Olympiad problems

2007 Moldova Team Selection Test, 1

Tags: inequalities , circumcircle , trigonometry , function , triangle inequality , geometry

Let $ABC$ be a triangle and $M,N,P$ be the midpoints of sides $BC, CA, AB$. The lines $AM, BN, CP$ meet the circumcircle of $ABC$ in the points $A_{1}, B_{1}, C_{1}$. Show that the area of triangle $ABC$ is at most the sum of areas of triangles $BCA_{1}, CAB_{1}, ABC_{1}$.

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 $x=\sin t.$

2005 Cono Sur Olympiad, 2

Tags: circumcircle , trigonometry , geometry

Let $ABC$ be an acute-angled triangle and let $AN$, $BM$ and $CP$ the altitudes with respect to the sides $BC$, $CA$ and $AB$, respectively. Let $R$, $S$ be the pojections of $N$ on the sides $AB$, $CA$, respectively, and let $Q$, $W$ be the projections of $N$ on the altitudes $BM$ and $CP$, respectively. (a) Show that $R$, $Q$, $W$, $S$ are collinear. (b) Show that $MP=RS-QW$.

1987 IMO Longlists, 38

Tags: 3d geometry , sphere , trigonometry , function , geometry

Let $S_1$ and $S_2$ be two spheres with distinct radii that touch externally. The spheres lie inside a cone $C$, and each sphere touches the cone in a full circle. Inside the cone there are $n$ additional solid spheres arranged in a ring in such a way that each solid sphere touches the cone $C$, both of the spheres $S_1$ and $S_2$ externally, as well as the two neighboring solid spheres. What are the possible values of $n$? [i]Proposed by Iceland.[/i]

2005 Estonia National Olympiad, 1

Tags: trigonometry , system of equations , algebra

Real numbers $x$ and $y$ satisfy the system of equalities $$\begin{cases} \sin x + \cos y = 1 \\ \cos x + \sin y = -1 \end{cases}$$ Prove that $\cos 2x = \cos 2y$.

2005 Peru MO (ONEM), 1

Tags: trigonometry , algebra

If $p = (1- \cos x)(1+ \sin x)$ and $q = (1+ \cos x)(1- \sin x)$, write the expression $$\cos^2 x - \cos^4 x - \sin2x + 2$$ in terms of $p$ and $q$.

2002 China Team Selection Test, 1

Tags: inequalities , trigonometry , geometry

In acute triangle $ ABC$, show that: $ \sin^3{A}\cos^2{(B \minus{} C)} \plus{} \sin^3{B}\cos^2{(C \minus{} A)} \plus{} \sin^3{C}\cos^2{(A \minus{} B)} \leq 3\sin{A} \sin{B} \sin{C}$ and find out when the equality holds.

1994 Korea National Olympiad, Problem 3

Tags: trigonometric identities , trigonometry , geometry

Let $\alpha,\beta ,\gamma$ be the angles of $\triangle ABC$. a) Show that $cos^2\alpha +cos^2\beta +cos^2 \gamma =1-2cos\alpha cos\beta cos\gamma$ . b) Given that $cos\alpha : cos\beta : cos\gamma = 39 : 33 : 25$, find $sin\alpha : sin\beta : sin\gamma$ .

2000 USA Team Selection Test, 6

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

Let $ ABC$ be a triangle inscribed in a circle of radius $ R$, and let $ P$ be a point in the interior of triangle $ ABC$. Prove that \[ \frac {PA}{BC^{2}} \plus{} \frac {PB}{CA^{2}} \plus{} \frac {PC}{AB^{2}}\ge \frac {1}{R}. \] [i]Alternative formulation:[/i] If $ ABC$ is a triangle with sidelengths $ BC\equal{}a$, $ CA\equal{}b$, $ AB\equal{}c$ and circumradius $ R$, and $ P$ is a point inside the triangle $ ABC$, then prove that $ \frac {PA}{a^{2}} \plus{} \frac {PB}{b^{2}} \plus{} \frac {PC}{c^{2}}\ge \frac {1}{R}$.

2010 Harvard-MIT Mathematics Tournament, 9

Tags: calculus , trigonometry

Let $x(t)$ be a solution to the differential equation \[\left(x+x^\prime\right)^2+x\cdot x^{\prime\prime}=\cos t\] with $x(0)=x^\prime(0)=\sqrt{\frac{2}{5}}$. Compute $x\left(\dfrac{\pi}{4}\right)$.

1987 India National Olympiad, 9

Tags: trigonometry , angle bisector , geometry

Prove that any triangle having two equal internal angle bisectors (each measured from a vertex to the opposite side) is isosceles.

1947 Putnam, B1

Tags: function , trigonometry

Let $f(x)$ be a function such that $f(1)=1$ and for $x \geq 1$ $$f'(x)= \frac{1}{x^2 +f(x)^{2}}.$$ Prove that $$\lim_{x\to \infty} f(x)$$ exists and is less than $1+ \frac{\pi}{4}.$

1970 Czech and Slovak Olympiad III A, 6

Tags: inequalities , trigonometry , algebra , logarithm

Determine all real $x$ such that \[\sqrt{\tan(x)-1}\,\Bigl(\log_{\tan(x)}\bigl(2+4\cos^2(x)-2\bigr)\Bigr)\ge0.\]

2016 German National Olympiad, 5

Tags: geometry , cyclic quadrilateral , trigonometry , angle

Let $A,B,C,D$ be points on a circle with radius $r$ in this order such that $|AB|=|BC|=|CD|=s$ and $|AD|=s+r$. Find all possible values of the interior angles of the quadrilateral $ABCD$.

1966 IMO Shortlist, 5

Tags: inequalities , trigonometry , geometric inequality

Prove the inequality \[\tan \frac{\pi \sin x}{4\sin \alpha} + \tan \frac{\pi \cos x}{4\cos \alpha} >1\] for any $x, \alpha$ with $0 \leq x \leq \frac{\pi }{2}$ and $\frac{\pi}{6} < \alpha < \frac{\pi}{3}.$

2010 Contests, 2

Tags: inequalities , trigonometry , calculus , derivative , algebra

Prove that for any real number $ x$ the following inequality is true: $ \max\{|\sin x|, |\sin(x\plus{}2010)|\}>\dfrac1{\sqrt{17}}$

2012 Online Math Open Problems, 8

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

In triangle $ABC$ let $D$ be the foot of the altitude from $A$. Suppose that $AD = 4$, $BD = 3$, $CD = 2$, and $AB$ is extended past $B$ to a point $E$ such that $BE = 5$. Determine the value of $CE^2$. [i]Ray Li.[/i] [hide="Clarifications"][list=1][*]Triangle $ABC$ is acute.[/list][/hide]

1979 Chisinau City MO, 176

Tags: trigonometry , algebra

Indicate all the roots of the equation $x^2+1 = \cos x$.

1990 IMO Longlists, 82

Tags: geometry , circumcircle , trigonometry

In a triangle, a symmedian is a line through a vertex that is symmetric to the median with the respect to the internal bisector (all relative to the same vertex). In the triangle $ABC$, the median $m_a$ meets $BC$ at $A'$ and the circumcircle again at $A_1$. The symmedian $s_a$ meets $BC$ at $M$ and the circumcircle again at $A_2$. Given that the line $A_1A_2$ contains the circumcenter $O$ of the triangle, prove that: [i](a) [/i]$\frac{AA'}{AM} = \frac{b^2+c^2}{2bc} ;$ [i](b) [/i]$1+4b^2c^2 = a^2(b^2+c^2)$

2010 AMC 10, 19

Tags: geometry , trigonometry , quadratic , pythagorean theorem , trig identities , law of cosines , perpendicular bisector

A circle with center $ O$ has area $ 156\pi$. Triangle $ ABC$ is equilateral, $ \overline{BC}$ is a chord on the circle, $ OA \equal{} 4\sqrt3$, and point $ O$ is outside $ \triangle ABC$. What is the side length of $ \triangle ABC$? $ \textbf{(A)}\ 2\sqrt3 \qquad\textbf{(B)}\ 6 \qquad\textbf{(C)}\ 4\sqrt3 \qquad\textbf{(D)}\ 12 \qquad\textbf{(E)}\ 18$