Olympiad problems

1987 IMO, 2

In an acute-angled triangle $ABC$ the interior bisector of angle $A$ meets $BC$ at $L$ and meets the circumcircle of $ABC$ again at $N$. From $L$ perpendiculars are drawn to $AB$ and $AC$, with feet $K$ and $M$ respectively. Prove that the quadrilateral $AKNM$ and the triangle $ABC$ have equal areas.

2012 Today's Calculation Of Integral, 787

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

Take two points $A\ (-1,\ 0),\ B\ (1,\ 0)$ on the $xy$-plane. Let $F$ be the figure by which the whole points $P$ on the plane satisfies $\frac{\pi}{4}\leq \angle{APB}\leq \pi$ and the figure formed by $A,\ B$. Answer the following questions: (1) Illustrate $F$. (2) Find the volume of the solid generated by a rotation of $F$ around the $x$-axis.

2010 Today's Calculation Of Integral, 619

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

Consider a function $f(x)=\frac{\sin x}{9+16\sin ^ 2 x}\ \left(0\leq x\leq \frac{\pi}{2}\right).$ Let $a$ be the value of $x$ for which $f(x)$ is maximized. Evaluate $\int_a^{\frac{\pi}{2}} f(x)\ dx.$ [i]2010 Saitama University entrance exam/Mathematics[/i] Last Edited

1977 Polish MO Finals, 2

Tags: trigonometry , ratio , geometric sequence , geometry

Let $s \geq 3$ be a given integer. A sequence $K_n$ of circles and a sequence $W_n$ of convex $s$-gons satisfy: \[ K_n \supset W_n \supset K_{n+1} \] for all $n = 1, 2, ...$ Prove that the sequence of the radii of the circles $K_n$ converges to zero.

2009 Jozsef Wildt International Math Competition, W. 15

Tags: trigonometry , inequalities

Let a triangle $\triangle ABC$ and the real numbers $x$, $y$, $z>0$. Prove that $$x^n\cos\frac{A}{2}+y^n\cos\frac{B}{2}+z^n\cos\frac{C}{2}\geq (yz)^{\frac{n}{2}}\sin A +(zx)^{\frac{n}{2}}\sin B +(xy)^{\frac{n}{2}}\sin C$$

2002 Austrian-Polish Competition, 8

Tags: algebra , limit , trigonometry , inequalities

Determine the number of real solutions of the system \[\left\{ \begin{aligned}\cos x_{1}&= x_{2}\\ &\cdots \\ \cos x_{n-1}&= x_{n}\\ \cos x_{n}&= x_{1}\\ \end{aligned}\right.\]

1996 IMO Shortlist, 8

Tags: inequalities , trigonometry , circumcircle , geometry

Let $ ABCD$ be a convex quadrilateral, and let $ R_A, R_B, R_C, R_D$ denote the circumradii of the triangles $ DAB, ABC, BCD, CDA,$ respectively. Prove that $ R_A \plus{} R_C > R_B \plus{} R_D$ if and only if $ \angle A \plus{} \angle C > \angle B \plus{} \angle D.$

2013 AIME Problems, 13

Tags: trigonometry , number theory , trig identities , geometry , similar triangles , relatively prime , geometric sequence

Triangle $AB_0C_0$ has side lengths $AB_0 = 12$, $B_0C_0 = 17$, and $C_0A = 25$. For each positive integer $n$, points $B_n$ and $C_n$ are located on $\overline{AB_{n-1}}$ and $\overline{AC_{n-1}}$, respectively, creating three similar triangles $\triangle AB_nC_n \sim \triangle B_{n-1}C_nC_{n-1} \sim \triangle AB_{n-1}C_{n-1}$. The area of the union of all triangles $B_{n-1}C_nB_n$ for $n\geq1$ can be expressed as $\tfrac pq$, where $p$ and $q$ are relatively prime positive integers. Find $q$.

1969 Canada National Olympiad, 5

Tags: trigonometry , search , angle bisector , geometry , ratio

Let $ABC$ be a triangle with sides of length $a$, $b$ and $c$. Let the bisector of the angle $C$ cut $AB$ in $D$. Prove that the length of $CD$ is \[ \frac{2ab\cos \frac{C}{2}}{a+b}. \]

2010 Today's Calculation Of Integral, 532

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

For a curve $ C: y \equal{} x\sqrt {9 \minus{} x^2}\ (x\geq 0)$, (1) Find the maximum value of the function. (2) Find the area of the figure bounded by the curve $ C$ and the $ x$-axis. (3) Find the volume of the solid by revolution of the figure in (2) around the $ y$-axis. Please find the volume without using cylindrical shells for my students. Last Edited.

2008 Germany Team Selection Test, 2

Tags: geometry , trigonometry , inequalities , circumcircle

For three points $ X,Y,Z$ let $ R_{XYZ}$ be the circumcircle radius of the triangle $ XYZ.$ If $ ABC$ is a triangle with incircle centre $ I$ then we have: \[ \frac{1}{R_{ABI}} \plus{} \frac{1}{R_{BCI}} \plus{} \frac{1}{R_{CAI}} \leq \frac{1}{\bar{AI}} \plus{} \frac{1}{\bar{BI}} \plus{} \frac{1}{\bar{CI}}.\]

Today's calculation of integrals, 863

Tags: trigonometry , limit , derivative , calculus , integration , calculus computations , inequalities

For $0<t\leq 1$, let $F(t)=\frac{1}{t}\int_0^{\frac{\pi}{2}t} |\cos 2x|\ dx.$ (1) Find $\lim_{t\rightarrow 0} F(t).$ (2) Find the range of $t$ such that $F(t)\geq 1.$

1995 VJIMC, Problem 4

Tags: limit , real analysis , trigonometry

Let $\{x_n\}_{n=1}^\infty$ be a sequence such that $x_1=25$, $x_n=\operatorname{arctan}(x_{n-1})$. Prove that this sequence has a limit and find it.

1981 Vietnam National Olympiad, 1

Tags: geometry , perimeter , circumcircle , trigonometry , inradius

Prove that a triangle $ABC$ is right-angled if and only if \[\sin A + \sin B + \sin C = \cos A + \cos B + \cos C + 1\]

2016 NIMO Problems, 7

Tags: trigonometry

Determine the number of odd integers $1 \le n \le 100$ with the property that \[ \sum_{\substack{1 \le k \le n \\ \gcd(k,n) = 1}} \cos\left(\frac{2\pi k}{n} \right) = 1 \quad\text{and}\quad \sum_{\substack{1 \le k \le n \\ \gcd(k,n) = 1}} \sin\left(\frac{2\pi k}{n} \right) = 0. \] [i]Based on a proposal by Mayank Pandey[/i]

2012 Romania National Olympiad, 1

Tags: algebra , trigonometry

[color=darkred]Let $M=\{x\in\mathbb{C}\, |\, |z|=1,\ \text{Re}\, z\in\mathbb{Q}\}\, .$ Prove that there exist infinitely many equilateral triangles in the complex plane having all affixes of their vertices in the set $M$ .[/color]

PEN S Problems, 38

Tags: integration , function , calculus , trigonometry

The function $\mu: \mathbb{N}\to \mathbb{C}$ is defined by \[\mu(n) = \sum^{}_{k \in R_{n}}\left( \cos \frac{2k\pi}{n}+i \sin \frac{2k\pi}{n}\right),\] where $R_{n}=\{ k \in \mathbb{N}\vert 1 \le k \le n, \gcd(k, n)=1 \}$. Show that $\mu(n)$ is an integer for all positive integer $n$.

1996 AIME Problems, 10

Tags: trigonometry , algorithm , modular arithmetic , function

Find the smallest positive integer solution to $\tan 19x^\circ=\frac{\cos 96^\circ+\sin 96^\circ}{\cos 96^\circ-\sin 96^\circ}.$

2008 Greece National Olympiad, 3

Tags: trigonometry , geometry

A triangle $ABC$ with orthocenter $H$ is inscribed in a circle with center $K$ and radius $1$, where the angles at $B$ and $C$ are non-obtuse. If the lines $HK$ and $BC$ meet at point $S$ such that $SK(SK -SH) = 1$, compute the area of the concave quadrilateral $ABHC$.