Found problems: 3349
2003 AMC 12-AHSME, 24
If $ a\ge b>1$, what is the largest possible value of $ \log_a(a/b)\plus{}\log_b(b/a)$?
$ \textbf{(A)}\ \minus{}2 \qquad
\textbf{(B)}\ 0 \qquad
\textbf{(C)}\ 2 \qquad
\textbf{(D)}\ 3 \qquad
\textbf{(E)}\ 4$
2005 Bulgaria Team Selection Test, 2
Find the number of the subsets $B$ of the set $\{1,2,\cdots, 2005 \}$ such that the sum of the elements of $B$ is congruent to $2006$ modulo $2048$
2008 Bulgaria Team Selection Test, 2
In the triangle $ABC$, $AM$ is median, $M \in BC$, $BB_{1}$ and $CC_{1}$ are altitudes, $C_{1} \in AB$, $B_{1} \in AC$. The line through $A$ which is perpendicular to $AM$ cuts the lines $BB_{1}$ and $CC_{1}$ at points $E$ and $F$, respectively. Let $k$ be the circumcircle of $\triangle EFM$. Suppose also that $k_{1}$ and $k_{2}$ are circles touching both $EF$ and the arc $EF$ of $k$ which does not contain $M$. If $P$ and $Q$ are the points at which $k_{1}$ intersects $k_{2}$, prove that $P$, $Q$, and $M$ are collinear.
2001 Brazil National Olympiad, 4
A calculator treats angles as radians. It initially displays 1. What is the largest value that can be achieved by pressing the buttons cos or sin a total of 2001 times? (So you might press cos five times, then sin six times and so on with a total of 2001 presses.)
1972 IMO Longlists, 6
Prove the inequality
\[(n + 1)\cos\frac{\pi}{n + 1}- n\cos\frac{\pi}{n}> 1\]
for all natural numbers $n \ge 2.$
2011 Today's Calculation Of Integral, 688
For a real number $x$, let $f(x)=\int_0^{\frac{\pi}{2}} |\cos t-x\sin 2t|\ dt$.
(1) Find the minimum value of $f(x)$.
(2) Evaluate $\int_0^1 f(x)\ dx$.
[i]2011 Tokyo Institute of Technology entrance exam, Problem 2[/i]
2021 AMC 12/AHSME Spring, 24
Semicircle $\Gamma$ has diameter $\overline{AB}$ of length $14$. Circle $\Omega$ lies tangent to $\overline{AB}$ at a point $P$ and intersects $\Gamma$ at points $Q$ and $R$. If $QR=3\sqrt3$ and $\angle QPR=60^\circ$, then the area of $\triangle PQR$ is $\frac{a\sqrt{b}}{c}$, where $a$ and $c$ are relatively prime positive integers, and $b$ is a positive integer not divisible by the square of any prime. What is $a+b+c$?
$\textbf{(A) }110 \qquad \textbf{(B) }114 \qquad \textbf{(C) }118 \qquad \textbf{(D) }122\qquad \textbf{(E) }126$
2012 India National Olympiad, 1
Let $ABCD$ be a quadrilateral inscribed in a circle. Suppose $AB=\sqrt{2+\sqrt{2}}$ and $AB$ subtends $135$ degrees at center of circle . Find the maximum possible area of $ABCD$.
2008 Harvard-MIT Mathematics Tournament, 32
Cyclic pentagon $ ABCDE$ has side lengths $ AB\equal{}BC\equal{}5$, $ CD\equal{}DE\equal{}12$, and $ AE \equal{} 14$. Determine the radius of its circumcircle.
1981 Miklós Schweitzer, 6
Let $ f$ be a strictly increasing, continuous function mapping $ I=[0,1]$ onto itself. Prove that the following inequality holds for all pairs $ x,y \in I$: \[ 1-\cos (xy) \leq \int_0^xf(t) \sin (tf(t))dt + \int_0^y f^{-1}(t) \sin (tf^{-1}(t)) dt .\]
[i]Zs. Pales[/i]
2013 ELMO Shortlist, 6
Let $ABCDEF$ be a non-degenerate cyclic hexagon with no two opposite sides parallel, and define $X=AB\cap DE$, $Y=BC\cap EF$, and $Z=CD\cap FA$. Prove that
\[\frac{XY}{XZ}=\frac{BE}{AD}\frac{\sin |\angle{B}-\angle{E}|}{\sin |\angle{A}-\angle{D}|}.\][i]Proposed by Victor Wang[/i]
2008 South africa National Olympiad, 6
Find all function pairs $(f,g)$ where each $f$ and $g$ is a function defined on the integers and with values, such that, for all integers $a$ and $b$,
\[f(a+b)=f(a)g(b)+g(a)f(b)\\
g(a+b)=g(a)g(b)-f(a)f(b).\]
2009 Indonesia TST, 4
Given triangle $ ABC$. Let the tangent lines of the circumcircle of $ AB$ at $ B$ and $ C$ meet at $ A_0$. Define $ B_0$ and $ C_0$ similarly.
a) Prove that $ AA_0,BB_0,CC_0$ are concurrent.
b) Let $ K$ be the point of concurrency. Prove that $ KG\parallel BC$ if and only if $ 2a^2\equal{}b^2\plus{}c^2$.
2010 AIME Problems, 9
Let $ ABCDEF$ be a regular hexagon. Let $ G$, $ H$, $ I$, $ J$, $ K$, and $ L$ be the midpoints of sides $ AB$, $ BC$, $ CD$, $ DE$, $ EF$, and $ AF$, respectively. The segments $ AH$, $ BI$, $ CJ$, $ DK$, $ EL$, and $ FG$ bound a smaller regular hexagon. Let the ratio of the area of the smaller hexagon to the area of $ ABCDEF$ be expressed as a fraction $ \frac {m}{n}$ where $ m$ and $ n$ are relatively prime positive integers. Find $ m \plus{} n$.
1967 IMO Longlists, 42
Decompose the expression into real factors:
\[E = 1 - \sin^5(x) - \cos^5(x).\]
1998 Niels Henrik Abels Math Contest (Norwegian Math Olympiad) Round 2, 9
In the triangle $ ABC$ we have $ AB \equal{} 5$ and $ AC \equal{} 6$. The area of the triangle when the $ \angle ACB$ is as large as possible is
$ \text{(A)}\ 15 \qquad \text{(B)}\ 5 \sqrt{7} \qquad \text{(C)}\ \frac{7}{2} \sqrt{7} \qquad \text{(D)}\ 3 \sqrt{11} \qquad \text{(E)}\ \frac{5}{2} \sqrt{11}$
1998 Poland - First Round, 3
In the isosceles triangle $ ABC$ the angle $ BAC$ is a right angle. Point $ D$ lies on the side $ BC$ and satisfies $ BD \equal{} 2 \cdot CD$. Point $ E$ is the foot of the perpendicular of the point $ B$ on the line $ AD$. Find the angle $ CED$.
1985 Balkan MO, 2
Let $a,b,c,d \in [-\frac{\pi}{2}, \frac{\pi}{2}]$ be real numbers such that
$\sin{a}+\sin{b}+\sin{c}+\sin{d}=1$ and $\cos{2a}+\cos{2b}+\cos{2c}+\cos{2d}\geq \frac{10}{3}$.
Prove that $a,b,c,d \in [0, \frac{\pi}{6}]$
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$
1994 Hong Kong TST, 1
In a $\triangle ABC$, $\angle C=2 \angle B$. $P$ is a point in the interior of $\triangle ABC$ satisfying that $AP=AC$ and $PB=PC$. Show that $AP$ trisects the angle $\angle A$.
1980 Putnam, A3
Evaluate
$$\int_{0}^{ \pi \slash 2} \frac{ dx}{1+( \tan x)^{\sqrt{2}} }\;.$$
2010 Contests, 2
Let $ABC$ be an acute triangle, $H$ its orthocentre, $D$ a point on the side $[BC]$, and $P$ a point such that $ADPH$ is a parallelogram.
Show that $\angle BPC > \angle BAC$.
1966 German National Olympiad, 5
Prove that \[\tan 7 30^{\prime }=\sqrt{6}+\sqrt{2}-\sqrt{3}-2.\]
2005 Today's Calculation Of Integral, 22
Evaluate
\[\int_0^1 (1-x^2)^n dx\ (n=0,1,2,\cdots)\]
2007 Princeton University Math Competition, 10
In triangle $ABC$ with $AB \neq AC$, points $N \in CA$, $M \in AB$, $P \in BC$, and $Q \in BC$ are chosen such that $MP \parallel AC$, $NQ \parallel AB$, $\frac{BP}{AB} = \frac{CQ}{AC}$, and $A, M, Q, P, N$ are concyclic. Find $\angle BAC$.