Found problems: 3349
2012 Brazil Team Selection Test, 4
Let $ ABC $ be an acute triangle. Denote by $ D $ the foot of the perpendicular line drawn from the point $ A $ to the side $ BC $, by $M$ the midpoint of $ BC $, and by $ H $ the orthocenter of $ ABC $. Let $ E $ be the point of intersection of the circumcircle $ \Gamma $ of the triangle $ ABC $ and the half line $ MH $, and $ F $ be the point of intersection (other than $E$) of the line $ ED $ and the circle $ \Gamma $. Prove that $ \tfrac{BF}{CF} = \tfrac{AB}{AC} $ must hold.
(Here we denote $XY$ the length of the line segment $XY$.)
V Soros Olympiad 1998 - 99 (Russia), 11.3
For what a from the interval $[0,\pi]$ do there exist $a$ and $b$ that are not simultaneously equal to zero, for which the inequality
$$a \cos x + b \cos 2x \le 0$$ is satisfied for all $x$ belonging to the segment $[a, \pi]$?
1995 AIME Problems, 12
Pyramid $OABCD$ has square base $ABCD,$ congruent edges $\overline{OA}, \overline{OB}, \overline{OC},$ and $\overline{OD},$ and $\angle AOB=45^\circ.$ Let $\theta$ be the measure of the dihedral angle formed by faces $OAB$ and $OBC.$ Given that $\cos \theta=m+\sqrt{n},$ where $m$ and $n$ are integers, find $m+n.$
1998 IMO Shortlist, 6
Let $ABCDEF$ be a convex hexagon such that $\angle B+\angle D+\angle F=360^{\circ }$ and \[ \frac{AB}{BC} \cdot \frac{CD}{DE} \cdot \frac{EF}{FA} = 1. \] Prove that \[ \frac{BC}{CA} \cdot \frac{AE}{EF} \cdot \frac{FD}{DB} = 1. \]
2010 ELMO Shortlist, 4
Let $-2 < x_1 < 2$ be a real number and define $x_2, x_3, \ldots$ by $x_{n+1} = x_n^2-2$ for $n \geq 1$. Assume that no $x_n$ is $0$ and define a number $A$, $0 \leq A \leq 1$ in the following way: The $n^{\text{th}}$ digit after the decimal point in the binary representation of $A$ is a $0$ if $x_1x_2\cdots x_n$ is positive and $1$ otherwise. Prove that $A = \frac{1}{\pi}\cos^{-1}\left(\frac{x_1}{2}\right)$.
[i]Evan O' Dorney.[/i]
1971 Spain Mathematical Olympiad, 4
Prove that in every triangle with sides $a, b, c$ and opposite angles $A, B, C$, is fulfilled (measuring the angles in radians) $$\frac{a A+bB+cC}{a+b+c} \ge \frac{\pi}{3}$$
Hint: Use $a \ge b \ge c \Rightarrow A \ge B \ge C$.
1983 Vietnam National Olympiad, 2
$(a)$ Prove that $\sqrt{2}(\sin t + \cos t) \ge 2\sqrt[4]{\sin 2t}$ for $0 \le t \le\frac{\pi}{2}.$
$(b)$ Find all $y, 0 < y < \pi$, such that $1 +\frac{2 \cot 2y}{\cot y} \ge \frac{\tan 2y}{\tan y}$.
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1970 IMO Shortlist, 1
Consider a regular $2n$-gon and the $n$ diagonals of it that pass through its center. Let $P$ be a point of the inscribed circle and let $a_1, a_2, \ldots , a_n$ be the angles in which the diagonals mentioned are visible from the point $P$. Prove that
\[\sum_{i=1}^n \tan^2 a_i = 2n \frac{\cos^2 \frac{\pi}{2n}}{\sin^4 \frac{\pi}{2n}}.\]
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 $\angle A$.
2012 Turkmenistan National Math Olympiad, 1
Find the max and min value of $a\cos^2 x+b\sin x\cos x+c\sin^2 x$.
1949 Putnam, A6
Prove that for every real or complex $x$
$$\prod_{k=1}^{\infty} \frac{1+2\cos \frac{2x}{3^{k}}}{3} =\frac{\sin x}{x}.$$
2010 Today's Calculation Of Integral, 644
For a constant $p$ such that $\int_1^p e^xdx=1$, prove that
\[\left(\int_1^p e^x\cos x\ dx\right)^2+\left(\int_1^p e^x\sin x\ dx\right)^2>\frac 12.\]
Own
2003 India National Olympiad, 1
Let $P$ be an interior point of an acute-angled triangle $ABC$. The line $BP$ meets the line $AC$ at $E$, and the line $CP$ meets the line $AB$ at $F$. The lines $AP$ and $EF$ intersect each other at $D$. Let $K$ be the foot of the perpendicular from the point $D$ to the line $BC$. Show that the line $KD$ bisects the angle $\angle EKF$.
2005 Today's Calculation Of Integral, 18
Calculate the following indefinite integrals.
[1] $\int (\sin x+\cos x)^4 dx$
[2] $\int \frac{e^{2x}}{e^x+1}dx$
[3] $\int \sin ^ 4 xdx$
[4] $\int \sin 6x\cos 2xdx$
[5] $\int \frac{x^2}{\sqrt{(x+1)^3}}dx$
2010 IberoAmerican Olympiad For University Students, 2
Calculate the sum of the series $\sum_{-\infty}^{\infty}\frac{\sin^33^k}{3^k}$.
2002 South africa National Olympiad, 5
In acute-angled triangle $ABC$, a semicircle with radius $r_a$ is constructed with its base on $BC$ and tangent to the other two sides. $r_b$ and $r_c$ are defined similarly. $r$ is the radius of the incircle of $ABC$. Show that \[ \frac{2}{r} = \frac{1}{r_a} + \frac{1}{r_b} + \frac{1}{r_c}. \]
2005 Danube Mathematical Olympiad, 1
Prove that the equation $4x^3-3x+1=2y^2$ has at least $31$ solutions in positive integers $x$ and $y$ with $x\leq 2005$.
2006 Stanford Mathematics Tournament, 10
Evaluate: $ \sum\limits_{n\equal{}1}^\infty \arctan{\left(\frac{1}{n^2\minus{}n\plus{}1}\right)}$
1950 AMC 12/AHSME, 49
A triangle has a fixed base $AB$ that is $2$ inches long. The median from $A$ to side $BC$ is $ 1\frac{1}{2}$ inches long and can have any position emanating from $A$. The locus of the vertex $C$ of the triangle is:
$\textbf{(A)}\ \text{A straight line }AB,1\dfrac{1}{2}\text{ inches from }A \qquad\\
\textbf{(B)}\ \text{A circle with }A\text{ as center and radius }2\text{ inches} \qquad\\
\textbf{(C)}\ \text{A circle with }A\text{ as center and radius }3\text{ inches} \qquad\\
\textbf{(D)}\ \text{A circle with radius }3\text{ inches and center }4\text{ inches from }B\text{ along } BA \qquad\\
\textbf{(E)}\ \text{An ellipse with }A\text{ as focus}$
2008 Kyiv Mathematical Festival, 4
Let $ K,L,M$ and $ N$ be the midpoints of sides $ AB,$ $ BC,$ $ CD$ and $ AD$ of the convex quadrangle $ ABCD.$ Is it possible that points $ A,B,L,M,D$ lie on the same circle and points $ K,B,C,D,N$ lie on the same circle?
2006 Thailand Mathematical Olympiad, 4
Let $P$ be a point outside a circle centered at $O$. From $P$, tangent lines are drawn to the circle, touching the circle at points $A$ and $B$. Ray $\overrightarrow{BO}$ is drawn intersecting the circle again at $C$ and intersecting ray $\overrightarrow{PA}$ at $Q$. If $3QA = 2AP$, what is the value of $\sin \angle CAQ$?
2010 Contests, 2
Let $ABCD$ be a convex quadrilateral. Assume line $AB$ and $CD$ intersect at $E$, and $B$ lies between $A$ and $E$. Assume line $AD$ and $BC$ intersect at $F$, and $D$ lies between $A$ and $F$. Assume the circumcircles of $\triangle BEC$ and $\triangle CFD$ intersect at $C$ and $P$. Prove that $\angle BAP=\angle CAD$ if and only if $BD\parallel EF$.
2013 Sharygin Geometry Olympiad, 15
(a) Triangles $A_1B_1C_1$ and $A_2B_2C_2$ are inscribed into triangle $ABC$ so that $C_1A_1 \perp BC$, $A_1B_1 \perp CA$, $B_1C_1 \perp AB$, $B_2A_2 \perp BC$, $C_2B_2 \perp CA$, $A_2C_2 \perp AB$. Prove that these triangles are equal.
(b) Points $A_1$, $B_1$, $C_1$, $A_2$, $B_2$, $C_2$ lie inside a triangle $ABC$ so that $A_1$ is on segment $AB_1$, $B_1$ is on segment $BC_1$, $C_1$ is on segment $CA_1$, $A_2$ is on segment $AC_2$, $B_2$ is on segment $BA_2$, $C_2$ is on segment $CB_2$, and the angles $BAA_1$, $CBB_2$, $ACC_1$, $CAA_2$, $ABB_2$, $BCC_2$ are equal. Prove that the triangles $A_1B_1C_1$ and $A_2B_2C_2$ are equal.
2014 IPhOO, 10
An electric field varies according the the relationship, \[ \textbf{E} = \left( 0.57 \, \dfrac{\text{N}}{\text{C}} \right) \cdot \sin \left[ \left( 1720 \, \text{s}^{-1} \right) \cdot t \right]. \]Find the maximum displacement current through a $ 1.0 \, \text{m}^2 $ area perpendicular to $\vec{\mathbf{E}}$. Assume the permittivity of free space to be $ 8.85 \times 10^{-12} \, \text{F}/\text{m} $. Round to two significant figures.
[i]Problem proposed by Kimberly Geddes[/i]
2013 Middle European Mathematical Olympiad, 5
Let $ABC$ be and acute triangle. Construct a triangle $PQR$ such that $ AB = 2PQ $, $ BC = 2QR $, $ CA = 2 RP $, and the lines $ PQ, QR,$ and $RP$ pass through the points $ A, B , $ and $ C $, respectively. (All six points $ A, B, C, P, Q, $ and $ R $ are distinct.)