Found problems: 3349
1996 USAMO, 1
Prove that the average of the numbers $n \sin n^{\circ} \; (n = 2,4,6,\ldots,180)$ is $\cot 1^{\circ}$.
1996 APMO, 1
Let $ABCD$ be a quadrilateral $AB = BC = CD = DA$. Let $MN$ and $PQ$ be two segments perpendicular to the diagonal $BD$ and such that the distance between them is $d > \frac{BD}{2}$, with $M \in AD$, $N \in DC$, $P \in AB$, and $Q \in BC$. Show that the perimeter of hexagon $AMNCQP$ does not depend on the position of $MN$ and $PQ$ so long as the distance between them remains constant.
2007 AMC 12/AHSME, 19
Triangles $ ABC$ and $ ADE$ have areas $ 2007$ and $ 7002,$ respectively, with $ B \equal{} (0,0),$ $ C \equal{} (223,0),$ $ D \equal{} (680,380),$ and $ E \equal{} (689,389).$ What is the sum of all possible x-coordinates of $ A?$
$ \textbf{(A)}\ 282 \qquad \textbf{(B)}\ 300 \qquad \textbf{(C)}\ 600 \qquad \textbf{(D)}\ 900 \qquad \textbf{(E)}\ 1200$
1997 China Team Selection Test, 1
Given a real number $\lambda > 1$, let $P$ be a point on the arc $BAC$ of the circumcircle of $\bigtriangleup ABC$. Extend $BP$ and $CP$ to $U$ and $V$ respectively such that $BU = \lambda BA$, $CV = \lambda CA$. Then extend $UV$ to $Q$ such that $UQ = \lambda UV$. Find the locus of point $Q$.
2006 IberoAmerican Olympiad For University Students, 3
Let $p_1(x)=p(x)=4x^3-3x$ and $p_{n+1}(x)=p(p_n(x))$ for each positive integer $n$. Also, let $A(n)$ be the set of all the real roots of the equation $p_n(x)=x$.
Prove that $A(n)\subseteq A(2n)$ and that the product of the elements of $A(n)$ is the average of the elements of $A(2n)$.
1961 IMO, 3
Solve the equation $\cos^n{x}-\sin^n{x}=1$ where $n$ is a natural number.
2006 IberoAmerican Olympiad For University Students, 7
Consider the multiplicative group $A=\{z\in\mathbb{C}|z^{2006^k}=1, 0<k\in\mathbb{Z}\}$ of all the roots of unity of degree $2006^k$ for all positive integers $k$.
Find the number of homomorphisms $f:A\to A$ that satisfy $f(f(x))=f(x)$ for all elements $x\in A$.
2003 Tuymaada Olympiad, 1
Prove that for every $\alpha_{1}, \alpha_{2}, \ldots, \alpha_{n}$ in the interval $(0,\pi/2)$
\[\left({1\over \sin \alpha_{1}}+{1\over \sin \alpha_{2}}+\ldots+{1\over \sin \alpha_{n}}\right) \left({1\over \cos \alpha_{1}}+{1\over \cos \alpha_{2}}+\ldots+{1\over \cos \alpha_{n}}\right) \leq\]
\[\leq 2 \left({1\over \sin 2\alpha_{1}}+{1\over \sin 2\alpha_{2}}+\ldots+{1\over \sin 2\alpha_{n}}\right)^{2}.\]
[i]Proposed by A. Khrabrov[/i]
1997 Slovenia Team Selection Test, 4
Let $ABC$ be an equilateral triangle and let $P$ be a point in its interior. Let the lines $AP$, $BP$, $CP$ meet the sides $BC$, $CA$, $AB$ at the points $A_1$, $B_1$, $C_1$, respectively. Prove that
$A_1B_1 \cdot B_1C_1 \cdot C_1A_1 \ge A_1B \cdot B_1C \cdot C_1A$.
1971 Canada National Olympiad, 9
Two flag poles of height $h$ and $k$ are situated $2a$ units apart on a level surface. Find the set of all points on the surface which are so situated that the angles of elevation of the tops of the poles are equal.
2002 India IMO Training Camp, 15
Let $x_1,x_2,\ldots,x_n$ be arbitrary real numbers. Prove the inequality
\[
\frac{x_1}{1+x_1^2} + \frac{x_2}{1+x_1^2 + x_2^2} + \cdots +
\frac{x_n}{1 + x_1^2 + \cdots + x_n^2} < \sqrt{n}.
\]
1999 China Second Round Olympiad, 2
Let $a$,$b$,$c$ be real numbers.
Let $z_{1}$,$z_{2}$,$z_{3}$ be complex numbers such that $|z_{k}|=1$ $(k=1,2,3)$ $~$ and $~$ $\frac{z_{1}}{z_{2}}+\frac{z_{2}}{z_{3}}+\frac{z_{3}}{z_{1}}=1$
Find $|az_{1}+bz_{2}+cz_{3}|$.
1984 IMO Longlists, 3
The opposite sides of the reentrant hexagon $AFBDCE$ intersect at the points $K,L,M$ (as shown in the figure). It is given that $AL = AM = a, BM = BK = b$, $CK = CL = c, LD = DM = d, ME = EK = e, FK = FL = f$.
[img]http://imgur.com/LUFUh.png[/img]
$(a)$ Given length $a$ and the three angles $\alpha, \beta$ and $\gamma$ at the vertices $A, B,$ and $C,$ respectively, satisfying the condition $\alpha+\beta+\gamma<180^{\circ}$, show that all the angles and sides of the hexagon are thereby uniquely determined.
$(b)$ Prove that
\[\frac{1}{a}+\frac{1}{c}=\frac{1}{b}+\frac{1}{d}\]
Easier version of $(b)$. Prove that
\[(a + f)(b + d)(c + e)= (a + e)(b + f)(c + d)\]
2011 ELMO Shortlist, 2
Let $\omega,\omega_1,\omega_2$ be three mutually tangent circles such that $\omega_1,\omega_2$ are externally tangent at $P$, $\omega_1,\omega$ are internally tangent at $A$, and $\omega,\omega_2$ are internally tangent at $B$. Let $O,O_1,O_2$ be the centers of $\omega,\omega_1,\omega_2$, respectively. Given that $X$ is the foot of the perpendicular from $P$ to $AB$, prove that $\angle{O_1XP}=\angle{O_2XP}$.
[i]David Yang.[/i]
2009 Today's Calculation Of Integral, 410
Evaluate $ \int_0^{\frac{\pi}{4}} \frac{1}{\cos \theta}\sqrt{\frac{1\plus{}\sin \theta}{\cos \theta}}\ d\theta$.
1996 AIME Problems, 15
In parallelogram $ABCD,$ let $O$ be the intersection of diagonals $\overline{AC}$ and $\overline{BD}.$ Angles $CAB$ and $DBC$ are each twice as large as angle $DBA,$ and angle $ACB$ is $r$ times as large as angle $AOB.$ Find the greatest integer that does not exceed $1000r.$
2007 Germany Team Selection Test, 3
Let $ ABC$ be a triangle and $ P$ an arbitrary point in the plane. Let $ \alpha, \beta, \gamma$ be interior angles of the triangle and its area is denoted by $ F.$ Prove:
\[ \ov{AP}^2 \cdot \sin 2\alpha + \ov{BP}^2 \cdot \sin 2\beta + \ov{CP}^2 \cdot \sin 2\gamma \geq 2F
\]
When does equality occur?
2014 PUMaC Geometry A, 3
Let $O$ be the circumcenter of triangle $ABC$ with circumradius $15$. Let $G$ be the centroid of $ABC$ and let $M$ be the midpoint of $BC$. If $BC=18$ and $\angle MOA=150^\circ$, find the area of $OMG$.
2012 Math Prize For Girls Problems, 20
There are 6 distinct values of $x$ strictly between $0$ and $\frac{\pi}{2}$ that satisfy the equation
\[
\tan(15 x) = 15 \tan(x) .
\]
Call these 6 values $r_1$, $r_2$, $r_3$, $r_4$, $r_5$, and $r_6$. What is the value of the sum
\[
\frac{1}{\tan^2 r_1} +
\frac{1}{\tan^2 r_2} +
\frac{1}{\tan^2 r_3} +
\frac{1}{\tan^2 r_4} +
\frac{1}{\tan^2 r_5} +
\frac{1}{\tan^2 r_6} \, ?
\]
2005 Taiwan TST Round 1, 1
Consider a circle $O_1$ with radius $R$ and a point $A$ outside the circle. It is known that $\angle BAC=60^\circ$, where $AB$ and $AC$ are tangent to $O_1$. We construct infinitely many circles $O_i$ $(i=1,2,\dots\>)$ such that for $i>1$, $O_i$ is tangent to $O_{i-1}$ and $O_{i+1}$, that they share the same tangent lines $AB$ and $AC$ with respect to $A$, and that none of the $O_i$ are larger than $O_1$. Find the total area of these circles.
I know this problem was easy, but it still appeared in the TST, and so I posted it. It was kind of a disappointment for me.
2013 Math Prize For Girls Problems, 17
Let $f$ be the function defined by $f(x) = -2 \sin(\pi x)$. How many values of $x$ such that $-2 \le x \le 2$ satisfy the equation $f(f(f(x))) = f(x)$?
2006 Turkey MO (2nd round), 2
$ABC$ be a triangle. Its incircle touches the sides $CB, AC, AB$ respectively at $N_{A},N_{B},N_{C}$. The orthic triangle of $ABC$ is $H_{A}H_{B}H_{C}$ with $H_{A}, H_{B}, H_{C}$ are respectively on $BC, AC, AB$. The incenter of $AH_{C}H_{B}$ is $I_{A}$; $I_{B}$ and $I_{C}$ were defined similarly.
Prove that the hexagon $I_{A}N_{B}I_{C}N_{A}I_{B}N_{C}$ has all sides equal.
1998 Harvard-MIT Mathematics Tournament, 1
Evaluate $\sin(1998^\circ+237^\circ)\sin(1998^\circ-1653^\circ)$.
2019 Purple Comet Problems, 15
Suppose $a$ is a real number such that $\sin(\pi \cdot \cos a) = \cos(\pi \cdot \sin a)$. Evaluate $35 \sin^2(2a) + 84 \cos^2(4a)$.
2009 Jozsef Wildt International Math Competition, W. 20
If $x \in \mathbb{R}\backslash \left \{\frac{k\pi}{2}\ |\ k\in \mathbb{Z} \right \}$, then $$\left (\sum \limits_{0\leq j<k\leq n} \sin (2(j+k)x)\right )^2 + \left (\sum \limits_{0\leq j<k\leq n} \cos (2(j+k)x)\right )^2 = \frac{\sin ^2 nx \sin ^2 (n+1)x}{\sin ^2x \sin^22x}$$