Olympiad problems

1998 IMO Shortlist, 7

Let $ABC$ be a triangle such that $\angle ACB=2\angle ABC$. Let $D$ be the point on the side $BC$ such that $CD=2BD$. The segment $AD$ is extended to $E$ so that $AD=DE$. Prove that \[ \angle ECB+180^{\circ }=2\angle EBC. \]

1991 Denmark MO - Mohr Contest, 2

Tags: trigonometry

Prove that for $0<x<\frac{\pi}{2}$, $$\sin x + \tan x > 2x$$

2013 NIMO Problems, 13

Tags: geometry , trapezoid , trigonometry

In trapezoid $ABCD$, $AD \parallel BC$ and $\angle ABC + \angle CDA = 270^{\circ}$. Compute $AB^2$ given that $AB \cdot \tan(\angle BCD) = 20$ and $CD = 13$. [i]Proposed by Lewis Chen[/i]

1991 Arnold's Trivium, 63

Tags: trigonometry

Is there a solution of the Cauchy problem $y\partial u/\partial x+\sin x\partial u/\partial y=y$, $u|_{x=0}=y^4$ on the whole $(x,y)$ plane? Is it unique?

2007 Moldova Team Selection Test, 2

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

If $I$ is the incenter of a triangle $ABC$ and $R$ is the radius of its circumcircle then \[AI+BI+CI\leq 3R\]

2008 All-Russian Olympiad, 6

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

In a scalene triangle $ ABC$ the altitudes $ AA_{1}$ and $ CC_{1}$ intersect at $ H, O$ is the circumcenter, and $ B_{0}$ the midpoint of side $ AC$. The line $ BO$ intersects side $ AC$ at $ P$, while the lines $ BH$ and $ A_{1}C_{1}$ meet at $ Q$. Prove that the lines $ HB_{0}$ and $ PQ$ are parallel.

2011 Spain Mathematical Olympiad, 1

Tags: ratio , trapezoid , trigonometry , perpendicular bisector , angle bisector , geometry

In triangle $ABC$, $\angle B=2\angle C$ and $\angle A>90^\circ$. Let $D$ be the point on the line $AB$ such that $CD$ is perpendicular to $AC$, and let $M$ be the midpoint of $BC$. Prove that $\angle AMB=\angle DMC$.

2010 Today's Calculation Of Integral, 668

Tags: calculus , integration , trigonometry , geometry , geometric transformation , rotation , calculus computations

Consider two curves $y=\sin x,\ y=\sin 2x$ in $0\leq x\leq 2\pi$. (1) Let $(\alpha ,\ \beta)\ (0<\alpha <\pi)$ be the intersection point of the curves. If $\sin x-\sin 2x$ has a local minimum at $x=x_1$ and a local maximum at $x=x_2$, then find the values of $\cos x_1,\ \cos x_1\cos x_2$. (2) Find the area enclosed by the curves, then find the volume of the part generated by a rotation of the part of $\alpha \leq x\leq \pi$ for the figure about the line $y=-1$. [i]2011 Kyorin　University entrance exam/Medicine [/i]

2003 AIME Problems, 11

Tags: probability , trigonometry , symmetry , inequalities , algebra , function , domain

An angle $x$ is chosen at random from the interval $0^\circ < x < 90^\circ$. Let $p$ be the probability that the numbers $\sin^2 x$, $\cos^2 x$, and $\sin x \cos x$ are not the lengths of the sides of a triangle. Given that $p = d/n$, where $d$ is the number of degrees in $\arctan m$ and $m$ and $n$ are positive integers with $m + n < 1000$, find $m + n$.

1985 IMO, 1

Tags: trigonometry , geometry , circumcircle , cyclic quadrilateral

A circle has center on the side $AB$ of the cyclic quadrilateral $ABCD$. The other three sides are tangent to the circle. Prove that $AD+BC=AB$.

2019 Moldova Team Selection Test, 2

Tags: trigonometry , easy , geometry

Prove that $E_n=\frac{\arccos {\frac{n-1}{n}} } {\text{arccot} {\sqrt{2n-1} }}$ is a natural number for any natural number $n$. (A natural number is a positive integer)

2003 IMC, 2

Tags: calculus , integration , limit , trigonometry , inequalities , real analysis , logarithm

Evaluate $\lim_{x\rightarrow 0^+}\int^{2x}_x\frac{\sin^m(t)}{t^n}dt$. ($m,n\in\mathbb{N}$)

2012 Today's Calculation Of Integral, 833

Tags: calculus , integration , trigonometry , derivative , complex number , calculus computations

Let $f(x)=\int_0^{x} e^{t} (\cos t+\sin t)\ dt,\ g(x)=\int_0^{x} e^{t} (\cos t-\sin t)\ dt.$ For a real number $a$, find $\sum_{n=1}^{\infty} \frac{e^{2a}}{\{f^{(n)}(a)\}^2+\{g^{(n)}(a)\}^2}.$

2008 AIME Problems, 13

Tags: geometry , geometric transformation , reflection , calculus , trigonometry , integration , analytic geometry

A regular hexagon with center at the origin in the complex plane has opposite pairs of sides one unit apart. One pair of sides is parallel to the imaginary axis. Let $ R$ be the region outside the hexagon, and let $ S\equal{}\{\frac{1}{z}|z\in R\}$. Then the area of $ S$ has the form $ a\pi\plus{}\sqrt{b}$, where $ a$ and $ b$ are positive integers. Find $ a\plus{}b$.

2010 Today's Calculation Of Integral, 567

Tags: calculus , integration , analytic geometry , geometry , trigonometry , calculus computations

Let $ a$ be a positive real numbers. In the coordinate plane denote by $ S$ the area of the figure bounded by the curve $ y=\sin x\ (0\leq x\leq \pi)$ and the $x$-axis and denote $T$ by the area of the figure bounded by the curves $y=\sin x\ \left(0\leq x\leq \frac{\pi}{2}\right),\ y=a\cos x\ \left(0\leq x\leq \frac{\pi}{2}\right)$ and the $x$-axis. Find the value of $a$ such that $ S: T=3: 1$.

1975 IMO Shortlist, 8

Tags: geometry , trigonometry , triangle , angle

In the plane of a triangle $ABC,$ in its exterior$,$ we draw the triangles $ABR, BCP, CAQ$ so that $\angle PBC = \angle CAQ = 45^{\circ}$, $\angle BCP = \angle QCA = 30^{\circ}$, $\angle ABR = \angle RAB = 15^{\circ}$. Prove that [b]a.)[/b] $\angle QRP = 90\,^{\circ},$ and [b]b.)[/b] $QR = RP.$

2007 Today's Calculation Of Integral, 210

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

Evaluate $\int_{1}^{\pi}\left(x^{3}\ln x-\frac{6}{x}\right)\sin x\ dx$.

2019 IFYM, Sozopol, 4

Tags: geometry , trigonometry

For a quadrilateral $ABCD$ is given that $\angle CBD=2\angle ADB$, $\angle ABD=2\angle CDB$, and $AB=CB$. Prove that $AD=CD$.

1984 AIME Problems, 8

Tags: trigonometry

The equation $z^6 + z^3 + 1$ has one complex root with argument $\theta$ between $90^\circ$ and $180^\circ$ in the complex plane. Determine the degree measure of $\theta$.

1971 Vietnam National Olympiad, 1

Tags: number theory , relatively prime , trigonometry

Consider positive integers $m <n,p < q$ such that $(m, n) = 1, (p, q) = 1$ and satisfy the condition that if $\frac{m}{n}= tan\alpha$ and $\frac{p}{q} = tan\beta$, then $\alpha + \beta = 45^o$. i) Given $m, n$, find $p, q$. ii) Given $n, q$, find $m, p$. ii) Given $m, q$, find $n, p$.

1978 Romania Team Selection Test, 4

Tags: equation , trigonometry , trigonometric equation , algebra

Solve the equation $ \sin x\sin 2x\cdots\sin nx+\cos x\cos 2x\cdots\cos nx =1, $ for $ n\in\mathbb{N} $ and $ x\in\mathbb{R} . $

2013 Turkey Team Selection Test, 2

Tags: incenter , ratio , trigonometry , geometry

Let the incircle of the triangle $ABC$ touch $[BC]$ at $D$ and $I$ be the incenter of the triangle. Let $T$ be midpoint of $[ID]$. Let the perpendicular from $I$ to $AD$ meet $AB$ and $AC$ at $K$ and $L$, respectively. Let the perpendicular from $T$ to $AD$ meet $AB$ and $AC$ at $M$ and $N$, respectively. Show that $|KM|\cdot |LN|=|BM|\cdot|CN|$.

2022 ISI Entrance Examination, 8

Tags: trigonometry , algebra , inequalities

Find the minimum value of $$\big|\sin x+\cos x+\tan x+\cot x+\sec x+\operatorname{cosec}x\big|$$ for real numbers $x$ not multiple of $\frac{\pi}{2}$.

1994 Korea National Olympiad, Problem 2

Tags: trigonometric inequality , trigonometry , inequality , geometry , algebra , inequalities

Let $ \alpha,\beta,\gamma$ be the angles of a triangle. Prove that $csc^2\frac{\alpha}{2}+csc^2\frac{\beta}{2}+csc^2\frac{\gamma}{2} \ge 12$ and find the conditions for equality.

2007 Purple Comet Problems, 23

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

Two circles with radius $2$ and radius $4$ have a common center at P. Points $A, B,$ and $C$ on the larger circle are the vertices of an equilateral triangle. Point $D$ is the intersection of the smaller circle and the line segment $PB$. Find the square of the area of triangle $ADC$.