Found problems: 3349
2019 India PRMO, 11
How many distinct triangles $ABC$ are tjere, up to simplilarity, such that the magnitudes of the angles $A, B$ and $C$ in degrees are positive integers and satisfy
$$\cos{A}\cos{B} + \sin{A}\sin{B}\sin{kC} = 1$$
for some positive integer $k$, where $kC$ does not exceet $360^{\circ}$?
2014 ELMO Shortlist, 4
Let $ABCD$ be a quadrilateral inscribed in circle $\omega$. Define $E = AA \cap CD$, $F = AA \cap BC$, $G = BE \cap \omega$, $H = BE \cap AD$, $I = DF \cap \omega$, and $J = DF \cap AB$. Prove that $GI$, $HJ$, and the $B$-symmedian are concurrent.
[i]Proposed by Robin Park[/i]
1988 IMO Shortlist, 18
Consider 2 concentric circle radii $ R$ and $ r$ ($ R > r$) with centre $ O.$ Fix $ P$ on the small circle and consider the variable chord $ PA$ of the small circle. Points $ B$ and $ C$ lie on the large circle; $ B,P,C$ are collinear and $ BC$ is perpendicular to $ AP.$
[b]i.)[/b] For which values of $ \angle OPA$ is the sum $ BC^2 \plus{} CA^2 \plus{} AB^2$ extremal?
[b]ii.)[/b] What are the possible positions of the midpoints $ U$ of $ BA$ and $ V$ of $ AC$ as $ \angle OPA$ varies?
2010 AMC 12/AHSME, 22
Let $ ABCD$ be a cyclic quadrilateral. The side lengths of $ ABCD$ are distinct integers less than $ 15$ such that $ BC\cdot CD\equal{}AB\cdot DA$. What is the largest possible value of $ BD$?
$ \textbf{(A)}\ \sqrt{\frac{325}{2}} \qquad \textbf{(B)}\ \sqrt{185} \qquad \textbf{(C)}\ \sqrt{\frac{389}{2}} \qquad \textbf{(D)}\ \sqrt{\frac{425}{2}} \qquad \textbf{(E)}\ \sqrt{\frac{533}{2}}$
2005 All-Russian Olympiad Regional Round, 11.1
Find all pairs of numbers $x, y \in \left( 0, \frac{\pi}{2}\right)$ , satisfying the equality
$$\sin x + \sin y = \sin (xy)$$
2014 Turkey Team Selection Test, 1
Let $P$ be a point inside the acute triangle $ABC$ with $m(\widehat{PAC})=m(\widehat{PCB})$. $D$ is the midpoint of the segment $PC$. $AP$ and $BC$ intersect at $E$, and $BP$ and $DE$ intersect at $Q$. Prove that $\sin\widehat{BCQ}=\sin\widehat{BAP}$.
2011 Tokio University Entry Examination, 1
On the coordinate plane, let $C$ be a circle centered $P(0,\ 1)$ with radius 1. let $a$ be a real number $a$ satisfying $0<a<1$. Denote by $Q,\ R$ intersection points of the line $y=a(x+1) $ and $C$.
(1) Find the area $S(a)$ of $\triangle{PQR}$.
(2) When $a$ moves in the range of $0<a<1$, find the value of $a$ for which $S(a)$ is maximized.
[i]2011 Tokyo University entrance exam/Science, Problem 1[/i]
2010 AMC 12/AHSME, 25
Two quadrilaterals are considered the same if one can be obtained from the other by a rotation and a translation. How many different convex cyclic quadrilaterals are there with integer sides and perimeter equal to $ 32$?
$ \textbf{(A)}\ 560 \qquad \textbf{(B)}\ 564 \qquad \textbf{(C)}\ 568 \qquad \textbf{(D)}\ 1498 \qquad \textbf{(E)}\ 2255$
1985 IMO Longlists, 7
A convex quadrilateral is inscribed in a circle of radius $1$. Prove that the difference between its perimeter and the sum of the lengths of its diagonals is greater than zero and less than $2.$
2012 China Second Round Olympiad, 9
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 $a$.
2009 Today's Calculation Of Integral, 449
Evaluate $ \sum_{k\equal{}1}^n \int_0^{\pi} (\sin x\minus{}\cos kx)^2dx.$
2001 Croatia National Olympiad, Problem 2
In a triangle $ABC$ with $AC\ne BC$, $M$ is the midpoint of $AB$ and $\angle A=\alpha$, $\angle B=\beta$, $\angle ACM=\varphi$ and $\angle BSM=\Psi$. Prove that
$$\frac{\sin\alpha\sin\beta}{\sin(\alpha-\beta)}=\frac{\sin\varphi\sin\Psi}{\sin(\varphi-\Psi)}.$$
2011 Today's Calculation Of Integral, 673
Let $f(x)=\int_0^ x \frac{1}{1+t^2}dt.$ For $-1\leq x<1$, find $\cos \left\{2f\left(\sqrt{\frac{1+x}{1-x}}\right)\right\}.$
[i]2011 Ritsumeikan University entrance exam/Science and Technology[/i]
2010 AIME Problems, 9
Let $ ABCDEF$ be a regular hexagon. Let $ G$, $ H$, $ I$, $ J$, $ K$, and $ L$ be the midpoints of sides $ AB$, $ BC$, $ CD$, $ DE$, $ EF$, and $ AF$, respectively. The segments $ AH$, $ BI$, $ CJ$, $ DK$, $ EL$, and $ FG$ bound a smaller regular hexagon. Let the ratio of the area of the smaller hexagon to the area of $ ABCDEF$ be expressed as a fraction $ \frac {m}{n}$ where $ m$ and $ n$ are relatively prime positive integers. Find $ m \plus{} n$.
2010 Moldova Team Selection Test, 2
Prove that for any real number $ x$ the following inequality is true:
$ \max\{|\sin x|, |\sin(x\plus{}2010)|\}>\dfrac1{\sqrt{17}}$
2013 ELMO Shortlist, 3
In $\triangle ABC$, a point $D$ lies on line $BC$. The circumcircle of $ABD$ meets $AC$ at $F$ (other than $A$), and the circumcircle of $ADC$ meets $AB$ at $E$ (other than $A$). Prove that as $D$ varies, the circumcircle of $AEF$ always passes through a fixed point other than $A$, and that this point lies on the median from $A$ to $BC$.
[i]Proposed by Allen Liu[/i]
1972 AMC 12/AHSME, 27
If the area of $\triangle ABC$ is $64$ square units and the geometric mean (mean proportional) between sides $AB$ and $AC$ is $12$ inches, then $\sin A$ is equal to
$\textbf{(A) }\dfrac{\sqrt{3}}{2}\qquad\textbf{(B) }\frac{3}{5}\qquad\textbf{(C) }\frac{4}{5}\qquad\textbf{(D) }\frac{8}{9}\qquad \textbf{(E) }\frac{15}{17}$
2001 AIME Problems, 5
An equilateral triangle is inscribed in the ellipse whose equation is $x^2+4y^2=4.$ One vertex of the triangle is $(0,1),$ one altitude is contained in the $y$-axis, and the length of each side is $\sqrt{\frac mn},$ where $m$ and $n$ are relatively prime positive integers. Find $m+n.$
1992 Taiwan National Olympiad, 1
Let $A,B$ be two points on a give circle, and $M$ be the midpoint of one of the arcs $AB$ . Point $C$ is the orthogonal projection of $B$ onto the tangent $l$ to the circle at $A$. The tangent at $M$ to the circle meets $AC,BC$ at $A',B'$ respectively. Prove that if $\hat{BAC}<\frac{\pi}{8}$ then $S_{ABC}<2S_{A'B'C'}$.
2002 China Team Selection Test, 1
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.
2023 All-Russian Olympiad, 1
If $x\in\mathbb{R}$ satisfy $sin$ $x+tan$ $x\in\mathbb{Q}$, $cos$ $x+cot$ $x\in\mathbb{Q}$
Prove that $sin$ $2x$ is a root of an integral coefficient quadratic function
Ukrainian TYM Qualifying - geometry, XII.15
Given a triangular pyramid $SABC$, in which $\angle BSC = \alpha$, $\angle CSA =\beta$, $\angle ASB = \gamma$, and the dihedral angles at the edges $SA$ and $SB$ have the value of $\phi$ and $\delta$, respectively. Prove that $\gamma > \alpha \cdot \cos \delta +\beta \cdot \cos \phi.$$
2003 France Team Selection Test, 3
$M$ is an arbitrary point inside $\triangle ABC$. $AM$ intersects the circumcircle of the triangle again at $A_1$. Find the points $M$ that minimise $\frac{MB\cdot MC}{MA_1}$.
2003 AIME Problems, 15
In $\triangle ABC$, $AB = 360$, $BC = 507$, and $CA = 780$. Let $M$ be the midpoint of $\overline{CA}$, and let $D$ be the point on $\overline{CA}$ such that $\overline{BD}$ bisects angle $ABC$. Let $F$ be the point on $\overline{BC}$ such that $\overline{DF} \perp \overline{BD}$. Suppose that $\overline{DF}$ meets $\overline{BM}$ at $E$. The ratio $DE: EF$ can be written in the form $m/n$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.
2004 National Olympiad First Round, 17
Let $R$ and $T$ be points respectively on sides $[BC]$ and $[CD]$ of a square $ABCD$ with side length $6$ such that $|CR|+|RT|+|TC|=12$. What is $\tan (\widehat{RAT})$
$
\textbf{(A)}\ 2\sqrt 3
\qquad\textbf{(B)}\ \sqrt 3
\qquad\textbf{(C)}\ \dfrac 13
\qquad\textbf{(D)}\ \dfrac 12
\qquad\textbf{(E)}\ 1
$