Found problems: 3349
2010 Princeton University Math Competition, 7
The expression $\sin2^\circ\sin4^\circ\sin6^\circ\cdots\sin90^\circ$ is equal to $p\sqrt{5}/2^{50}$, where $p$ is an integer. Find $p$.
2008 China Team Selection Test, 3
Find all positive integers $ n$ having the following properties:in two-dimensional Cartesian coordinates, there exists a convex $ n$ lattice polygon whose lengths of all sides are odd numbers, and unequal to each other. (where lattice polygon is defined as polygon whose coordinates of all vertices are integers in Cartesian coordinates.)
2019 LIMIT Category A, Problem 11
$\angle A,\angle B,\angle C$ are angles of a triangle such that $\sin^2A+\sin^2B=\sin^2C$, then $\angle C$ in degrees is equal to
$\textbf{(A)}~30$
$\textbf{(B)}~90$
$\textbf{(C)}~45$
$\textbf{(D)}~\text{none of the above}$
2001 Saint Petersburg Mathematical Olympiad, 11.1
Do there exist distinct numbers $x,y,z$ from $[0,\dfrac{\pi}{2}]$, such that six number $\sin x$, $\sin y$,$\sin z$, $\cos x$, $\cos y$, $\cos z$ could be partitioned into 3 pairs with equal sums?
[I]Proposed by A. Golovanov[/i]
1966 German National Olympiad, 3
Consider all segments dividing the area of a triangle $ABC$ in two equal parts. Find the length of the shortest segment among them, if the side lengths $a,$ $b,$ $c$ of triangle $ABC$ are given. How many of these shortest segments exist ?
1991 AIME Problems, 15
For positive integer $n$, define $S_n$ to be the minimum value of the sum \[ \sum_{k=1}^n \sqrt{(2k-1)^2+a_k^2}, \] where $a_1,a_2,\ldots,a_n$ are positive real numbers whose sum is 17. There is a unique positive integer $n$ for which $S_n$ is also an integer. Find this $n$.
2011 NIMO Problems, 14
In circle $\theta_1$ with radius $1$, circles $\phi_1, \phi_2, \dots, \phi_8$, with equal radii, are drawn such that for $1 \le i \le 8$, $\phi_i$ is tangent to $\omega_1$, $\phi_{i-1}$, and $\phi_{i+1}$, where $\phi_0 = \phi_8$ and $\phi_1 = \phi_9$. There exists a circle $\omega_2$ such that $\omega_1 \neq \omega_2$ and $\omega_2$ is tangent to $\phi_i$ for $1 \le i \le 8$. The radius of $\omega_2$ can be expressed in the form $a - b\sqrt{c} -d\sqrt{e - \sqrt{f}} + g \sqrt{h - j \sqrt{k}}$ such that $a, b, \dots, k$ are positive integers and the numbers $e, f, k, \gcd(h, j)$ are squarefree. What is $a+b+c+d+e+f+g+h+j+k$.
[i]Proposed by Eugene Chen
[/i]
2005 AMC 10, 25
In $ ABC$ we have $ AB \equal{} 25$, $ BC \equal{} 39$, and $ AC \equal{} 42$. Points $ D$ and $ E$ are on $ AB$ and $ AC$ respectively, with $ AD \equal{} 19$ and $ AE \equal{} 14$. What is the ratio of the area of triangle $ ADE$ to the area of quadrilateral $ BCED$?
$ \textbf{(A)}\ \frac{266}{1521}\qquad
\textbf{(B)}\ \frac{19}{75}\qquad
\textbf{(C)}\ \frac{1}{3}\qquad
\textbf{(D)}\ \frac{19}{56}\qquad
\textbf{(E)}\ 1$
Indonesia MO Shortlist - geometry, g7
In triangle $ABC$, find the smallest possible value of $$|(\cot A + \cot B)(\cot B +\cot C)(\cot C + \cot A)|$$
2005 MOP Homework, 7
Let $ABCD$ be a cyclic quadrilateral who interior angle at $B$ is $60$ degrees. Show that if $BC=CD$, then $CD+DA=AB$. Does the converse hold?
2008 Harvard-MIT Mathematics Tournament, 8
Compute $ \arctan\left(\tan65^\circ \minus{} 2\tan40^\circ\right)$. (Express your answer in degrees.)
Today's calculation of integrals, 863
For $0<t\leq 1$, let $F(t)=\frac{1}{t}\int_0^{\frac{\pi}{2}t} |\cos 2x|\ dx.$
(1) Find $\lim_{t\rightarrow 0} F(t).$
(2) Find the range of $t$ such that $F(t)\geq 1.$
2014 Uzbekistan National Olympiad, 4
A circle passes through the points $A,C$ of triangle $ABC$ intersects with the sides $AB,BC$ at points $D,E$ respectively. Let $ \frac{BD}{CE}=\frac{3}{2}$, $BE=4$, $AD=5$ and $AC=2\sqrt{7} $.
Find the angle $ \angle BDC$.
2009 Today's Calculation Of Integral, 429
Find the length of the curve expressed by the polar equation: $ r\equal{}1\plus{}\cos \theta \ (0\leq \theta \leq \pi)$.
1987 IMO Longlists, 16
Let $ABC$ be a triangle. For every point $M$ belonging to segment $BC$ we denote by $B'$ and $C'$ the orthogonal projections of $M$ on the straight lines $AC$ and $BC$. Find points $M$ for which the length of segment $B'C'$ is a minimum.
2010 Today's Calculation Of Integral, 664
For a positive integer $n$, let $I_n=\int_{-\pi}^{\pi} \left(\frac{\pi}{2}-|x|\right)\cos nx\ dx$.
Find $I_1+I_2+I_3+I_4$.
[i]1992 University of Fukui entrance exam/Medicine[/i]
1982 All Soviet Union Mathematical Olympiad, 335
Three numbers $a,b,c$ belong to $[0,\pi /2]$ interval with $$\cos a = a, \sin(\cos b) = b, \cos(\sin c ) = c$$ Sort those numbers in increasing order.
2005 Thailand Mathematical Olympiad, 18
Compute the sum $$\sum_{k=0}^{1273}\frac{1}{1 + tan^{2548}\left(\frac{k\pi}{2548}\right)}$$
2009 India Regional Mathematical Olympiad, 1
Let $ ABC$ be a triangle in which $ AB \equal{} AC$ and let $ I$ be its in-centre. Suppose $ BC \equal{} AB \plus{} AI$. Find $ \angle{BAC}$
2010 Today's Calculation Of Integral, 575
For a function $ f(x)\equal{}\int_x^{\frac{\pi}{4}\minus{}x} \log_4 (1\plus{}\tan t)dt\ \left(0\leq x\leq \frac{\pi}{8}\right)$, answer the following questions.
(1) Find $ f'(x)$.
(2) Find the $ n$ th term of the sequence $ a_n$ such that $ a_1\equal{}f(0),\ a_{n\plus{}1}\equal{}f(a_n)\ (n\equal{}1,\ 2,\ 3,\ \cdots)$.
2013 NIMO Problems, 6
Let $ABC$ be a triangle with $AB = 42$, $AC = 39$, $BC = 45$. Let $E$, $F$ be on the sides $\overline{AC}$ and $\overline{AB}$ such that $AF = 21, AE = 13$. Let $\overline{CF}$ and $\overline{BE}$ intersect at $P$, and let ray $AP$ meet $\overline{BC}$ at $D$. Let $O$ denote the circumcenter of $\triangle DEF$, and $R$ its circumradius. Compute $CO^2-R^2$.
[i]Proposed by Yang Liu[/i]
1962 IMO, 4
Solve the equation $\cos^2{x}+\cos^2{2x}+\cos^2{3x}=1$
2011 ELMO Shortlist, 7
Determine whether there exist two reals $x,y$ and a sequence $\{a_n\}_{n=0}^{\infty}$ of nonzero reals such that $a_{n+2}=xa_{n+1}+ya_n$ for all $n\ge0$ and for every positive real number $r$, there exist positive integers $i,j$ such that $|a_i|<r<|a_j|$.
[i]Alex Zhu.[/i]
1953 Putnam, A7
Assuming that the roots of $x^3 +px^2 +qx +r=0$ are all real and positive, find the relation between $p,q,r$ which is a necessary and sufficient condition that the roots are the cosines of the angles of a triangle.
1994 AMC 12/AHSME, 29
Points $A, B$ and $C$ on a circle of radius $r$ are situated so that $AB=AC, AB>r$, and the length of minor arc $BC$ is $r$. If angles are measured in radians, then $AB/BC=$
[asy]
draw(Circle((0,0), 13));
draw((-13,0)--(12,5)--(12,-5)--cycle);
dot((-13,0));
dot((12,5));
dot((12,-5));
label("A", (-13,0), W);
label("B", (12,5), NE);
label("C", (12,-5), SE);
[/asy]
$ \textbf{(A)}\ \frac{1}{2}\csc{\frac{1}{4}} \qquad\textbf{(B)}\ 2\cos{\frac{1}{2}} \qquad\textbf{(C)}\ 4\sin{\frac{1}{2}} \qquad\textbf{(D)}\ \csc{\frac{1}{2}} \qquad\textbf{(E)}\ 2\sec{\frac{1}{2}} $