Found problems: 3349
1949 Kurschak Competition, 1
Prove that $\sin x + \frac12 \sin 2x + \frac13 \sin 3x > 0$ for $0 < x < 180^o$.
1966 IMO, 4
Prove that for every natural number $n$, and for every real number $x \neq \frac{k\pi}{2^t}$ ($t=0,1, \dots, n$; $k$ any integer) \[ \frac{1}{\sin{2x}}+\frac{1}{\sin{4x}}+\dots+\frac{1}{\sin{2^nx}}=\cot{x}-\cot{2^nx} \]
2001 Brazil National Olympiad, 3
$ABC$ is a triangle
$E, F$ are points in $AB$, such that $AE = EF = FB$
$D$ is a point at the line $BC$ such that $ED$ is perpendiculat to $BC$
$AD$ is perpendicular to $CF$.
The angle CFA is the triple of angle BDF. ($3\angle BDF = \angle CFA$)
Determine the ratio $\frac{DB}{DC}$.
%Edited!%
2012 Online Math Open Problems, 16
Let $ABC$ be a triangle with $AB = 4024$, $AC = 4024$, and $BC=2012$. The reflection of line $AC$ over line $AB$ meets the circumcircle of $\triangle{ABC}$ at a point $D\ne A$. Find the length of segment $CD$.
[i]Ray Li.[/i]
1991 AIME Problems, 14
A hexagon is inscribed in a circle. Five of the sides have length 81 and the sixth, denoted by $\overline{AB}$, has length 31. Find the sum of the lengths of the three diagonals that can be drawn from $A$.
2012 Germany Team Selection Test, 2
Let $ABC$ be an acute triangle. Let $\omega$ be a circle whose centre $L$ lies on the side $BC$. Suppose that $\omega$ is tangent to $AB$ at $B'$ and $AC$ at $C'$. Suppose also that the circumcentre $O$ of triangle $ABC$ lies on the shorter arc $B'C'$ of $\omega$. Prove that the circumcircle of $ABC$ and $\omega$ meet at two points.
[i]Proposed by Härmel Nestra, Estonia[/i]
2004 Tournament Of Towns, 2
The incircle of the triangle ABC touches the sides BC, AC, and AB at points A', B', and C', respectively. It is known that AA'=BB'=CC'. Does the triangle ABC have to be equilateral?
(I am very interested in ingenious solution of this problem, because I found an ugly one using Stewart's theorem and tons of algebra during the contest).
2012 AMC 10, 21
Let points $A=(0,0,0)$, $B=(1,0,0)$, $C=(0,2,0)$, and $D=(0,0,3)$. Points $E,F,G$, and $H$ are midpoints of line segments $\overline{BD},\overline{AB},\overline{AC}$, and $\overline{DC}$ respectively. What is the area of $EFGH$?
$ \textbf{(A)}\ \sqrt2
\qquad\textbf{(B)}\ \frac{2\sqrt5}{3}
\qquad\textbf{(C)}\ \frac{3\sqrt5}{4}
\qquad\textbf{(D)}\ \sqrt3
\qquad\textbf{(E)}\ \frac{2\sqrt7}{3}
$
2005 BAMO, 3
Let $ n\ge12$ be an integer, and let $ P_1,P_2,...P_n, Q$ be distinct points in a plane. Prove that for some $ i$, at least $ \frac{n}{6}\minus{}1$ of the distances $ P_1P_i,P_2P_i,...P_{i\minus{}1}P_i,P_{i\plus{}1}P_i,...P_nP_i$ are less than $ P_iQ$.
2004 Romania National Olympiad, 3
Let $ABCD A^{\prime} B^{\prime} C^{\prime} D^{\prime}$ be a truncated regular pyramid in which $BC^{\prime}$ and $DA^{\prime}$ are perpendicular.
(a) Prove that $\measuredangle \left( AB^{\prime},DA^{\prime} \right) = 60^{\circ}$;
(b) If the projection of $B^{\prime}$ on $(ABC)$ is the center of the incircle of $ABC$, then prove that $d \left( CB^{\prime},AD^{\prime} \right) = \frac12 BC^{\prime}$.
[i]Mircea Fianu[/i]
2004 Austrian-Polish Competition, 3
Solve the following system of equations in $\mathbb{R}$ where all square roots are non-negative:
$
\begin{matrix}
a - \sqrt{1-b^2} + \sqrt{1-c^2} = d \\
b - \sqrt{1-c^2} + \sqrt{1-d^2} = a \\
c - \sqrt{1-d^2} + \sqrt{1-a^2} = b \\
d - \sqrt{1-a^2} + \sqrt{1-b^2} = c \\
\end{matrix}
$
1974 IMO Shortlist, 9
Let $x, y, z$ be real numbers each of whose absolute value is different from $\frac{1}{\sqrt 3}$ such that $x + y + z = xyz$. Prove that
\[\frac{3x - x^3}{1-3x^2} + \frac{3y - y^3}{1-3y^2} + \frac{3z -z^3}{1-3z^2} = \frac{3x - x^3}{1-3x^2} \cdot \frac{3y - y^3}{1-3y^2} \cdot \frac{3z - z^3}{1-3z^2}\]
2022 Auckland Mathematical Olympiad, 10
It is known that $\frac{7}{13} + \sin \phi = \cos \phi$ for some real $\phi$. What is sin $2\phi$?
Today's calculation of integrals, 853
Let $0<a<\frac {\pi}2.$ Find $\lim_{a\rightarrow +0} \frac{1}{a^3}\int_0^a \ln\ (1+\tan a\tan x)\ dx.$
2005 Today's Calculation Of Integral, 37
Evaluate
\[\int_{\frac{\pi}{2}}^{\frac{2\pi}{3}} \frac{1}{\sin x \sqrt{1-\cos x}}dx\]
2000 Vietnam National Olympiad, 1
For every integer $ n \ge 3$ and any given angle $ \alpha$ with $ 0 < \alpha < \pi$, let $ P_n(x) \equal{} x^n \sin\alpha \minus{} x \sin n\alpha \plus{} \sin(n \minus{} 1)\alpha$.
(a) Prove that there is a unique polynomial of the form $ f(x) \equal{} x^2 \plus{} ax \plus{} b$ which divides $ P_n(x)$ for every $ n \ge 3$.
(b) Prove that there is no polynomial $ g(x) \equal{} x \plus{} c$ which divides $ P_n(x)$ for every $ n \ge 3$.
2014 Harvard-MIT Mathematics Tournament, 4
In quadrilateral $ABCD$, $\angle DAC = 98^{\circ}$, $\angle DBC = 82^\circ$, $\angle BCD = 70^\circ$, and $BC = AD$. Find $\angle ACD.$
2010 Kazakhstan National Olympiad, 6
Let $ABCD$ be convex quadrilateral, such that exist $M,N$ inside $ABCD$ for which $\angle NAD= \angle MAB; \angle NBC= \angle MBA; \angle MCB=\angle NCD; \angle NDA=\angle MDC$
Prove, that $S_{ABM}+S_{ABN}+S_{CDM}+S_{CDN}=S_{BCM}+S_{BCN}+S_{ADM}+S_{ADN}$, where $S_{XYZ}$-area of triangle $XYZ$
2013 Princeton University Math Competition, 16
Is $\cos 1^\circ$ rational? Prove.
2017 Latvia Baltic Way TST, 2
Find all pairs of real numbers $(x, y)$ that satisfy the equation
$$\frac{(x+y)(2-\sin(x+y))}{4\sin^2(x+y)}=\frac{xy}{x+y}$$
2012 AMC 12/AHSME, 23
Let $S$ be the square one of whose diagonals has endpoints $(0.1,0.7)$ and $(-0.1,-0.7)$. A point $v=(x,y)$ is chosen uniformly at random over all pairs of real numbers $x$ and $y$ such that $0\le x \le 2012$ and $0 \le y \le 2012$. Let $T(v)$ be a translated copy of $S$ centered at $v$. What is the probability that the square region determined by $T(v)$ contains exactly two points with integer coordinates in its interior?
$ \textbf{(A)}\ 0.125\qquad\textbf{(B)}\ 0.14\qquad\textbf{(C)}\ 0.16\qquad\textbf{(D)}\ 0.25\qquad\textbf{(E)}\ 0.32 $
V Soros Olympiad 1998 - 99 (Russia), 10.2
On the coordinate plane, draw all points$ M(x, y)$, the coordinates of which satisfy the inequalities
$$\cos(x + y)^2 \le \cos(x - y)^2, \,\,\, 0 \le x^3, \,\,\, 0 \le y^3.$$
2005 USA Team Selection Test, 2
Let $A_{1}A_{2}A_{3}$ be an acute triangle, and let $O$ and $H$ be its circumcenter and orthocenter, respectively. For $1\leq i \leq 3$, points $P_{i}$ and $Q_{i}$ lie on lines $OA_{i}$ and $A_{i+1}A_{i+2}$ (where $A_{i+3}=A_{i}$), respectively, such that $OP_{i}HQ_{i}$ is a parallelogram. Prove that
\[\frac{OQ_{1}}{OP_{1}}+\frac{OQ_{2}}{OP_{2}}+\frac{OQ_{3}}{OP_{3}}\geq 3.\]
2009 Princeton University Math Competition, 1
Find 100 times the area of a regular dodecagon inscribed in a unit circle. Round your answer to the nearest integer if necessary.
[asy]
defaultpen(linewidth(0.7)); real theta = 17; pen dr = rgb(0.8,0,0), dg = rgb(0,0.6,0), db = rgb(0,0,0.6)+linewidth(1);
draw(unitcircle,dg);
for(int i = 0; i < 12; ++i) {
draw(dir(30*i+theta)--dir(30*(i+1)+theta), db);
dot(dir(30*i+theta),Fill(rgb(0.8,0,0)));
} dot(dir(theta),Fill(dr)); dot((0,0),Fill(dr));
[/asy]
1978 IMO Shortlist, 16
Determine all the triples $(a, b, c)$ of positive real numbers such that the system
\[ax + by -cz = 0,\]\[a \sqrt{1-x^2}+b \sqrt{1-y^2}-c \sqrt{1-z^2}=0,\]
is compatible in the set of real numbers, and then find all its real solutions.