Found problems: 3349
1968 Bulgaria National Olympiad, Problem 6
Find the kind of a triangle if
$$\frac{a\cos\alpha+b\cos\beta+c\cos\gamma}{a\sin\alpha+b\sin\beta+c\sin\gamma}=\frac{2p}{9R}.$$
($\alpha,\beta,\gamma$ are the measures of the angles, $a,b,c$ are the respective lengths of the sides, $p$ the semiperimeter, $R$ is the circumradius)
[i]K. Petrov[/i]
1967 IMO Shortlist, 3
Find all $x$ for which, for all $n,$ \[\sum^n_{k=1} \sin {k x} \leq \frac{\sqrt{3}}{2}.\]
IV Soros Olympiad 1997 - 98 (Russia), 10.7
How many different solutions on the interval $[0, \pi]$ does the equation $$6\sqrt2 \sin x \cdot tgx - 2\sqrt2 tgx +3\sin x -1=0$$ have?
2013 Today's Calculation Of Integral, 871
Define sequences $\{a_n\},\ \{b_n\}$ by
\[a_n=\int_{-\frac {\pi}6}^{\frac{\pi}6} e^{n\sin \theta}d\theta,\ b_n=\int_{-\frac {\pi}6}^{\frac{\pi}6} e^{n\sin \theta}\cos \theta d\theta\ (n=1,\ 2,\ 3,\ \cdots).\]
(1) Find $b_n$.
(2) Prove that for each $n$, $b_n\leq a_n\leq \frac 2{\sqrt{3}}b_n.$
(3) Find $\lim_{n\to\infty} \frac 1{n}\ln (na_n).$
1998 IMO Shortlist, 3
Let $I$ be the incenter of triangle $ABC$. Let $K,L$ and $M$ be the points of tangency of the incircle of $ABC$ with $AB,BC$ and $CA$, respectively. The line $t$ passes through $B$ and is parallel to $KL$. The lines $MK$ and $ML$ intersect $t$ at the points $R$ and $S$. Prove that $\angle RIS$ is acute.
1988 IMO Shortlist, 27
Let $ ABC$ be an acute-angled triangle. Let $ L$ be any line in the plane of the triangle $ ABC$. Denote by $ u$, $ v$, $ w$ the lengths of the perpendiculars to $ L$ from $ A$, $ B$, $ C$ respectively. Prove the inequality $ u^2\cdot\tan A \plus{} v^2\cdot\tan B \plus{} w^2\cdot\tan C\geq 2\cdot S$, where $ S$ is the area of the triangle $ ABC$. Determine the lines $ L$ for which equality holds.
1988 AIME Problems, 7
In triangle $ABC$, $\tan \angle CAB = 22/7$, and the altitude from $A$ divides $BC$ into segments of length 3 and 17. What is the area of triangle $ABC$?
2015 Mathematical Talent Reward Programme, MCQ: P 10
If $\sum_{i=1}^{n} \cos ^{-1}\left(\alpha_{i}\right)=0,$ then find $\sum_{i=1}^{n} \alpha_{i}$
[list=1]
[*] $\frac{n}{2} $
[*] $n $
[*] $n\pi $
[*] $\frac{n\pi}{2} $
[/list]
2014 PUMaC Geometry B, 8
$ABCD$ is a cyclic quadrilateral with circumcenter $O$ and circumradius $7$. $AB$ intersects $CD$ at $E$, $DA$ intersects $CB$ at $F$. $OE=13$, $OF=14$. Let $\cos\angle FOE=\dfrac pq$, with $p$, $q$ coprime. Find $p+q$.
2005 Today's Calculation Of Integral, 20
Calculate the following indefinite integrals.
[1] $\int \ln (x^2-1)dx$
[2] $\int \frac{1}{e^x+1}dx$
[3] $\int (ax^2+bx+c)e^{mx}dx\ (abcm\neq 0)$
[4] $\int \left(\tan x+\frac{1}{\tan x}\right)^2 dx$
[5] $\int \sqrt{1-\sin x}dx$
2006 AMC 12/AHSME, 24
Let $ S$ be the set of all points $ (x,y)$ in the coordinate plane such that $ 0\le x\le \frac \pi2$ and $ 0\le y\le \frac \pi2$. What is the area of the subset of $ S$ for which
\[ \sin^2 x \minus{} \sin x\sin y \plus{} \sin^2 y\le \frac 34?
\]$ \textbf{(A) } \frac {\pi^2}9 \qquad \textbf{(B) } \frac {\pi^2}8 \qquad \textbf{(C) } \frac {\pi^2}6\qquad \textbf{(D) } \frac {3\pi^2}{16} \qquad \textbf{(E) } \frac {2\pi^2}9$
2012 AIME Problems, 12
Let $\triangle ABC$ be a right triangle with right angle at $C$. Let $D$ and $E$ be points on $\overline{AB}$ with $D$ between $A$ and $E$ such that $\overline{CD}$ and $\overline{CE}$ trisect $\angle C$. If $\frac{DE}{BE} = \frac{8}{15}$, then $\tan B$ can be written as $\frac{m\sqrt{p}}{n}$, where $m$ and $n$ are relatively prime positive integers, and $p$ is a positive integer not divisible by the square of any prime. Find $m+n+p$.
1984 AMC 12/AHSME, 30
For any complex number $w = a + bi$, $|w|$ is defined to be the real number $\sqrt{a^2 + b^2}$. If $w = \cos{40^\circ} + i\sin{40^\circ}$, then
\[ |w + 2w^2 + 3w^3 + \cdots + 9w^9|^{-1} \]
equals
$\textbf{(A)}\ \frac{1}{9}\sin{40^\circ} \qquad \textbf{(B)}\ \frac{2}{9}\sin{20^\circ} \qquad \textbf{(C)}\ \frac{1}{9}\cos{40^\circ} \qquad \textbf{(D)}\ \frac{1}{18}\cos{20^\circ} \qquad \textbf{(E)}\text{ none of these}$
1999 Baltic Way, 15
Let $ABC$ be a triangle with $\angle C=60^\circ$ and $AC<BC$. The point $D$ lies on the side $BC$ and satisfies $BD=AC$. The side $AC$ is extended to the point $E$ where $AC=CE$. Prove that $AB=DE$.
2000 Harvard-MIT Mathematics Tournament, 14
$ABCD$ is a cyclic quadrilateral inscribed in a circle of radius $5$, with $AB=6$, $BC=7$, $CD=8$. Find $AD$.
1969 IMO Longlists, 29
$(GDR 1)$ Find all real numbers $\lambda$ such that the equation $\sin^4 x - \cos^4 x = \lambda(\tan^4 x - \cot^4 x)$
$(a)$ has no solution,
$(b)$ has exactly one solution,
$(c)$ has exactly two solutions,
$(d)$ has more than two solutions (in the interval $(0, \frac{\pi}{4}).$
2007 Pre-Preparation Course Examination, 2
Let $C_{1}$, $C_{2}$ and $C_{3}$ be three circles that does not intersect and non of them is inside another. Suppose $(L_{1},L_{2})$, $(L_{3},L_{4})$ and $(L_{5},L_{6})$ be internal common tangents of $(C_{1}, C_{2})$, $(C_{1}, C_{3})$, $(C_{2}, C_{3})$. Let $L_{1},L_{2},L_{3},L_{4},L_{5},L_{6}$ be sides of polygon $AC'BA'CB'$. Prove that $AA',BB',CC'$ are concurrent.
1990 APMO, 3
Consider all the triangles $ABC$ which have a fixed base $AB$ and whose altitude from $C$ is a constant $h$. For which of these triangles is the product of its altitudes a maximum?
2015 India IMO Training Camp, 3
Prove that for any triangle $ABC$, the inequality $\displaystyle\sum_{\text{cyclic}}\cos A\le\sum_{\text{cyclic}}\sin (A/2)$ holds.
2012 Today's Calculation Of Integral, 805
Prove the following inequalities:
(1) For $0\leq x\leq 1$,
\[1-\frac 13x\leq \frac{1}{\sqrt{1+x^2}}\leq 1.\]
(2) $\frac{\pi}{3}-\frac 16\leq \int_0^{\frac{\sqrt{3}}{2}} \frac{1}{\sqrt{1-x^4}}dx\leq \frac{\pi}{3}.$
2009 Kyiv Mathematical Festival, 1
Solve the equation $\big(2cos(x-\frac{\pi}{4})+tgx\big)^3=54 sin^2x$, $x\in \big[0,\frac{\pi}{2}\big)$
2011 Northern Summer Camp Of Mathematics, 1
Solve the system of equations
\[(x+\sqrt{x^2+1})(y+\sqrt{y^2+1})=1,\]\[y+\frac{y}{\sqrt{x^2-1}}+\frac{35}{12}=0.\]
VI Soros Olympiad 1999 - 2000 (Russia), 9.8
Let $a_n$ denote an angle from the interval for each $\left( 0, \frac{\pi}{2}\right)$ , the tangent of which is equal to $n$ . Prove that
$$\sqrt{1+1^2} \sin(a_1-a_{1000}) + \sqrt{1+2^2} \sin(a_2-a_{1000})+...+\sqrt{1+2000^2} \sin(a_{2000}-a_{1000}) = \sin a_{1000} $$
2007 Korea - Final Round, 5
For the vertex $ A$ of a triangle $ ABC$, let $ l_a$ be the distance between the projections on $ AB$ and $ AC$ of the intersection of the angle bisector of ∠$ A$ with side $ BC$. Define $ l_b$ and $ l_c$ analogously. If $ l$ is the perimeter of triangle $ ABC$, prove that $ \frac{l_a l_b l_c}{l^3}\le\frac{1}{64}$.
2005 Serbia Team Selection Test, 4
Let $T$ be the centroid of triangle $ABC$. Prove that \[ \frac 1{\sin \angle TAC} + \frac 1{\sin \angle TBC} \geq 4 \]