Olympiad problems

1967 IMO Longlists, 49

Let $n$ and $k$ be positive integers such that $1 \leq n \leq N+1$, $1 \leq k \leq N+1$. Show that: \[ \min_{n \neq k} |\sin n - \sin k| < \frac{2}{N}. \]

1961 Poland - Second Round, 3

Tags: trigonometry

Prove that for any angles $x,y,z$ holds the equality $$1-\cos^2x-\cos^2y- y-\cos^2z +2 \cos x \cos y \cos z= 4 \sin \frac{x+y+z}{2} \sin \frac{x+y-z}{2} \sin \frac{x-y+z}{2} \sin\frac{-x-y+z}{2}. $$

2000 Flanders Math Olympiad, 4

Tags: trigonometry

Solve for $x \in [0,2\pi[$: \[\sin x < \cos x < \tan x < \cot x\]

1998 APMO, 3

Tags: inequalities , trigonometry , algebra

Let $a$, $b$, $c$ be positive real numbers. Prove that \[ \biggl(1+\frac{a}{b}\biggr) \biggl(1+\frac{b}{c}\biggr) \biggl(1+\frac{c}{a}\biggr) \ge 2 \biggl(1+\frac{a+b+c}{\sqrt[3]{abc}}\biggr). \]

2002 USA Team Selection Test, 1

Tags: inequalities , trigonometry , function , trigonometric inequality

Let $ ABC$ be a triangle, and $ A$, $ B$, $ C$ its angles. Prove that \[ \sin\frac{3A}{2}+\sin\frac{3B}{2}+\sin\frac{3C}{2}\leq \cos\frac{A-B}{2}+\cos\frac{B-C}{2}+\cos\frac{C-A}{2}. \]

2010 ISI B.Math Entrance Exam, 7

Tags: polynomial , trigonometry , algebra

We are given $a,b,c \in \mathbb{R}$ and a polynomial $f(x)=x^3+ax^2+bx+c$ such that all roots (real or complex) of $f(x)$ have same absolute value. Show that $a=0$ iff $b=0$.

2009 Today's Calculation Of Integral, 509

Tags: calculus , integration , trigonometry , algebra , partial fractions , calculus computations , logarithm

Evaluate $ \int_0^{\frac{\pi}{4}} \frac{\tan x}{1\plus{}\sin x}\ dx$.

2010 Today's Calculation Of Integral, 666

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

Let $f(x)$ be a function defined in $0<x<\frac{\pi}{2}$ satisfying: (i) $f\left(\frac{\pi}{6}\right)=0$ (ii) $f'(x)\tan x=\int_{\frac{\pi}{6}}^x \frac{2\cos t}{\sin t}dt$. Find $f(x)$. [i]1987 Sapporo Medical University entrance exam[/i]

1978 IMO Longlists, 6

Tags: algebra , polynomial , function , trigonometry , geometry

Prove that for all $X > 1$, there exists a triangle whose sides have lengths $P_1(X) = X^4+X^3+2X^2+X+1, P_2(X) = 2X^3+X^2+2X+1$, and $P_3(X) = X^4-1$. Prove that all these triangles have the same greatest angle and calculate it.

1992 India Regional Mathematical Olympiad, 4

Tags: geometry , circumcircle , trigonometry , parallelogram , rectangle , cyclic quadrilateral , power of a point

$ABCD$ is a cyclic quadrilateral with $AC \perp BD$; $AC$ meets $BD$ at $E$. Prove that \[ EA^2 + EB^2 + EC^2 + ED^2 = 4 R^2 \] where $R$ is the radius of the circumscribing circle.

1996 Turkey Team Selection Test, 2

Tags: parallelogram , trigonometry , geometry

In a parallelogram $ABCD$ with $\angle A < 90$, the circle with diameter $AC$ intersects the lines $CB$ and $CD$ again at $E$ and $F$ , and the tangent to this circle at $A$ meets the line $BD$ at $P$ . Prove that the points $P$, $E$, $F$ are collinear.

2000 IMO Shortlist, 4

Tags: trigonometry , geometry , cyclic , polygon

Let $ A_1A_2 \ldots A_n$ be a convex polygon, $ n \geq 4.$ Prove that $ A_1A_2 \ldots A_n$ is cyclic if and only if to each vertex $ A_j$ one can assign a pair $ (b_j, c_j)$ of real numbers, $ j = 1, 2, \ldots, n,$ so that $ A_iA_j = b_jc_i - b_ic_j$ for all $ i, j$ with $ 1 \leq i < j \leq n.$

2014 PUMaC Geometry A, 7

Tags: geometry , trigonometry , analytic geometry , trig identities , law of cosines

Let $O$ be the center of a circle of radius $26$, and let $A$, $B$ be two distinct points on the circle, with $M$ being the midpoint of $AB$. Consider point $C$ for which $CO=34$ and $\angle COM=15^\circ$. Let $N$ be the midpoint of $CO$. Suppose that $\angle ACB=90^\circ$. Find $MN$.

2011 Graduate School Of Mathematical Sciences, The Master Cource, The University Of Tokyo, 2

Tags: trigonometry , calculus , inequalities

Let $f(x,\ y)=\frac{x+y}{(x^2+1)(y^2+1)}.$ (1) Find the maximum value of $f(x,\ y)$ for $0\leq x\leq 1,\ 0\leq y\leq 1.$ (2) Find the maximum value of $f(x,\ y),\ \forall{x,\ y}\in{\mathbb{R}}.$

2013 Lusophon Mathematical Olympiad, 2

Tags: geometry , circumcircle , trigonometry , perpendicular bisector

Let $ABC$ be an acute triangle. The circumference with diameter $AB$ intersects sides $AC$ and $BC$ at $E$ and $F$ respectively. The tangent lines to the circumference at the points $E$ and $F$ meet at $P$. Show that $P$ belongs to the altitude from $C$ of triangle $ABC$.

2000 Estonia National Olympiad, 4

Tags: trigonometry , sidelenghts , angle

Prove that for any triangle the equation holds $a \cdot \cos (\beta + \gamma ) + b \cdot \cos (\gamma +\alpha) + c\cdot \cos (\alpha -\beta) = 0$, where $a, b, c$ are the sides of the triangle and $\alpha, \beta,\gamma$ according to their angles sizes of opposite angles.

2015 AMC 12/AHSME, 24

Tags: probability , trigonometry , function , complex number

Rational numbers $a$ and $b$ are chosen at random among all rational numbers in the interval $[0,2)$ that can be written as fractions $\tfrac nd$ where $n$ and $d$ are integers with $1\leq d\leq 5$. What is the probability that \[(\cos(a\pi)+i\sin(b\pi))^4\] is a real number? $\textbf{(A) }\dfrac3{50}\qquad\textbf{(B) }\dfrac4{25}\qquad\textbf{(C) }\dfrac{41}{200}\qquad\textbf{(D) }\dfrac6{25}\qquad\textbf{(E) }\dfrac{13}{50}$

2013 Irish Math Olympiad, 6

Tags: trigonometry

The three distinct points$ B, C, D$ are collinear with C between B and D. Another point A not on the line BD is such that $|AB| = |AC| = |CD|.$ Prove that ∠$BAC = 36$ if and only if $1/|CD|-1/|BD|=1/(|CD| + |BD|)$ .

2009 Serbia Team Selection Test, 1

Tags: trigonometry , calculus , derivative , angle bisector , geometry , inequalities

Let $ \alpha$ and $ \beta$ be the angles of a non-isosceles triangle $ ABC$ at points $ A$ and $ B$, respectively. Let the bisectors of these angles intersect opposing sides of the triangle in $ D$ and $ E$, respectively. Prove that the acute angle between the lines $ DE$ and $ AB$ isn't greater than $ \frac{|\alpha\minus{}\beta|}3$.

2011 Today's Calculation Of Integral, 693

Tags: calculus , integration , trigonometry , calculus computations

Evaluate $\int_0^{\pi} \sqrt[4]{1+|\cos x|}\ dx.$ created by kunny

2007 Vietnam Team Selection Test, 3

Tags: trigonometry , search , geometry , inequalities

Given a triangle $ABC$. Find the minimum of \[\frac{\cos^{2}\frac{A}{2}\cos^{2}\frac{B}{2}}{\cos^{2}\frac{C}{2}}+\frac{\cos^{2}\frac{B}{2}\cos^{2}\frac{C}{2}}{\cos^{2}\frac{A}{2}}+\frac{\cos^{2}\frac{C}{2}\cos^{2}\frac{A}{2}}{\cos^{2}\frac{B}{2}}. \]

2003 Purple Comet Problems, 24

Tags: trigonometry

In $\triangle ABC$, $\angle A = 30^{\circ}$ and $AB = AC = 16$ in. Let $D$ lie on segment $BC$ such that $\frac{DB}{DC} = \frac23$ . Let $E$ and $F$ be the orthogonal projections of $D$ onto $AB$ and $AC$, respectively. Find $DE + DF$ in inches.

2004 Thailand Mathematical Olympiad, 1

Tags: trigonometry

Given that $\cos 4A =\frac13$ and $-\frac{\pi}{4} \le A \le \frac{\pi}{4}$ , find the value of $\cos^8 A - \sin^8 A$.

2011 Tokyo Instutute Of Technology Entrance Examination, 1

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

Consider a curve $C$ on the $x$-$y$ plane expressed by $x=\tan \theta ,\ y=\frac{1}{\cos \theta}\left (0\leq \theta <\frac{\pi}{2}\right)$. For a constant $t>0$, let the line $l$ pass through the point $P(t,\ 0)$ and is perpendicular to the $x$-axis,intersects with the curve $C$ at $Q$. Denote by $S_1$ the area of the figure bounded by the curve $C$, the $x$-axis, the $y$-axis and the line $l$, and denote by $S_2$ the area of $\triangle{OPQ}$. Find $\lim_{t\to\infty} \frac{S_1-S_2}{\ln t}.$

2000 Baltic Way, 20

Tags: integration , trigonometry , algebra , logarithm

For every positive integer $n$, let \[x_n=\frac{(2n+1)(2n+3)\cdots (4n-1)(4n+1)}{(2n)(2n+2)\cdots (4n-2)(4n)}\] Prove that $\frac{1}{4n}<x_n-\sqrt{2}<\frac{2}{n}$.