Found problems: 3349
1997 Czech And Slovak Olympiad IIIA, 5
For a given integer $n \ge 2$, find the maximum possible value of $V_n = \sin x_1 \cos x_2 +\sin x_2 \cos x_3 +...+\sin x_n \cos x_1$, where $x_1,x_2,...,x_n$ are real numbers.
2013 ELMO Shortlist, 10
Let $AB=AC$ in $\triangle ABC$, and let $D$ be a point on segment $AB$. The tangent at $D$ to the circumcircle $\omega$ of $BCD$ hits $AC$ at $E$. The other tangent from $E$ to $\omega$ touches it at $F$, and $G=BF \cap CD$, $H=AG \cap BC$. Prove that $BH=2HC$.
[i]Proposed by David Stoner[/i]
2007 AMC 12/AHSME, 24
For each integer $ n > 1,$ let $ F(n)$ be the number of solutions of the equation $ \sin x \equal{} \sin nx$ on the interval $ [0,\pi].$ What is $ \sum_{n \equal{} 2}^{2007}F(n)?$
$ \textbf{(A)}\ 2,014,524 \qquad \textbf{(B)}\ 2,015,028 \qquad \textbf{(C)}\ 2,015,033 \qquad \textbf{(D)}\ 2,016,532 \qquad \textbf{(E)}\ 2,017,033$
2012 Singapore MO Open, 1
The incircle with centre $I$ of the triangle $ABC$ touches the sides $BC, CA$ and $AB$ at $D, E, F$ respectively. The line $ID$ intersects the segment $EF$ at $K$. Proof that $A, K, M$ collinear, where $M$ is the midpoint of $BC$.
2024 Moldova EGMO TST, 7
$ \frac{\sqrt{10+\sqrt{1}}+\sqrt{10+\sqrt{2}}+...+\sqrt{10+\sqrt{99}}}{\sqrt{10-\sqrt{1}}+\sqrt{10-\sqrt{2}}+...+\sqrt{10-\sqrt{99}}}=? $
2011 Graduate School Of Mathematical Sciences, The Master Cource, The University Of Tokyo, 3
Let $a$ be a positive real number.
Evaluate $I=\int_0^{+\infty} \frac{\sin x\cos x}{x(x^2+a^2)}dx.$
2003 France Team Selection Test, 3
$M$ is an arbitrary point inside $\triangle ABC$. $AM$ intersects the circumcircle of the triangle again at $A_1$. Find the points $M$ that minimise $\frac{MB\cdot MC}{MA_1}$.
2019 Jozsef Wildt International Math Competition, W. 6
Compute$$\int \limits_{\frac{\pi}{6}}^{\frac{\pi}{4}}\frac{(1+\ln x)\cos x+x\sin x\ln x}{\cos^2 x + x^2 \ln^2 x}dx$$
1998 AIME Problems, 5
Given that $A_k=\frac{k(k-1)}2\cos\frac{k(k-1)\pi}2,$ find $|A_{19}+A_{20}+\cdots+A_{98}|.$
2009 AIME Problems, 13
Let $ A$ and $ B$ be the endpoints of a semicircular arc of radius $ 2$. The arc is divided into seven congruent arcs by six equally spaced points $ C_1,C_2,\ldots,C_6$. All chords of the form $ \overline{AC_i}$ or $ \overline{BC_i}$ are drawn. Let $ n$ be the product of the lengths of these twelve chords. Find the remainder when $ n$ is divided by $ 1000$.
Today's calculation of integrals, 884
Prove that :
\[\pi (e-1)<\int_0^{\pi} e^{|\cos 4x|}dx<2(e^{\frac{\pi}{2}}-1)\]
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}$.
2011 Today's Calculation Of Integral, 720
Evaluate $\int_0^{2\pi} |x^2-\pi ^ 2 -\sin ^ 2 x|\ dx$.
1990 IMO Longlists, 4
Find the minimal value of the function
\[\begin{array}{c}\ f(x) =\sqrt{15 - 12 \cos x} + \sqrt{4 -2 \sqrt 3 \sin x}+\sqrt{7-4\sqrt 3 \sin x} +\sqrt{10-4 \sqrt 3 \sin x - 6 \cos x}\end{array}\]
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$.
2003 France Team Selection Test, 3
$M$ is an arbitrary point inside $\triangle ABC$. $AM$ intersects the circumcircle of the triangle again at $A_1$. Find the points $M$ that minimise $\frac{MB\cdot MC}{MA_1}$.
2019 All-Russian Olympiad, 2
Is it true, that for all pairs of non-negative integers $a$ and $b$ , the system
\begin{align*}
\tan{13x} \tan{ay} =& 1 \\
\tan{21x} \tan{by}= & 1
\end{align*}
has at least one solution?
1974 IMO Longlists, 15
Let $ABC$ be a triangle. Prove that there exists a point $D$ on the side $AB$ of the triangle $ABC$, such that $CD$ is the geometric mean of $AD$ and $DB$, iff the triangle $ABC$ satisfies the inequality $\sin A\sin B\le\sin^2\frac{C}{2}$.
[hide="Comment"][i]Alternative formulation, from IMO ShortList 1974, Finland 2:[/i] We consider a triangle $ABC$. Prove that: $\sin(A) \sin(B) \leq \sin^2 \left( \frac{C}{2} \right)$ is a necessary and sufficient condition for the existence of a point $D$ on the segment $AB$ so that $CD$ is the geometrical mean of $AD$ and $BD$.[/hide]
PEN G Problems, 7
Show that $ \pi$ is irrational.
2005 Today's Calculation Of Integral, 63
For a positive number $x$, let $f(x)=\lim_{n\to\infty} \sum_{k=1}^n \left|\cos \left(\frac{2k+1}{2n}x\right)-\cos \left(\frac{2k-1}{2n}x\right)\right|$
Evaluate
\[\lim_{x\rightarrow\infty}\frac{f(x)}{x}\]
1986 Canada National Olympiad, 1
In the diagram line segments $AB$ and $CD$ are of length 1 while angles $ABC$ and $CBD$ are $90^\circ$ and $30^\circ$ respectively. Find $AC$.
[asy]
import geometry;
import graph;
unitsize(1.5 cm);
pair A, B, C, D;
B = (0,0);
D = (3,0);
A = 2*dir(120);
C = extension(B,dir(30),A,D);
draw(A--B--D--cycle);
draw(B--C);
draw(arc(B,0.5,0,30));
label("$A$", A, NW);
label("$B$", B, SW);
label("$C$", C, NE);
label("$D$", D, SE);
label("$30^\circ$", (0.8,0.2));
label("$90^\circ$", (0.1,0.5));
perpendicular(B,NE,C-B);
[/asy]
2007 Bulgaria Team Selection Test, 1
Let $ABC$ is a triangle with $\angle BAC=\frac{\pi}{6}$ and the circumradius equal to 1. If $X$ is a point inside or in its boundary let $m(X)=\min(AX,BX,CX).$ Find all the angles of this triangle if $\max(m(X))=\frac{\sqrt{3}}{3}.$
2004 Estonia Team Selection Test, 4
Denote $f(m) =\sum_{k=1}^m (-1)^k cos \frac{k\pi}{2 m + 1}$
For which positive integers $m$ is $f(m)$ rational?
2005 MOP Homework, 7
Let $ABC$ be a triangle. Prove that \[\frac{a^2}{bc}+\frac{b^2}{ca}+\frac{c^2}{ab} \ge 4\left(\sin^2\frac{A}{2}+\sin^2\frac{B}{2}+\sin^2\frac{C}{2}\right).\]
1987 Iran MO (2nd round), 1
Calculate the product:
\[A=\sin 1^\circ \times \sin 2^\circ \times \sin 3^\circ \times \cdots \times \sin 89^\circ\]