Olympiad problems

2005 China National Olympiad, 1

Tags: trigonometry , inequalities

Suppose $\theta_{i}\in(-\frac{\pi}{2},\frac{\pi}{2}), i = 1,2,3,4$. Prove that, there exist $x\in \mathbb{R}$, satisfying two inequalities \begin{eqnarray*} \cos^2\theta_1\cos^2\theta_2-(\sin\theta\sin\theta_2-x)^2 &\geq& 0, \\ \cos^2\theta_3\cos^2\theta_4-(\sin\theta_3\sin\theta_4-x)^2 & \geq & 0 \end{eqnarray*} if and only if \[ \sum^4_{i=1}\sin^2\theta_i\leq2(1+\prod^4_{i=1}\sin\theta_i + \prod^4_{i=1}\cos\theta_i). \]

2009 China Team Selection Test, 2

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

In acute triangle $ ABC,$ points $ P,Q$ lie on its sidelines $ AB,AC,$ respectively. The circumcircle of triangle $ ABC$ intersects of triangle $ APQ$ at $ X$ (different from $ A$). Let $ Y$ be the reflection of $ X$ in line $ PQ.$ Given $ PX>PB.$ Prove that $ S_{\bigtriangleup XPQ}>S_{\bigtriangleup YBC}.$ Where $ S_{\bigtriangleup XYZ}$ denotes the area of triangle $ XYZ.$

2005 AIME Problems, 12

Tags: analytic geometry , trigonometry , quadratic , geometry , geometric transformation , rotation , reflection

Square $ABCD$ has center $O$, $AB=900$, $E$ and $F$ are on $AB$ with $AE<BF$ and $E$ between $A$ and $F$, $m\angle EOF =45^\circ$, and $EF=400$. Given that $BF=p+q\sqrt{r}$, wherer $p,q,$ and $r$ are positive integers and $r$ is not divisible by the square of any prime, find $p+q+r$.

2002 Romania National Olympiad, 3

Tags: trigonometry , algebra

Find all real numbers $a,b,c,d,e$ in the interval $[-2,2]$, that satisfy: \begin{align*}a+b+c+d+e &= 0\\ a^3+b^3+c^3+d^3+e^3&= 0\\ a^5+b^5+c^5+d^5+e^5&=10 \end{align*}

2008 IberoAmerican, 2

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

Given a triangle $ ABC$, let $ r$ be the external bisector of $ \angle ABC$. $ P$ and $ Q$ are the feet of the perpendiculars from $ A$ and $ C$ to $ r$. If $ CP \cap BA \equal{} M$ and $ AQ \cap BC\equal{}N$, show that $ MN$, $ r$ and $ AC$ concur.

2005 National Olympiad First Round, 33

Tags: geometry , trigonometry , cyclic quadrilateral

Let $K$ be the intersection of diagonals of cyclic quadrilateral $ABCD$, where $|AB|=|BC|$, $|BK|=b$, and $|DK|=d$. What is $|AB|$? $ \textbf{(A)}\ \sqrt{d^2 + bd} \qquad\textbf{(B)}\ \sqrt{b^2+bd} \qquad\textbf{(C)}\ \sqrt{2bd} \qquad\textbf{(D)}\ \sqrt{2(b^2+d^2-bd)} \qquad\textbf{(E)}\ \sqrt{bd} $

Today's calculation of integrals, 856

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

On the coordinate plane, find the area of the part enclosed by the curve $C: (a+x)y^2=(a-x)x^2\ (x\geq 0)$ for $a>0$.

2010 Today's Calculation Of Integral, 637

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

For a non negative integer $n$, set t $I_n=\int_0^{\frac{\pi}{4}} \tan ^ n x\ dx$ to answer the following questions: (1) Calculate $I_{n+2}+I_n.$ (2) Evaluate the values of $I_1,\ I_2$ and $I_3.$ 1978 Niigata university entrance exam

2010 Postal Coaching, 6

Tags: trigonometry , geometry

Let $a,b,c$ denote the sides of a triangle and $[ABC]$ the area of the triangle as usual. $(a)$ If $6[ABC] = 2a^2+bc$, determine $A,B,C$. $(b)$ For all triangles, prove that $3a^2+3b^2 - c^2 \ge 4 \sqrt{3} [ABC]$.

2005 iTest, 19

Tags: trigonometry

Find the amplitude of $y = 4 \sin (x) + 3 \cos (x)$.

2005 IMO Shortlist, 5

Tags: geometry , circumcircle , trigonometry , spiral similarity , miquel point

Let $\triangle ABC$ be an acute-angled triangle with $AB \not= AC$. Let $H$ be the orthocenter of triangle $ABC$, and let $M$ be the midpoint of the side $BC$. Let $D$ be a point on the side $AB$ and $E$ a point on the side $AC$ such that $AE=AD$ and the points $D$, $H$, $E$ are on the same line. Prove that the line $HM$ is perpendicular to the common chord of the circumscribed circles of triangle $\triangle ABC$ and triangle $\triangle ADE$.

1984 Canada National Olympiad, 5

Tags: trigonometry , inequalities , algebra

Given any $7$ real numbers, prove that there are two of them $x,y$ such that $0\le\frac{x-y}{1+xy}\le\frac{1}{\sqrt{3}}$.

2014 AIME Problems, 9

Tags: algebra , polynomial , vieta , quadratic , trigonometry

Let $x_1<x_2<x_3$ be three real roots of equation $\sqrt{2014}x^3-4029x^2+2=0$. Find $x_2(x_1+x_3)$.

2013 China Western Mathematical Olympiad, 6

Tags: trigonometry , geometry

Let $PA, PB$ be tangents to a circle centered at $O$, and $C$ a point on the minor arc $AB$. The perpendicular from $C$ to $PC$ intersects internal angle bisectors of $AOC,BOC$ at $D,E$. Show that $CD=CE$

2012 IMO Shortlist, G4

Tags: geometry , circumcircle , trigonometry , triangle , reflection

Let $ABC$ be a triangle with $AB \neq AC$ and circumcenter $O$. The bisector of $\angle BAC$ intersects $BC$ at $D$. Let $E$ be the reflection of $D$ with respect to the midpoint of $BC$. The lines through $D$ and $E$ perpendicular to $BC$ intersect the lines $AO$ and $AD$ at $X$ and $Y$ respectively. Prove that the quadrilateral $BXCY$ is cyclic.

2014 Math Prize For Girls Problems, 8

Tags: geometry , trigonometry , area of a triangle , heron's formula , trig identities , law of cosines

A triangle has sides of length $\sqrt{13}$, $\sqrt{17}$, and $2 \sqrt{5}$. Compute the area of the triangle.

2006 ISI B.Stat Entrance Exam, 1

Tags: trigonometry , analytic geometry , graphing lines , slope , calculus , calculus computations

If the normal to the curve $x^{\frac{2}{3}}+y^{\frac23}=a^{\frac23}$ at some point makes an angle $\theta$ with the $X$-axis, show that the equation of the normal is \[y\cos\theta-x\sin\theta=a\cos 2\theta\]

1981 IMO Shortlist, 11

Tags: inequalities , trigonometry , geometry , law of sines , quadrilateral

On a semicircle with unit radius four consecutive chords $AB,BC, CD,DE$ with lengths $a, b, c, d$, respectively, are given. Prove that \[a^2 + b^2 + c^2 + d^2 + abc + bcd < 4.\]

1973 AMC 12/AHSME, 4

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

Two congruent $ 30^{\circ}$-$ 60^{\circ}$-$ 90^{\circ}$ are placed so that they overlap partly and their hypotenuses coincide. If the hypotenuse of each triangle is 12, the area common to both triangles is $ \textbf{(A)}\ 6\sqrt3 \qquad \textbf{(B)}\ 8\sqrt3 \qquad \textbf{(C)}\ 9\sqrt3 \qquad \textbf{(D)}\ 12\sqrt3 \qquad \textbf{(E)}\ 24$

2017 Korea USCM, 8

Tags: differential equation , ordinary differential equation , trigonometry

$u(t)$ is solution of the following initial value problem. $$\begin{cases} u''(t) + u'(t) = \sin u(t) &\;\;(t>0),\\ u(0)=1,\;\; u'(0)=0 & \end{cases}$$ (1) Show that $u(t)$ and $u'(t)$ are bounded on $t>0$. (2) Find $\lim\limits_{t\to\infty} u(t)$ with proof.

2010 Belarus Team Selection Test, 8.3

Tags: geometry , incenter , circumcircle , trigonometry , inradius

Let $ABCD$ be a circumscribed quadrilateral. Let $g$ be a line through $A$ which meets the segment $BC$ in $M$ and the line $CD$ in $N$. Denote by $I_1$, $I_2$ and $I_3$ the incenters of $\triangle ABM$, $\triangle MNC$ and $\triangle NDA$, respectively. Prove that the orthocenter of $\triangle I_1I_2I_3$ lies on $g$. [i]Proposed by Nikolay Beluhov, Bulgaria[/i]

2013 Middle European Mathematical Olympiad, 6

Tags: geometry , circumcircle , trigonometry , geometric transformation , reflection , perpendicular bisector , power of a point

Let $K$ be a point inside an acute triangle $ ABC $, such that $ BC $ is a common tangent of the circumcircles of $ AKB $ and $ AKC$. Let $ D $ be the intersection of the lines $ CK $ and $ AB $, and let $ E $ be the intersection of the lines $ BK $ and $ AC $ . Let $ F $ be the intersection of the line $BC$ and the perpendicular bisector of the segment $DE$. The circumcircle of $ABC$ and the circle $k$ with centre $ F$ and radius $FD$ intersect at points $P$ and $Q$. Prove that the segment $PQ$ is a diameter of $k$.