Olympiad problems

2009 Today's Calculation Of Integral, 474

Calculate the following indefinite integrals. (1) $ \int \frac {3x \plus{} 4}{x^2 \plus{} 3x \plus{} 2}dx$ (2) $ \int \sin 2x\cos 2x\cos 4x\ dx$ (3) $ \int xe^{x}dx$ (4) $ \int 5^{x}dx$

2014 ELMO Shortlist, 9

Tags: geometry , circumcircle , trigonometry , parallelogram , homothety , geometric transformation

Let $P$ be a point inside a triangle $ABC$ such that $\angle PAC= \angle PCB$. Let the projections of $P$ onto $BC$, $CA$, and $AB$ be $X,Y,Z$ respectively. Let $O$ be the circumcenter of $\triangle XYZ$, $H$ be the foot of the altitude from $B$ to $AC$, $N$ be the midpoint of $AC$, and $T$ be the point such that $TYPO$ is a parallelogram. Show that $\triangle THN$ is similar to $\triangle PBC$. [i]Proposed by Sammy Luo[/i]

Today's calculation of integrals, 883

Tags: trigonometry , logarithm , calculus computations , integral inequality , integration , calculus

Prove that for each positive integer $n$ \[\frac{4n^2+1}{4n^2-1}\int_0^{\pi} (e^{x}-e^{-x})\cos 2nx\ dx>\frac{e^{\pi}-e^{-\pi}-2}{4}\ln \frac{(2n+1)^2}{(2n-1)(n+3)}.\]

2003 China National Olympiad, 3

Tags: trigonometry , inequalities

Given a positive integer $n$, find the least $\lambda>0$ such that for any $x_1,\ldots x_n\in \left(0,\frac{\pi}{2}\right)$, the condition $\prod_{i=1}^{n}\tan x_i=2^{\frac{n}{2}}$ implies $\sum_{i=1}^{n}\cos x_i\le\lambda$. [i]Huang Yumin[/i]

2013 AMC 12/AHSME, 21

Tags: trigonometry , logarithm , calculus

Consider \[A = \log (2013 + \log (2012 + \log (2011 + \log (\cdots + \log (3 + \log 2) \cdots )))).\] Which of the following intervals contains $ A $? $ \textbf{(A)} \ (\log 2016, \log 2017) $ $ \textbf{(B)} \ (\log 2017, \log 2018) $ $ \textbf{(C)} \ (\log 2018, \log 2019) $ $ \textbf{(D)} \ (\log 2019, \log 2020) $ $ \textbf{(E)} \ (\log 2020, \log 2021) $

2011 Uzbekistan National Olympiad, 3

Tags: trigonometry , circumcircle , trig identities , law of sines , geometry

Given an acute triangle $ABC$ with altituties AD and BE. O circumcinter of $ABC$.If o lies on the segment DE then find the value of $sinAsinBcosC$

1966 IMO Shortlist, 9

Tags: equation , trigonometry , trigonometric equation , algebra

Find $x$ such that trigonometric \[\frac{\sin 3x \cos (60^\circ -4x)+1}{\sin(60^\circ - 7x) - \cos(30^\circ + x) + m}=0\] where $m$ is a fixed real number.

1960 IMO, 3

Tags: trigonometry , geometry , circumcircle , trigonometric identities

In a given right triangle $ABC$, the hypotenuse $BC$, of length $a$, is divided into $n$ equal parts ($n$ and odd integer). Let $\alpha$ be the acute angel subtending, from $A$, that segment which contains the mdipoint of the hypotenuse. Let $h$ be the length of the altitude to the hypotenuse fo the triangle. Prove that: \[ \tan{\alpha}=\dfrac{4nh}{(n^2-1)a}. \]

1985 IMO Longlists, 39

Tags: geometry , trig identities , law of sines , trigonometry

Given a triangle $ABC$ and external points $X, Y$ , and $Z$ such that $\angle BAZ = \angle CAY , \angle CBX = \angle ABZ$, and $\angle ACY = \angle BCX$, prove that $AX,BY$ , and $CZ$ are concurrent.

2021 BMT, 15

Tags: trigonometry

Compute $$\frac{\cos \left(\frac{\pi}{12}\right)\cos \left(\frac{\pi}{24}\right)\cos \left(\frac{\pi}{48}\right)\cos \left(\frac{\pi}{96}\right)...}{\cos \left(\frac{\pi}{4}\right)\cos \left(\frac{\pi}{8}\right)\cos \left(\frac{\pi}{16}\right)\cos \left(\frac{\pi}{32}\right)...}$$

1997 Turkey Team Selection Test, 1

Tags: trigonometry , geometry

A convex $ABCDE$ is inscribed in a unit circle, $AE$ being its diameter. If $AB = a$, $BC = b$, $CD = c$, $DE = d$ and $ab = cd =\frac{1}{4}$, compute $AC + CE$ in terms of $a, b, c, d.$

2011 Iran Team Selection Test, 1

Tags: circumcircle , trigonometry , geometry

In acute triangle $ABC$ angle $B$ is greater than$C$. Let $M$ is midpoint of $BC$. $D$ and $E$ are the feet of the altitude from $C$ and $B$ respectively. $K$ and $L$ are midpoint of $ME$ and $MD$ respectively. If $KL$ intersect the line through $A$ parallel to $BC$ in $T$, prove that $TA=TM$.

2011 Bosnia Herzegovina Team Selection Test, 2

Tags: number theory , trigonometry , geometry

On semicircle, with diameter $|AB|=d$, are given points $C$ and $D$ such that: $|BC|=|CD|=a$ and $|DA|=b$ where $a, b, d$ are different positive integers. Find minimum possible value of $d$

2010 IberoAmerican Olympiad For University Students, 2

Tags: limit , real analysis , trigonometry

Calculate the sum of the series $\sum_{-\infty}^{\infty}\frac{\sin^33^k}{3^k}$.

2013 India IMO Training Camp, 2

Tags: parallelogram , geometric transformation , trigonometry , homothety , reflection , circumcircle , geometry

In a triangle $ABC$, with $\widehat{A} > 90^\circ$, let $O$ and $H$ denote its circumcenter and orthocenter, respectively. Let $K$ be the reflection of $H$ with respect to $A$. Prove that $K, O$ and $C$ are collinear if and only if $\widehat{A} - \widehat{B} = 90^\circ$.

Today's calculation of integrals, 871

Tags: calculus computations , logarithm , limit , trigonometry , definite integral , integration , calculus

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 $\lim_{n\to\infty} \frac 1{n}\ln (na_n).$

2009 Today's Calculation Of Integral, 499

Tags: function , symmetry , integration , trigonometry , calculus , calculus computations

Evaluate \[ \int_0^{\pi} (\sqrt[2009]{\cos x}\plus{}\sqrt[2009]{\sin x}\plus{}\sqrt[2009]{\tan x})\ dx.\]

III Soros Olympiad 1996 - 97 (Russia), 11.2

Tags: inequalities , trigonometry , algebra

Find the smallest value of the expression: $$y=\frac{x^2}{8}+x \cos x +\cos 2x$$

2009 Sharygin Geometry Olympiad, 15

Tags: geometry , trigonometry

Given a circle and a point $ C$ not lying on this circle. Consider all triangles $ ABC$ such that points $ A$ and $ B$ lie on the given circle. Prove that the triangle of maximal area is isosceles.

2008 ISI B.Math Entrance Exam, 8

Tags: quadratic , trigonometry , algebra

Let $a^2+b^2=1$ , $c^2+d^2=1$ , $ac+bd=0$ Prove that $a^2+c^2=1$ , $b^2+d^2=1$ , $ab+cd=0$ .

2005 USA Team Selection Test, 6

Tags: symmetry , circumcircle , geometry , geometric transformation , trigonometry , reflection

Let $ABC$ be an acute scalene triangle with $O$ as its circumcenter. Point $P$ lies inside triangle $ABC$ with $\angle PAB = \angle PBC$ and $\angle PAC = \angle PCB$. Point $Q$ lies on line $BC$ with $QA = QP$. Prove that $\angle AQP = 2\angle OQB$.

2000 India National Olympiad, 1

Tags: reflection , parallelogram , geometric transformation , asymptote , trigonometry , circumcircle , geometry

The incircle of $ABC$ touches $BC$, $CA$, $AB$ at $K$, $L$, $M$ respectively. The line through $A$ parallel to $LK$ meets $MK$ at $P$, and the line through $A$ parallel to $MK$ meets $LK$ at $Q$. Show that the line $PQ$ bisects $AB$ and bisects $AC$.

1999 Polish MO Finals, 3

Tags: geometry , trigonometry , complex number , analytic geometry

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 \[ \frac{BC}{CA} \cdot \frac{AE}{EF} \cdot \frac{FD}{DB} = 1. \]

2015 Moldova Team Selection Test, 2

Tags: trigonometry , algebra

Prove the equality:\\ $\tan (\frac{3\pi}{7})-4\sin (\frac{\pi}{7})= \sqrt{7}$ .

2013 CentroAmerican, 2

Tags: trigonometry , angle bisector , trig identities , law of sines , geometry

Let $ABC$ be an acute triangle and let $\Gamma$ be its circumcircle. The bisector of $\angle{A}$ intersects $BC$ at $D$, $\Gamma$ at $K$ (different from $A$), and the line through $B$ tangent to $\Gamma$ at $X$. Show that $K$ is the midpoint of $AX$ if and only if $\frac{AD}{DC}=\sqrt{2}$.