17\qquad\textbf{(E)}\ 12 $

1966 IMO Shortlist, 18
Tags: parabola , parameterization , trigonometry , trigonometric equation
Solve the equation $\frac{1}{\sin x}+\frac{1}{\cos x}=\frac 1p$ where $p$ is a real parameter. Discuss for which values of $p$ the equation has at least one real solution and determine the number of solutions in $[0, 2\pi)$ for a given $p.$

1998 Spain Mathematical Olympiad, 3
Tags: circumcircle , geometry , trigonometry
Let $ABC$ be a triangle. Points $D$ and $E$ are taken on the line $BC$ such that $AD$ and $AE$ are parallel to the respective tangents to the circumcircle at $C$ and $B$. Prove that \[\frac{BE}{CD}=\left(\frac{AB}{AC}\right)^2 \]

Estonia Open Senior - geometry, 1996.2.4
Tags: geometry , area , square , trigonometry , circles
The figure shows a square and a circle with a common center $O$, with equal areas of striped shapes. Find the value of $\cos a$. [img]https://2.bp.blogspot.com/-7uwa0H42ELg/XnmsSoPMgcI/AAAAAAAALgk/pHNBqtbsdKgMhcvIRYLm_8JRpOeIYcUeACK4BGAYYCw/s400/96%2Bestonia%2Bopen%2Bs2.4.png[/img]

2012 Balkan MO Shortlist, A1
Tags: trigonometry , triangle inequality , rearrangement inequality , function
Prove that \[\sum_{cyc}(x+y)\sqrt{(z+x)(z+y)} \geq 4(xy+yz+zx),\] for all positive real numbers $x,y$ and $z$.

2000 AIME Problems, 15
Tags: trigonometry , induction , function
Find the least positive integer $n$ such that \[ \frac 1{\sin 45^\circ\sin 46^\circ}+\frac 1{\sin 47^\circ\sin 48^\circ}+\cdots+\frac 1{\sin 133^\circ\sin 134^\circ}=\frac 1{\sin n^\circ}. \]

V Soros Olympiad 1998 - 99 (Russia), 11.3
Tags: algebra , trigonometry , inequalities
For what a from the interval $[0,\pi]$ do there exist $a$ and $b$ that are not simultaneously equal to zero, for which the inequality $$a \cos x + b \cos 2x \le 0$$ is satisfied for all $x$ belonging to the segment $[a, \pi]$?

2020 Purple Comet Problems, 23
Tags: trigonometry
There is a real number $x$ between $0$ and $\frac{\pi}{2}$ such that $$\frac{\sin^3 x + \cos^3 x}{\sin^5 x + \cos^5 x}=\frac{12}{11}$$ and $\sin x + \cos x =\frac{\sqrt{m}}{n}$ , where $m$ and $n$ are positive integers, and $m$ is not divisible by the square of any prime. Find $m + n$.

2007 Purple Comet Problems, 10
Tags: trigonometry , complex number
For a particular value of the angle $\theta$ we can take the product of the two complex numbers $(8+i)\sin\theta+(7+4i)\cos\theta$ and $(1+8i)\sin\theta+(4+7i)\cos\theta$ to get a complex number in the form $a+bi$ where $a$ and $b$ are real numbers. Find the largest value for $a+b$.

1969 Vietnam National Olympiad, 2
Tags: algebra , trigonometric equation , trigonometry
Find all real $x$ such that $0 < x < \pi $ and $\frac{8}{3 sin x - sin 3x} + 3 sin^2x \le 5$.

2013 F = Ma, 11
Tags: geometry , 3d geometry , trigonometry
A right-triangular wooden block of mass $M$ is at rest on a table, as shown in figure. Two smaller wooden cubes, both with mass $m$, initially rest on the two sides of the larger block. As all contact surfaces are frictionless, the smaller cubes start sliding down the larger block while the block remains at rest. What is the normal force from the system to the table? $\textbf{(A) } 2mg\\ \textbf{(B) } 2mg + Mg\\ \textbf{(C) } mg + Mg\\ \textbf{(D) } Mg + mg( \sin \alpha + \sin \beta)\\ \textbf{(E) } Mg + mg( \cos \alpha + \cos \beta)$

2011 Harvard-MIT Mathematics Tournament, 8
Tags: hmmt , complex number , trigonometry
Let $z = \cos \frac{2\pi}{2011} + i\sin \frac{2\pi}{2011}$, and let \[ P(x) = x^{2008} + 3x^{2007} + 6x^{2006} + \cdots + \frac{2008 \cdot 2009}{2} x + \frac{2009 \cdot 2010}{2} \] for all complex numbers $x$. Evaluate $P(z)P(z^2)P(z^3) \cdots P(z^{2010})$.

Today's calculation of integrals, 848
Tags: trigonometry , calculus , logarithm , integration , calculus computations
Evaluate $\int_0^{\frac {\pi}{4}} \frac {\sin \theta -2\ln \frac{1-\sin \theta}{\cos \theta}}{(1+\cos 2\theta)\sqrt{\ln \frac{1+\sin \theta}{\cos \theta}}}d\theta .$

2006 Germany Team Selection Test, 1
Tags: trigonometry , function , percent , algebra
Find all real solutions $x$ of the equation $\cos\cos\cos\cos x=\sin\sin\sin\sin x$. (Angles are measured in radians.)

2010 Math Prize For Girls Problems, 15
Tags: trigonometry
Compute the value of the sum \begin{align*} \frac{1}{1 + \tan^3 0^\circ} &+ \frac{1}{1 + \tan^3 10^\circ} + \frac{1}{1 + \tan^3 20^\circ} + \frac{1}{1 + \tan^3 30^\circ} + \frac{1}{1 + \tan^3 40^\circ} \\ &+ \frac{1}{1 + \tan^3 50^\circ} + \frac{1}{1 + \tan^3 60^\circ} + \frac{1}{1 + \tan^3 70^\circ} + \frac{1}{1 + \tan^3 80^\circ} \, . \end{align*}

2009 Harvard-MIT Mathematics Tournament, 2
Tags: integration , calculus , function , trigonometry
The differentiable function $F:\mathbb{R}\to\mathbb{R}$ satisfies $F(0)=-1$ and \[\dfrac{d}{dx}F(x)=\sin (\sin (\sin (\sin(x))))\cdot \cos( \sin (\sin (x))) \cdot \cos (\sin(x))\cdot\cos(x).\] Find $F(x)$ as a function of $x$.

2014 NIMO Problems, 4
Tags: law of sines , trig identities , trigonometry , power of a point , ratio , trapezoid , geometry
Points $A$, $B$, $C$, and $D$ lie on a circle such that chords $\overline{AC}$ and $\overline{BD}$ intersect at a point $E$ inside the circle. Suppose that $\angle ADE =\angle CBE = 75^\circ$, $BE=4$, and $DE=8$. The value of $AB^2$ can be written in the form $a+b\sqrt{c}$ for positive integers $a$, $b$, and $c$ such that $c$ is not divisible by the square of any prime. Find $a+b+c$. [i]Proposed by Tony Kim[/i]

1976 AMC 12/AHSME, 17
Tags: trigonometry
If $\theta$ is an acute angle, and $\sin 2\theta=a$, then $\sin\theta+\cos\theta$ equals $\textbf{(A) }\sqrt{a+1}\qquad\textbf{(B) }(\sqrt{2}-1)a+1\qquad\textbf{(C) }\sqrt{a+1}-\sqrt{a^2-a}\qquad$ $\textbf{(D) }\sqrt{a+1}+\sqrt{a^2-a}\qquad \textbf{(E) }\sqrt{a+1}+a^2-a$

1988 Irish Math Olympiad, 2
Tags: trigonometry , circumcircle , geometry
A; B; C; D are the vertices of a square, and P is a point on the arc CD of its circumcircle. Prove that $ |PA|^2 - |PB|^2 = |PB|.|PD| -|PA|.|PC| $ Can anyone here find the solution? I'm not great with geometry, so i tried turning it into co-ordinate geometry equations, but sadly to no avail. Thanks in advance.

Swiss NMO - geometry, 2011.2
Tags: trigonometry , trig identities , geometry , law of sines , incenter
Let $\triangle{ABC}$ be an acute-angled triangle and let $D$, $E$, $F$ be points on $BC$, $CA$, $AB$, respectively, such that \[\angle{AFE}=\angle{BFD}\mbox{,}\quad\angle{BDF}=\angle{CDE}\quad\mbox{and}\quad\angle{CED}=\angle{AEF}\mbox{.}\] Prove that $D$, $E$ and $F$ are the feet of the perpendiculars through $A$, $B$ and $C$ on $BC$, $CA$ and $AB$, respectively. [i](Swiss Mathematical Olympiad 2011, Final round, problem 2)[/i]

2007 JBMO Shortlist, 2
Tags: trigonometry , geometry , incenter , cyclic quadrilateral , circumcircle , function , angle bisector
Let $ABCD$ be a convex quadrilateral with $\angle{DAC}= \angle{BDC}= 36^\circ$ , $\angle{CBD}= 18^\circ$ and $\angle{BAC}= 72^\circ$. The diagonals and intersect at point $P$ . Determine the measure of $\angle{APD}$.

2009 Today's Calculation Of Integral, 495
Tags: trigonometry , calculus , integration , logarithm , calculus computations
Evaluate the following definite integrals. (1) $ \int_0^{\frac {1}{2}} \frac {x^2}{\sqrt {1 \minus{} x^2}}\ dx$ (2) $ \int_0^1 \frac {1 \minus{} x}{(1 \plus{} x^2)^2}\ dx$ (3) $ \int_{ \minus{} 1}^7 \frac {dx}{1 \plus{} \sqrt [3]{1 \plus{} x}}$

1966 IMO Longlists, 9
Tags: trigonometry , algebra , trigonometric equation , equation
Find $x$ such that trigonometric \[\frac{\sin 3x \cos (60^\circ -4x)+1}{\sin(60^\circ - 7x) - \cos(30^\circ + x) + m}=0\] where $m$ is a fixed real number.

2010 Germany Team Selection Test, 3
Tags: algebra , function , trigonometry , functional equation
Find all functions $f: \mathbb{R} \to \mathbb{R}$ such that \[f(x)f(y) = (x+y+1)^2 \cdot f \left( \frac{xy-1}{x+y+1} \right)\] $\forall x,y \in \mathbb{R}$ with $x+y+1 \neq 0$ and $f(x) > 1$ $\forall x > 0.$