Found problems: 3349
2012 Baltic Way, 13
Let $ABC$ be an acute triangle, and let $H$ be its orthocentre. Denote by $H_A$, $H_B$, and $H_C$ the second intersection of the circumcircle with the altitudes from $A$, $B$, and $C$ respectively. Prove that the area of triangle $H_A H_B H_C$ does not exceed the area of triangle $ABC$.
2009 AIME Problems, 15
Let $ \overline{MN}$ be a diameter of a circle with diameter $ 1$. Let $ A$ and $ B$ be points on one of the semicircular arcs determined by $ \overline{MN}$ such that $ A$ is the midpoint of the semicircle and $ MB\equal{}\frac35$. Point $ C$ lies on the other semicircular arc. Let $ d$ be the length of the line segment whose endpoints are the intersections of diameter $ \overline{MN}$ with the chords $ \overline{AC}$ and $ \overline{BC}$. The largest possible value of $ d$ can be written in the form $ r\minus{}s\sqrt{t}$, where $ r$, $ s$, and $ t$ are positive integers and $ t$ is not divisible by the square of any prime. Find $ r\plus{}s\plus{}t$.
2011 Morocco National Olympiad, 4
The diagonals of a trapezoid $ ABCD $ whose bases are $ [AB] $ and $ [CD] $ intersect at $P.$ Prove that
\[S_{PAB} + S_{PCD} > S_{PBC} + S_{PDA},\]
Where $S_{XYZ} $ denotes the area of $\triangle XYZ $.
1998 AIME Problems, 13
If $\{a_1,a_2,a_3,\ldots,a_n\}$ is a set of real numbers, indexed so that $a_1<a_2<a_3<\cdots<a_n,$ its complex power sum is defined to be $a_1i+a_2i^2+a_3i^3+\cdots+a_ni^n,$ where $i^2=-1.$ Let $S_n$ be the sum of the complex power sums of all nonempty subsets of $\{1,2,\ldots,n\}.$ Given that $S_8=-176-64i$ and $S_9=p+qi,$ were $p$ and $q$ are integers, find $|p|+|q|.$
1981 All Soviet Union Mathematical Olympiad, 311
It is known about real $a$ and $b$ that the inequality $$a \cos x + b \cos (3x) > 1$$ has no real solutions. Prove that $|b|\le 1$.
2013 China Team Selection Test, 2
Let $P$ be a given point inside the triangle $ABC$. Suppose $L,M,N$ are the midpoints of $BC, CA, AB$ respectively and \[PL: PM: PN= BC: CA: AB.\] The extensions of $AP, BP, CP$ meet the circumcircle of $ABC$ at $D,E,F$ respectively. Prove that the circumcentres of $APF, APE, BPF, BPD, CPD, CPE$ are concyclic.
2005 Czech-Polish-Slovak Match, 5
Given a convex quadrilateral $ABCD$, find the locus of the points $P$ inside the quadrilateral such that
\[S_{PAB}\cdot S_{PCD} = S_{PBC}\cdot S_{PDA}\]
(where $S_X$ denotes the area of triangle $X$).
2006 Taiwan TST Round 1, 1
Let the three sides of $\triangle ABC$ be $a,b,c$. Prove that
$\displaystyle \frac{\sin^2A}{a}+\frac{\sin^2B}{b}+\frac{\sin^2C}{c} \le \frac{S^2}{abc}$
where $\displaystyle S=\frac{a+b+c}{2}$. Find the case where equality holds.
2007 Junior Balkan Team Selection Tests - Romania, 2
Let $ABCD$ be a trapezium $(AB \parallel CD)$ and $M,N$ be the intersection points of the circles of diameters $AD$ and $BC$. Prove that $O \in MN$, where $O \in AC \cap BD$.
2005 France Pre-TST, 5
Let $I$ be the incenter of the triangle $ABC$. Let $A_1,A_2$ be two distinct points on the line $BC$, let $B_1,B_2$ be two distinct points on the line $CA$, and let $C_1,C_2$ be two distinct points on the line $BA$ such that $AI = A_1I = A_2I$ and $BI = B_1I = B_2I$ and $CI = C_1I = C_2I$.
Prove that $A_1A_2+B_1B_2+C_1C_2 = p$ where $p$ denotes the perimeter of $ABC.$
Pierre.
2010 Today's Calculation Of Integral, 593
For a positive integer $m$, prove the following ineqaulity.
$0\leq \int_0^1 \left(x+1-\sqrt{x^2+2x\cos \frac{2\pi}{2m+1}+1\right)dx\leq 1.}$
1996 Osaka University entrance exam
2012 India IMO Training Camp, 1
Let $ABC$ be an isosceles triangle with $AB=AC$. Let $D$ be a point on the segment $BC$ such that $BD=2DC$. Let $P$ be a point on the segment $AD$ such that $\angle BAC=\angle BPD$. Prove that $\angle BAC=2\angle DPC$.
2013 AMC 12/AHSME, 24
Three distinct segments are chosen at random among the segments whose end-points are the vertices of a regular 12-gon. What is the probability that the lengths of these three segments are the three side lengths of a triangle with positive area?
$ \textbf{(A)} \ \frac{553}{715} \qquad \textbf{(B)} \ \frac{443}{572} \qquad \textbf{(C)} \ \frac{111}{143} \qquad \textbf{(D)} \ \frac{81}{104} \qquad \textbf{(E)} \ \frac{223}{286}$
1966 IMO Longlists, 61
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) \[ \frac{1}{\sin{2x}}+\frac{1}{\sin{4x}}+\dots+\frac{1}{\sin{2^nx}}=\cot{x}-\cot{2^nx} \]
2010 Today's Calculation Of Integral, 596
Find the minimum value of $\int_0^{\frac{\pi}{2}} |a\sin 2x-\cos ^ 2 x|dx\ (a>0).$
2009 Shimane University entrance exam/Medicine
Estonia Open Senior - geometry, 2003.1.2
Four rays spread out from point $O$ in a $3$-dimensional space in a way that the angle between every two rays is $a$. Find $\cos a$.
2012 AIME Problems, 6
The complex numbers $z$ and $w$ satisfy $z^{13} = w$, $w^{11} = z$, and the imaginary part of $z$ is $\sin\left(\frac{m\pi}n\right)$ for relatively prime positive integers $m$ and $n$ with $m < n$. Find $n$.
2004 India IMO Training Camp, 3
Determine all functionf $f : \mathbb{R} \mapsto \mathbb{R}$ such that
\[ f(x+y) = f(x)f(y) - c \sin{x} \sin{y} \] for all reals $x,y$ where $c> 1$ is a given constant.
2007 Purple Comet Problems, 4
To the nearest degree, find the measure of the largest angle in a triangle with side lengths $3$, $5$, and $7$.
2011 NIMO Summer Contest, 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]
2009 Today's Calculation Of Integral, 438
Evaluate $ \int_{\sqrt{2}\minus{}1}^{\sqrt{2}\plus{}1} \frac{x^4\plus{}x^2\plus{}2}{(x^2\plus{}1)^2}\ dx.$
2007 Peru MO (ONEM), 1
Find all values of $A$ such that $0^o < A < 360^o$ and also
$\frac{\sin A}{\cos A - 1} \ge 1$ and $\frac{3\cos A - 1}{\sin A} \ge 1.$
2004 Vietnam Team Selection Test, 2
Let us consider a convex hexagon ABCDEF. Let $A_1, B_1,C_1, D_1, E_1, F_1$ be midpoints of the sides $AB, BC, CD, DE, EF,FA$ respectively. Denote by $p$ and $p_1$, respectively, the perimeter of the hexagon $ A B C D E F $ and hexagon $ A_1B_1C_1D_1E_1F_1 $. Suppose that all inner angles of hexagon $ A_1B_1C_1D_1E_1F_1 $ are equal. Prove that \[ p \geq \frac{2 \cdot \sqrt{3}}{3} \cdot p_1 .\] When does equality hold ?
2006 National Olympiad First Round, 17
Let $D$ be a point on the side $[BC]$ of $\triangle ABC$ such that $|BD|=2$ and $|DC|=6$. If $|AB|=4$ and $m(\widehat{ACB})=20^\circ$, then what is $m(\widehat {BAD})$?
$
\textbf{(A)}\ 10^\circ
\qquad\textbf{(B)}\ 18^\circ
\qquad\textbf{(C)}\ 20^\circ
\qquad\textbf{(D)}\ 22^\circ
\qquad\textbf{(E)}\ 25^\circ
$
1954 AMC 12/AHSME, 40
If $ \left (a\plus{}\frac{1}{a} \right )^2\equal{}3$, then $ a^3\plus{}\frac{1}{a^3}$ equals:
$ \textbf{(A)}\ \frac{10\sqrt{3}}{3} \qquad
\textbf{(B)}\ 3\sqrt{3} \qquad
\textbf{(C)}\ 0 \qquad
\textbf{(D)}\ 7\sqrt{7} \qquad
\textbf{(E)}\ 6\sqrt{3}$