Found problems: 3349
1995 Baltic Way, 7
Prove that $\sin^318^{\circ}+\sin^218^{\circ}=\frac18$.
2011 AIME Problems, 10
A circle with center $O$ has radius 25. Chord $\overline{AB}$ of length 30 and chord $\overline{CD}$ of length 14 intersect at point $P$. The distance between the midpoints of the two chords is 12. The quantity $OP^2$ can be represented as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find the remainder where $m+n$ is divided by 1000.
2003 Bundeswettbewerb Mathematik, 3
Let $ABCD$ be a parallelogram. Let $M$ be a point on the side $AB$ and $N$ be a point on the side $BC$ such that the segments $AM$ and $CN$ have equal lengths and are non-zero. The lines $AN$ and $CM$ meet at $Q$.
Prove that the line $DQ$ is the bisector of the angle $\measuredangle ADC$.
[i]Alternative formulation.[/i] Let $ABCD$ be a parallelogram. Let $M$ and $N$ be points on the sides $AB$ and $BC$, respectively, such that $AM=CN\neq 0$. The lines $AN$ and $CM$ intersect at a point $Q$.
Prove that the point $Q$ lies on the bisector of the angle $\measuredangle ADC$.
1985 IMO Longlists, 39
Given a triangle $ABC$ and external points $X, Y$ , and $Z$ such that $\angle BAZ = \angle CAY , \angle CBX = \angle ABZ$, and $\angle ACY = \angle BCX$, prove that $AX,BY$ , and $CZ$ are concurrent.
1992 USAMO, 2
Prove
\[ \frac{1}{\cos 0^\circ \cos 1^\circ} + \frac{1}{\cos 1^\circ \cos 2^\circ} + \cdots + \frac{1}{\cos 88^\circ \cos 89^\circ} = \frac{\cos 1^\circ}{\sin^2 1^\circ}. \]
2003 Turkey Junior National Olympiad, 1
Let $ABCD$ be a cyclic quadrilateral, and $E$ be the intersection of its diagonals. If $m(\widehat{ADB}) = 22.5^\circ$, $|BD|=6$, and $|AD|\cdot|CE|=|DC|\cdot|AE|$, find the area of the quadrilateral $ABCD$.
2015 Postal Coaching, Problem 4
Let $ABC$ be at triangle with incircle $\Gamma$. Let $\Gamma_1$, $\Gamma_2$, $\Gamma_3$ be three circles inside $\triangle ABC$ each of which is tangent to $\Gamma$ and two sides of the triangle and their radii are $1,4,9$. Find the radius of $\Gamma$.
2013 National Olympiad First Round, 9
Let $ABC$ be a triangle with $|AB|=18$, $|AC|=24$, and $m(\widehat{BAC}) = 150^\circ$. Let $D$, $E$, $F$ be points on sides $[AB]$, $[AC]$, $[BC]$, respectively, such that $|BD|=6$, $|CE|=8$, and $|CF|=2|BF|$. Let $H_1$, $H_2$, $H_3$ be the reflections of the orthocenter of triangle $ABC$ over the points $D$, $E$, $F$, respectively. What is the area of triangle $H_1H_2H_3$?
$
\textbf{(A)}\ 70
\qquad\textbf{(B)}\ 72
\qquad\textbf{(C)}\ 84
\qquad\textbf{(D)}\ 96
\qquad\textbf{(E)}\ 108
$
2019 Jozsef Wildt International Math Competition, W. 46
Let $x$, $y$, $z > 0$ such that $x^2 + y^2 + z^2 = 3$. Then $$x^3\tan^{-1}\frac{1}{x}+y^3\tan^{-1}\frac{1}{y}+z^3\tan^{-1}\frac{1}{z}<\frac{\pi \sqrt{3}}{2}$$
2003 AIME Problems, 12
In convex quadrilateral $ABCD$, $\angle A \cong \angle C$, $AB = CD = 180$, and $AD \neq BC$. The perimeter of $ABCD$ is 640. Find $\lfloor 1000 \cos A \rfloor$. (The notation $\lfloor x \rfloor$ means the greatest integer that is less than or equal to $x$.)
2007 Romania National Olympiad, 3
Consider the triangle $ ABC$ with $ m(\angle BAC) \equal{} 90^\circ$ and $ AB < AC$.Let a point $ D$ on the side $ AC$ such that: $ m(\angle ACB) \equal{} m(\angle DBA)$.Let $ E$ be a point on the side $ BC$ such that $ DE\perp BC$.It is known that $ BD \plus{} DE \equal{} AC$. Find the measures of the angles in the triangle $ ABC$.
2011 Korea - Final Round, 2
$ABC$ is a triangle such that $AC<AB<BC$ and $D$ is a point on side $AB$ satisfying $AC=AD$.
The circumcircle of $ABC$ meets with the bisector of angle $A$ again at $E$ and meets with $CD$ again at $F$.
$K$ is an intersection point of $BC$ and $DE$. Prove that $CK=AC$ is a necessary and sufficient condition for $DK \cdot EF = AC \cdot DF$.
2001 JBMO ShortLists, 11
Consider a triangle $ABC$ with $AB=AC$, and $D$ the foot of the altitude from the vertex $A$. The point $E$ lies on the side $AB$ such that $\angle ACE= \angle ECB=18^{\circ}$.
If $AD=3$, find the length of the segment $CE$.
2008 Sharygin Geometry Olympiad, 9
(A.Zaslavsky, 9--10) The reflections of diagonal $ BD$ of a quadrilateral $ ABCD$ in the bisectors of angles $ B$ and $ D$ pass
through the midpoint of diagonal $ AC$. Prove that the reflections of diagonal $ AC$ in the bisectors of angles $ A$ and $ C$ pass
through the midpoint of diagonal $ BD$ (There was an error in published condition of this problem).
2019 EGMO, 3
Let $ABC$ be a triangle such that $\angle CAB > \angle ABC$, and let $I$ be its incentre. Let $D$ be the point on segment $BC$ such that $\angle CAD = \angle ABC$. Let $\omega$ be the circle tangent to $AC$ at $A$ and passing through $I$. Let $X$ be the second point of intersection of $\omega$ and the circumcircle of $ABC$. Prove that the angle bisectors of $\angle DAB$ and $\angle CXB$ intersect at a point on line $BC$.
2010 Canadian Mathematical Olympiad Qualification Repechage, 3
Prove that there is no real number $x$ satisfying both equations \begin{align*}2^x+1=2\sin x \\ 2^x-1=2\cos x.\end{align*}
2011 Today's Calculation Of Integral, 699
Find the volume of the part bounded by $z=x+y,\ z=x^2+y^2$ in the $xyz$ space.
1997 Junior Balkan MO, 4
Determine the triangle with sides $a,b,c$ and circumradius $R$ for which $R(b+c) = a\sqrt{bc}$.
[i]Romania[/i]
2014 Online Math Open Problems, 21
Consider a sequence $x_1,x_2,\cdots x_{12}$ of real numbers such that $x_1=1$ and for $n=1,2,\dots,10$ let \[ x_{n+2}=\frac{(x_{n+1}+1)(x_{n+1}-1)}{x_n}. \] Suppose $x_n>0$ for $n=1,2,\dots,11$ and $x_{12}=0$. Then the value of $x_2$ can be written as $\frac{\sqrt{a}+\sqrt{b}}{c}$ for positive integers $a,b,c$ with $a>b$ and no square dividing $a$ or $b$. Find $100a+10b+c$.
[i]Proposed by Michael Kural[/i]
1995 VJIMC, Problem 4
Let $\{x_n\}_{n=1}^\infty$ be a sequence such that $x_1=25$, $x_n=\operatorname{arctan}(x_{n-1})$. Prove that this sequence has a limit and find it.
2010 Today's Calculation Of Integral, 573
Find the area of the figure bounded by three curves
$ C_1: y\equal{}\sin x\ \left(0\leq x<\frac {\pi}{2}\right)$
$ C_2: y\equal{}\cos x\ \left(0\leq x<\frac {\pi}{2}\right)$
$ C_3: y\equal{}\tan x\ \left(0\leq x<\frac {\pi}{2}\right)$.
2013 Iran MO (3rd Round), 4
In a triangle $ABC$ with circumcircle $(O)$ suppose that $A$-altitude cut $(O)$ at $D$. Let altitude of $B,C$ cut $AC,AB$ at $E,F$. $H$ is orthocenter and $T$ is midpoint of $AH$. Parallel line with $EF$ passes through $T$ cut $AB,AC$ at $X,Y$. Prove that $\angle XDF = \angle YDE$.
2008 All-Russian Olympiad, 6
The incircle of a triangle $ABC$ touches the side $AB$ and $AC$ at respectively at $X$ and $Y$. Let $K$ be the midpoint of the arc $\widehat{AB}$ on the circumcircle of $ABC$. Assume that $XY$ bisects the segment $AK$. What are the possible measures of angle $BAC$?
2012 China Second Round Olympiad, 2
In $\triangle ABC$, the corresponding sides of angle $A,B,C$ are $a,b,c$ respectively. If $a\cos B-b\cos A=\frac{3}{5}c$, find the value of $\frac{\tan A}{\tan B}$.
2005 National High School Mathematics League, 3
$\triangle ABC$ is inscribed to unit circle. Bisector of $\angle A,\angle B,\angle C$ intersect the circle at $A_1,B_1,C_1$ respectively. The value of $\frac{\displaystyle AA_1\cdot\cos\frac{A}{2}+BB_1\cdot\cos\frac{B}{2}+CC_1\cdot\cos\frac{C}{2}}{\sin A+\sin B+\sin C}$ is
$\text{(A)}2\qquad\text{(B)}4\qquad\text{(C)}6\qquad\text{(D)}8$