Found problems: 3349
1998 USAMTS Problems, 4
Prove that if $0<x<\pi/2$, then $\sec^6 x+\csc^6 x+(\sec^6 x)(\csc^6 x)\geq 80$.
2003 Alexandru Myller, 2
Calculate $ \int_0^{2\pi }\prod_{i=1}^{2002} cos^i (it) dt. $
[i]Dorin Andrica[/i]
2008 AIME Problems, 14
Let $ a$ and $ b$ be positive real numbers with $ a\ge b$. Let $ \rho$ be the maximum possible value of $ \frac{a}{b}$ for which the system of equations
\[ a^2\plus{}y^2\equal{}b^2\plus{}x^2\equal{}(a\minus{}x)^2\plus{}(b\minus{}y)^2\]has a solution in $ (x,y)$ satisfying $ 0\le x<a$ and $ 0\le y<b$. Then $ \rho^2$ can be expressed as a fraction $ \frac{m}{n}$, where $ m$ and $ n$ are relatively prime positive integers. Find $ m\plus{}n$.
1988 IMO Shortlist, 30
A point $ M$ is chosen on the side $ AC$ of the triangle $ ABC$ in such a way that the radii of the circles inscribed in the triangles $ ABM$ and $ BMC$ are equal. Prove that
\[ BM^{2} \equal{} X \cot \left( \frac {B}{2}\right)
\]
where X is the area of triangle $ ABC.$
2024 JHMT HS, 15
Let $N_{14}$ be the answer to problem 14.
Rectangle $ABCD$ has area $\sqrt{2N_{14}}$. Points $E$, $F$, $G$, and $H$ lie on the rays $\overrightarrow{AB}$, $\overrightarrow{BC}$, $\overrightarrow{CD}$, and $\overrightarrow{DA}$, respectively, such that $EFGH$ is a rectangle with area $2\sqrt{2N_{14}}$ that contains all of $ABCD$ in its interior. If
\[ \tan\angle AEH = \tan\angle BFE = \tan\angle CGF = \tan\angle DHG = \sqrt{\frac{1}{48}}, \]
then $EG=\tfrac{m\sqrt{n}}{p}$, where $m$, $n$, and $p$ are positive integers, $m$ and $p$ are relatively prime, and $n$ is not divisible by the square of any prime. Compute $m + n + p$.
2008 IMC, 2
Two different ellipses are given. One focus of the first ellipse coincides with one focus of the second ellipse. Prove that the ellipses have at most two points in common.
1999 IMO Shortlist, 4
For a triangle $T = ABC$ we take the point $X$ on the side $(AB)$ such that $AX/AB=4/5$, the point $Y$ on the segment $(CX)$ such that $CY = 2YX$ and, if possible, the point $Z$ on the ray ($CA$ such that $\widehat{CXZ} = 180 - \widehat{ABC}$. We denote by $\Sigma$ the set of all triangles $T$ for which
$\widehat{XYZ} = 45$. Prove that all triangles from $\Sigma$ are similar and find the measure of their smallest angle.
2023 UMD Math Competition Part II, 4
Assume every side length of a triangle $ABC$ is more than $2$ and two of its angles are given by $\angle ABC = 57^\circ$ and $ACB = 63^\circ$. Point $P$ is chosen on side $BC$ with $BP:PC = 2:1$. Points $M,N$ are chosen on sides $AB$ and $AC$, respectively so that $BM = 2$ and $CN = 1$. Let $Q$ be the point on segment $MN$ for which $MQ:QN = 2:1$. Find the value of $PQ$. Your answer must be in simplest form.
2020 CHMMC Winter (2020-21), 4
Consider the minimum positive real number $\lambda$ such that for any two squares $A,B$ satisfying $\text{Area}(A) + \text{Area}(B)=1$, there always exists some rectangle $C$ of area $\lambda$, such that $A,B$ can be put inside $C$ and satisfy the following two constraints:
1. $A,B$ are non-overlapping;
2. the sides of $A$ and $B$ are parallel to some side of $C$.
$\lambda$ can be written as $\frac{\sqrt{m}+n}{p}$ for positive integers $m$, $n$, and $p$ where $n$ and $p$ are relatively prime. Find $m+n+p$.
2006 AIME Problems, 6
Square $ABCD$ has sides of length 1. Points $E$ and $F$ are on $\overline{BC}$ and $\overline{CD}$, respectively, so that $\triangle AEF$ is equilateral. A square with vertex $B$ has sides that are parallel to those of $ABCD$ and a vertex on $\overline{AE}$. The length of a side of this smaller square is $\displaystyle \frac{a-\sqrt{b}}{c}$, where $a$, $b$, and $c$ are positive integers and $b$ is not divisible by the square of any prime. Find $a+b+c$.
IV Soros Olympiad 1997 - 98 (Russia), 11.10
Let $a_n = \frac{\pi}{2n}$, where $n$ is a natural number. Prove that for any $k = 1$,$2$,$...$, $n$ holds the equality $$\frac{\sin ka_n}{1-\cos a_n}+\frac{\sin 5ka_n}{1-\cos 5a_n}+\frac{\sin 9ka_n}{1-\cos 9a_n}+...+\frac{\sin (4n-3)a_n}{1-\cos (4n-3)a_n}=kn$$
2006 Hong Kong TST., 3
In triangle ABC, the altitude, angle bisector and median from C divide the angle C into four equal angles. Find angle B.
2012 Flanders Math Olympiad, 3
(a) Show that for any angle $\theta$ and for any natural number $m$:
$$| \sin m\theta| \le m| \sin \theta|$$
(b) Show that for all angles $\theta_1$ and $\theta_2$ and for all even natural numbers $m$:
$$| \sin m \theta_2 - \sin m \theta_1| \le m| \sin (\theta_2 - \theta_1)|$$
(c) Show that for every odd natural number $m$ there are two angles, resp. $\theta_1$ and $\theta_2$, exist for which the inequality in (b) is not valid.
2005 Today's Calculation Of Integral, 3
Calculate the following indefinite integrals.
[1] $\int \sin x\sin 2x dx$
[2] $\int \frac{e^{2x}}{e^x-1}dx$
[3] $\int \frac{\tan ^2 x}{\cos ^2 x}dx$
[4] $\int \frac{e^x+e^{-x}}{e^x-e^{-x}}dx$
[5] $\int \frac{e^x}{e^x+1}dx$
1999 AMC 12/AHSME, 27
In triangle $ ABC$, $ 3\sin A \plus{} 4\cos B \equal{} 6$ and $ 4\sin B \plus{} 3\cos A \equal{} 1$. Then $ \angle C$ in degrees is
$ \textbf{(A)}\ 30\qquad
\textbf{(B)}\ 60\qquad
\textbf{(C)}\ 90\qquad
\textbf{(D)}\ 120\qquad
\textbf{(E)}\ 150$
2010 Today's Calculation Of Integral, 646
Evaluate
\[\int_0^{\pi} a^x\cos bx\ dx,\ \int_0^{\pi} a^x\sin bx\ dx\ (a>0,\ a\neq 1,\ b\in{\mathbb{N^{+}}})\]
Own
2014 Contests, 900
Find $\sum_{k=0}^n \frac{(-1)^k}{2k+1}\ _n C_k.$
1998 Baltic Way, 11
If $a,b,c$ be the lengths of the sides of a triangle. Let $R$ denote its circumradius. Prove that
\[ R\ge \frac{a^2+b^2}{2\sqrt{2a^2+2b^2-c^2}}\]
When does equality hold?
2019 CCA Math Bonanza, T3
What is the sum of all possible values of $\cos\left(2\theta\right)$ if $\cos\left(2\theta\right)=2\cos\left(\theta\right)$ for a real number $\theta$?
[i]2019 CCA Math Bonanza Team Round #3[/i]
2010 All-Russian Olympiad Regional Round, 11.5
The angles of the triangle $\alpha, \beta, \gamma$ satisfy the inequalities $$\sin \alpha > \cos \beta, \sin \beta > \cos \gamma, \sin \gamma > \cos \alpha. $$Prove that the trĪ±iangle is acute-angled.
2010 Math Prize For Girls Problems, 18
If $a$ and $b$ are positive integers such that
\[
\sqrt{8 + \sqrt{32 + \sqrt{768}}} = a \cos \frac{\pi}{b} \, ,
\]
compute the ordered pair $(a, b)$.
1992 IMO Longlists, 67
In a triangle, a symmedian is a line through a vertex that is symmetric to the median with the respect to the internal bisector (all relative to the same vertex). In the triangle $ABC$, the median $m_a$ meets $BC$ at $A'$ and the circumcircle again at $A_1$. The symmedian $s_a$ meets $BC$ at $M$ and the circumcircle again at $A_2$. Given that the line $A_1A_2$ contains the circumcenter $O$ of the triangle, prove that:
[i](a) [/i]$\frac{AA'}{AM} = \frac{b^2+c^2}{2bc} ;$
[i](b) [/i]$1+4b^2c^2 = a^2(b^2+c^2)$
1968 German National Olympiad, 5
Prove that for all real numbers $x$ of the interval $0 < x <\pi$ the inequality
$$\sin x +\frac12 \sin 2x +\frac13 \sin 3x > 0$$
holds.
2013 AMC 12/AHSME, 16
Let $ABCDE$ be an equiangular convex pentagon of perimeter $1$. The pairwise intersections of the lines that extend the side of the pentagon determine a five-pointed star polygon. Let $s$ be the perimeter of the star. What is the difference between the maximum and minimum possible perimeter of $s$?
$ \textbf{(A)}\ 0 \qquad \textbf{(B)}\ \frac{1}{2} \qquad\textbf{(C)}\ \frac{\sqrt{5}-1}{2} \qquad\textbf{(D)}\ \frac{\sqrt{5}+1}{2} \qquad\textbf{(E)}\ \sqrt{5} $
2013 Serbia National Math Olympiad, 3
Let $M$, $N$ and $P$ be midpoints of sides $BC, AC$ and $AB$, respectively, and let $O$ be circumcenter of acute-angled triangle $ABC$. Circumcircles of triangles $BOC$ and $MNP$ intersect at two different points $X$ and $Y$ inside of triangle $ABC$. Prove that \[\angle BAX=\angle CAY.\]