Olympiad problems

2014 Indonesia MO, 3

Tags: geometry , trapezoid , parallelogram , trigonometry , geometric transformation , dilation , logarithm

Let $ABCD$ be a trapezoid (quadrilateral with one pair of parallel sides) such that $AB < CD$. Suppose that $AC$ and $BD$ meet at $E$ and $AD$ and $BC$ meet at $F$. Construct the parallelograms $AEDK$ and $BECL$. Prove that $EF$ passes through the midpoint of the segment $KL$.

2014 AMC 10, 21

Tags: trapezoid , trigonometry , geometry

Trapezoid $ABCD$ has parallel sides $\overline{AB}$ or length $33$ and $\overline{CD}$ of length $21$. The other two sides are of lengths $10$ and $14$. The angles at $A$ and $B$ are acute. What is the length of the shorter diagonal of $ABCD$? $ \textbf {(A) } 10\sqrt{6} \qquad \textbf {(B) } 25 \qquad \textbf {(C) } 8\sqrt{10} \qquad \textbf {(D) } 18\sqrt{2} \qquad \textbf {(E) } 26 $

2024 JHMT HS, 14

Tags: trigonometry

Compute \[ \frac{1}{2}\sin\frac{3\pi}{7}+\sin\frac{2\pi}{7}\cos\frac{3\pi}{7}. \]

1988 IMO Longlists, 37

Tags: geometry , 3d geometry , tetrahedron , trigonometry , inradius , incenter , sphere

[b]i.)[/b] Four balls of radius 1 are mutually tangent, three resting on the floor and the fourth resting on the others. A tedrahedron, each of whose edges has length $ s,$ is circumscribed around the balls. Find the value of $ s.$ [b]ii.)[/b] Suppose that $ ABCD$ and $ EFGH$ are opposite faces of a retangular solid, with $ \angle DHC \equal{} 45^{\circ}$ and $ \angle FHB \equal{} 60^{\circ}.$ Find the cosine of $ \angle BHD.$

2014 Math Prize For Girls Problems, 20

Tags: trigonometry , ceiling function , absolute value

How many complex numbers $z$ such that $\left| z \right| < 30$ satisfy the equation \[ e^z = \frac{z - 1}{z + 1} \, ? \]

2011 ISI B.Stat Entrance Exam, 10

Tags: trigonometry , geometry , circumcircle

Show that the triangle whose angles satisfy the equality \[\frac{\sin^2A+\sin^2B+\sin^2C}{\cos^2A+\cos^2B+\cos^2C} = 2\] is right angled.

2002 National High School Mathematics League, 12

Tags: trigonometry

For all $x\in\mathbb{R}$, $\sin^2 x+a\cos x+a^2\geq 1+\cos x$, then the range value of negative number $a$ is________

2001 Brazil National Olympiad, 4

Tags: trigonometry , function , inequalities , algebra

A calculator treats angles as radians. It initially displays 1. What is the largest value that can be achieved by pressing the buttons cos or sin a total of 2001 times? (So you might press cos five times, then sin six times and so on with a total of 2001 presses.)

2007 Pre-Preparation Course Examination, 3

Tags: trigonometry , inequalities , circumcircle , geometry

$ABC$ is an arbitrary triangle. $A',B',C'$ are midpoints of arcs $BC, AC, AB$. Sides of triangle $ABC$, intersect sides of triangle $A'B'C'$ at points $P,Q,R,S,T,F$. Prove that \[\frac{S_{PQRSTF}}{S_{ABC}}=1-\frac{ab+ac+bc}{(a+b+c)^{2}}\]

2013 Stanford Mathematics Tournament, 9

Tags: calculus , integration , trigonometry

Evaluate $\int_{0}^{\pi/2}\frac{dx}{\left(\sqrt{\sin x}+\sqrt{\cos x}\right)^4}$.

2020 India National Olympiad, 2

Tags: polynomial , trigonometry

Suppose $P(x)$ is a polynomial with real coefficients, satisfying the condition $P(\cos \theta+\sin \theta)=P(\cos \theta-\sin \theta)$, for every real $\theta$. Prove that $P(x)$ can be expressed in the form$$P(x)=a_0+a_1(1-x^2)^2+a_2(1-x^2)^4+\dots+a_n(1-x^2)^{2n}$$for some real numbers $a_0, a_1, \dots, a_n$ and non-negative integer $n$. [i]Proposed by C.R. Pranesacher[/i]

2014 AIME Problems, 11

Tags: ratio , analytic geometry , trigonometry , number theory , relatively prime , pythagorean theorem , geometry

In $\triangle RED, RD =1, \angle DRE = 75^\circ$ and $\angle RED = 45^\circ$. Let $M$ be the midpoint of segment $\overline{RD}$. Point $C$ lies on side $\overline{ED}$ such that $\overline{RC} \perp \overline{EM}$. Extend segment $\overline{DE}$ through $E$ to point $A$ such that $CA = AR$. Then $AE = \tfrac{a-\sqrt{b}}{c},$ where $a$ and $c$ are relatively prime positive integers, and $b$ is a positive integer. Find $a+b+c$.

Today's calculation of integrals, 851

Tags: calculus , integration , function , trigonometry , limit , geometric series , calculus computations

Let $T$ be a period of a function $f(x)=|\cos x|\sin x\ (-\infty,\ \infty).$ Find $\lim_{n\to\infty} \int_0^{nT} e^{-x}f(x)\ dx.$

2005 France Team Selection Test, 2

Tags: geometry , circumcircle , perimeter , inradius , inequalities , trigonometry , pythagorean theorem

Two right angled triangles are given, such that the incircle of the first one is equal to the circumcircle of the second one. Let $S$ (respectively $S'$) be the area of the first triangle (respectively of the second triangle). Prove that $\frac{S}{S'}\geq 3+2\sqrt{2}$.

1998 Belarus Team Selection Test, 4

Tags: circumcircle , trigonometry , triangle , easy geometry , geometry

The altitudes through the vertices $ A,B,C$ of an acute-angled triangle $ ABC$ meet the opposite sides at $ D,E, F,$ respectively. The line through $ D$ parallel to $ EF$ meets the lines $ AC$ and $ AB$ at $ Q$ and $ R,$ respectively. The line $ EF$ meets $ BC$ at $ P.$ Prove that the circumcircle of the triangle $ PQR$ passes through the midpoint of $ BC.$

2012 National Olympiad First Round, 29

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

Let $D$ and $E$ be points on $[BC]$ and $[AC]$ of acute $\triangle ABC$, respectively. $AD$ and $BE$ meet at $F$. If $|AF|=|CD|=2|BF|=2|CE|$, and $Area(\triangle ABF) = Area(\triangle DEC)$, then $Area(\triangle AFC)/Area(\triangle BFC) = ?$ $ \textbf{(A)}\ 4 \qquad \textbf{(B)}\ 2\sqrt2 \qquad \textbf{(C)}\ 2 \qquad \textbf{(D)}\ \sqrt2 \qquad \textbf{(E)}\ 1$

1999 Baltic Way, 4

Tags: quadratic , trigonometry , algebra

For all positive real numbers $x$ and $y$ let \[f(x,y)=\min\left( x,\frac{y}{x^2+y^2}\right) \] Show that there exist $x_0$ and $y_0$ such that $f(x, y)\le f(x_0, y_0)$ for all positive $x$ and $y$, and find $f(x_0,y_0)$.

2013 Sharygin Geometry Olympiad, 6

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

The altitudes $AA_1, BB_1, CC_1$ of an acute triangle $ABC$ concur at $H$. The perpendicular lines from $H$ to $B_1C_1, A_1C_1$ meet rays $CA, CB$ at $P, Q$ respectively. Prove that the line from $C$ perpendicular to $A_1B_1$ passes through the midpoint of $PQ$.

Today's calculation of integrals, 866

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

Given a solid $R$ contained in a semi cylinder with the hight $1$ which has a semicircle with radius $1$ as the base. The cross section at the hight $x\ (0\leq x\leq 1)$ is the form combined with two right-angled triangles as attached figure as below. Answer the following questions. (1) Find the cross-sectional area $S(x)$ at the hight $x$. (2) Find the volume of $R$. If necessary, when you integrate, set $x=\sin t.$

1996 Taiwan National Olympiad, 1

Tags: trigonometry , number theory

Suppose that $a,b,c$ are real numbers in $(0,\frac{\pi}{2})$ such that $a+b+c=\frac{\pi}{4}$ and $\tan{a}=\frac{1}{x},\tan{b}=\frac{1}{y},\tan{c}=\frac{1}{z}$ , where $x,y,z$ are positive integer numbers. Find $x,y,z$.

2002 Iran MO (3rd Round), 7

Tags: trigonometry , geometry

In triangle $ABC$, $AD$ is angle bisector ($D$ is on $BC$) if $AB+AD=CD$ and $AC+AD=BC$, what are the angles of $ABC$?

2000 Polish MO Finals, 1

Tags: trigonometry , function , algebra

Find number of solutions in non-negative reals to the following equations: \begin{eqnarray*}x_1 + x_n ^2 = 4x_n \\ x_2 + x_1 ^2 = 4x_1 \\ ... \\ x_n + x_{n-1}^2 = 4x_{n-1} \end{eqnarray*}

2024 AMC 12/AHSME, 8

Tags: trigonometry , logarithm

How many angles $\theta$ with $0\le\theta\le2\pi$ satisfy $\log(\sin(3\theta))+\log(\cos(2\theta))=0$? $ \textbf{(A) }0 \qquad \textbf{(B) }1 \qquad \textbf{(C) }2 \qquad \textbf{(D) }3 \qquad \textbf{(E) }4 \qquad $

the 11th XMO, 9

Tags: inequalities , trigonometry , algebra

$x,y\in\mathbb{R},(4x^3-3x)^2+(4y^3-3y)^2=1.\text { Find the maximum of } x+y.$

2000 National Olympiad First Round, 21

Tags: geometry , trigonometry , cyclic quadrilateral , pythagorean theorem

Let $ABCD$ be a cyclic quadrilateral with $|AB|=26$, $|BC|=10$, $m(\widehat{ABD})=45^\circ$,$m(\widehat{ACB})=90^\circ$. What is the area of $\triangle DAC$ ? $ \textbf{(A)}\ 120 \qquad\textbf{(B)}\ 108 \qquad\textbf{(C)}\ 90 \qquad\textbf{(D)}\ 84 \qquad\textbf{(E)}\ 80 $