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

2012 Today's Calculation Of Integral, 851
Tags: calculus , integration , function , trigonometry , limit , geometric series , calculus computations
Let $T$ be a period of a function $f(x)=|\cos x|\sin x\ (-\infty,\ \infty).$ Find $\lim_{n\to\infty} \int_0^{nT} e^{-x}f(x)\ dx.$

2008 ITest, 57
Tags: trigonometry , number theory , relatively prime
Let $a$ and $b$ be the two possible values of $\tan\theta$ given that \[\sin\theta + \cos\theta = \dfrac{193}{137}.\] If $a+b=m/n$, where $m$ and $n$ are relatively prime positive integers, compute $m+n$.

2019 Jozsef Wildt International Math Competition, W. 47
Tags: inequalities , limit , trigonometry
[list=1] [*] If $a$, $b$, $c$, $d > 0$, show inequality:$$\left(\tan^{-1}\left(\frac{ad-bc}{ac+bd}\right)\right)^2\geq 2\left(1-\frac{ac+bd}{\sqrt{\left(a^2+b^2\right)\left(c^2+d^2\right)}}\right)$$ [*] Calculate $$\lim \limits_{n \to \infty}n^{\alpha}\left(n- \sum \limits_{k=1}^n\frac{n^+k^2-k}{\sqrt{\left(n^2+k^2\right)\left(n^2+(k-1)^2\right)}}\right)$$where $\alpha \in \mathbb{R}$ [/list]

2011 Iran MO (3rd Round), 3
Tags: circumcircle , ratio , trigonometry , projective geometry , geometry
In triangle $ABC$, $X$ and $Y$ are the tangency points of incircle (with center $I$) with sides $AB$ and $AC$ respectively. A tangent line to the circumcircle of triangle $ABC$ (with center $O$) at point $A$, intersects the extension of $BC$ at $D$. If $D,X$ and $Y$ are collinear then prove that $D,I$ and $O$ are also collinear. [i]proposed by Amirhossein Zabeti[/i]

1999 Vietnam National Olympiad, 1
Tags: trigonometry , algebra , inequalities
Given are three positive real numbers $ a,b,c$ satisfying $ abc \plus{} a \plus{} c \equal{} b$. Find the max value of the expression: \[ P \equal{} \frac {2}{a^2 \plus{} 1} \minus{} \frac {2}{b^2 \plus{} 1} \plus{} \frac {3}{c^2 \plus{} 1}.\]

2007 China Northern MO, 1
Tags: trigonometry , inequalities
Let $ \alpha$, $ \beta$ be acute angles. Find the maximum value of \[ \frac{\left(1-\sqrt{\tan\alpha\tan\beta}\right)^{2}}{\cot\alpha+\cot\beta}\]

2006 District Olympiad, 1
Tags: trigonometry , vector , geometry
On the plane of triangle $ABC$ with $\angle BAC = 90^\circ$ we raise perpendicular lines in $A$ and $B$, on the same side of the plane. On these two perpendicular lines we consider the points $M$ and $N$ respectively such that $BN < AM$. Knowing that $AC = 2a$, $AB = a\sqrt 3$, $AM=a$ and that the plane $MNC$ makes an angle of $30^\circ$ with the plane $ABC$ find a) the area of the triangle $MNC$; b) the distance from $B$ to the plane $MNC$.

2009 Serbia Team Selection Test, 1
Tags: inequalities , trigonometry , calculus , derivative , angle bisector , geometry
Let $ \alpha$ and $ \beta$ be the angles of a non-isosceles triangle $ ABC$ at points $ A$ and $ B$, respectively. Let the bisectors of these angles intersect opposing sides of the triangle in $ D$ and $ E$, respectively. Prove that the acute angle between the lines $ DE$ and $ AB$ isn't greater than $ \frac{|\alpha\minus{}\beta|}3$.

1986 AIME Problems, 15
Tags: geometry , analytic geometry , graphing lines , slope , trigonometry , pythagorean theorem , area of a triangle
Let triangle $ABC$ be a right triangle in the xy-plane with a right angle at $C$. Given that the length of the hypotenuse $AB$ is 60, and that the medians through $A$ and $B$ lie along the lines $y=x+3$ and $y=2x+4$ respectively, find the area of triangle $ABC$.

1989 IMO Longlists, 41
Tags: trigonometry , algebra
Let $ f(x) \equal{} a \sin^2x \plus{} b \sin x \plus{} c,$ where $ a, b,$ and $ c$ are real numbers. Find all values of $ a, b$ and $ c$ such that the following three conditions are satisfied simultaneously: [b](i)[/b] $ f(x) \equal{} 381$ if $ \sin x \equal{} \frac{1}{2}.$ [b](ii)[/b] The absolute maximum of $ f(x)$ is $ 444.$ [b](iii)[/b] The absolute minimum of $ f(x)$ is $ 364.$

1960 IMO Shortlist, 3
Tags: trigonometry , geometry , circumcircle , trigonometric identities
In a given right triangle $ABC$, the hypotenuse $BC$, of length $a$, is divided into $n$ equal parts ($n$ and odd integer). Let $\alpha$ be the acute angel subtending, from $A$, that segment which contains the mdipoint of the hypotenuse. Let $h$ be the length of the altitude to the hypotenuse fo the triangle. Prove that: \[ \tan{\alpha}=\dfrac{4nh}{(n^2-1)a}. \]

2005 Today's Calculation Of Integral, 23
Tags: calculus , integration , limit , trigonometry , calculus computations , logarithm
Evaluate \[\lim_{a\rightarrow \frac{\pi}{2}-0}\ \int_0^a\ (\cos x)\ln (\cos x)\ dx\ \left(0\leqq a <\frac{\pi}{2}\right)\]

2009 Harvard-MIT Mathematics Tournament, 4
Tags: trigonometry
If $\tan x + \tan y = 4$ and $\cot x + \cot y = 5$, compute $\tan(x + y)$.

1998 Harvard-MIT Mathematics Tournament, 1
Tags: calculus , search , trigonometry , function
Farmer Tim is lost in the densely-forested Cartesian plane. Starting from the origin he walks a sinusoidal path in search of home; that is, after $t$ minutes he is at position $(t,\sin t)$. Five minutes after he sets out, Alex enters the forest at the origin and sets out in search of Tim. He walks in such a way that after he has been in the forest for $m$ minutes, his position is $(m,\cos t)$. What is the greatest distance between Alex and Farmer Tim while they are walking in these paths?

1986 IMO Longlists, 11
Tags: geometry , 3d geometry , tetrahedron , trigonometry , congruent triangles
Prove that the sum of the face angles at each vertex of a tetrahedron is a straight angle if and only if the faces are congruent triangles.

1998 AMC 12/AHSME, 28
Tags: trigonometry , ratio , relatively prime , geometry , angle bisector , pythagorean theorem , number theory
In triangle $ ABC$, angle $ C$ is a right angle and $ CB > CA$. Point $ D$ is located on $ \overline{BC}$ so that angle $ CAD$ is twice angle $ DAB$. If $ AC/AD \equal{} 2/3$, then $ CD/BD \equal{} m/n$, where $ m$ and $ n$ are relatively prime positive integers. Find $ m \plus{} n$. $ \textbf{(A)}\ 10\qquad \textbf{(B)}\ 14\qquad \textbf{(C)}\ 18\qquad \textbf{(D)}\ 22\qquad \textbf{(E)}\ 26$

2013 Romania National Olympiad, 1
Tags: trigonometry , algebra
Solve the following equation ${{2}^{{{\sin }^{4}}x-{{\cos }^{2}}x}}-{{2}^{{{\cos }^{4}}x-{{\sin }^{2}}x}}=\cos 2x$

2010 Today's Calculation Of Integral, 533
Tags: calculus , integration , function , trigonometry , analytic geometry , parameterization , algebra
Let $ C$ be the circle with radius 1 centered on the origin. Fix the endpoint of the string with length $ 2\pi$ on the point $ A(1,\ 0)$ and put the other end point $ P$ on the point $ P_0(1,\ 2\pi)$. From this situation, when we twist the string around $ C$ by moving the point $ P$ in anti clockwise with the string streched tightly, find the length of the curve that the point $ P$ draws from sarting point $ P_0$ to reaching point $ A$.

2010 Today's Calculation Of Integral, 614
Tags: calculus , integration , ratio , trigonometry , calculus computations
Evaluate $\int_0^1 \{x(1-x)\}^{\frac 32}dx.$ [i]2010 Hirosaki University School of Medicine entrance exam[/i]

2014 AIME Problems, 7
Tags: trigonometry , calculus , derivative , vector , ratio , analytic geometry , graphing lines
Let $w$ and $z$ be complex numbers such that $|w| = 1$ and $|z| = 10$. Let $\theta = \arg\left(\tfrac{w-z}{z}\right)$. The maximum possible value of $\tan^2 \theta$ can be written as $\tfrac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p+q$. (Note that $\arg(w)$, for $w \neq 0$, denotes the measure of the angle that the ray from $0$ to $w$ makes with the positive real axis in the complex plane.

1979 IMO Longlists, 68
Tags: trigonometry , geometry , 3d geometry , maximization
We consider a point $P$ in a plane $p$ and a point $Q \not\in p$. Determine all the points $R$ from $p$ for which \[ \frac{QP+PR}{QR} \] is maximum.

2000 Putnam, 3
Tags: trigonometry , limit , calculus , derivative , algebra , polynomial
Let $f(t) = \displaystyle\sum_{j=1}^{N} a_j \sin (2\pi jt)$, where each $a_j$ is areal and $a_N$ is not equal to $0$. Let $N_k$ denote the number of zeroes (including multiplicites) of $\dfrac{d^k f}{dt^k}$. Prove that \[ N_0 \le N_1 \le N_2 \le \cdots \text { and } \lim_{k \rightarrow \infty} N_k = 2N. \] [color=green][Only zeroes in [0, 1) should be counted.][/color]

2009 All-Russian Olympiad, 7
Tags: geometry , parallelogram , circumcircle , trigonometry , function
Let be given a parallelogram $ ABCD$ and two points $ A_1$, $ C_1$ on its sides $ AB$, $ BC$, respectively. Lines $ AC_1$ and $ CA_1$ meet at $ P$. Assume that the circumcircles of triangles $ AA_1P$ and $ CC_1P$ intersect at the second point $ Q$ inside triangle $ ACD$. Prove that $ \angle PDA \equal{} \angle QBA$.

2009 India National Olympiad, 5
Tags: inequalities , geometry , trigonometry , circumcircle , geometric transformation , reflection , rectangle
Let $ ABC$ be an acute angled triangle and let $ H$ be its ortho centre. Let $ h_{max}$ denote the largest altitude of the triangle $ ABC$. Prove that: $AH \plus{} BH \plus{} CH\leq2h_{max}$

2011 China Team Selection Test, 1
Tags: geometry , rectangle , analytic geometry , trapezoid , geometric transformation , homothety , trigonometry
In $\triangle ABC$ we have $BC>CA>AB$. The nine point circle is tangent to the incircle, $A$-excircle, $B$-excircle and $C$-excircle at the points $T,T_A,T_B,T_C$ respectively. Prove that the segments $TT_B$ and lines $T_AT_C$ intersect each other.