Olympiad problems

2010 Contests, 3

Tags: geometry , 3d geometry , trigonometry , vector , linear algebra , matrix , inequalities

What is the biggest shadow that a cube of side length $1$ can have, with the sun at its peak? Note: "The biggest shadow of a figure with the sun at its peak" is understood to be the biggest possible area of the orthogonal projection of the figure on a plane.

2007 Today's Calculation Of Integral, 204

Tags: calculus , integration , trigonometry , special factorizations , calculus computations

Evaluate \[\int_{0}^{1}\frac{x\ dx}{(x^{2}+x+1)^{\frac{3}{2}}}\]

2008 IMAC Arhimede, 5

Tags: geometry , parallelogram , vector , trigonometry , geometric transformation , reflection , cyclic quadrilateral

The diagonals of the cyclic quadrilateral $ ABCD$ are intersecting at the point $ E$. $ K$ and $ M$ are the midpoints of $ AB$ and $ CD$, respectively. Let the points $ L$ on $ BC$ and $ N$ on $ AD$ s.t. $ EL\perp BC$ and $ EN\perp AD$.Prove that $ KM\perp LN$.

1969 IMO Shortlist, 37

Tags: trigonometric equation , algebra , trigonometry , equation

$(HUN 4)$IMO2 If $a_1, a_2, . . . , a_n$ are real constants, and if $y = \cos(a_1 + x) +2\cos(a_2+x)+ \cdots+ n \cos(a_n + x)$ has two zeros $x_1$ and $x_2$ whose difference is not a multiple of $\pi$, prove that $y = 0.$

2019 CCA Math Bonanza, L4.4

Tags: trigonometry

If an angle $0^\circ<\theta<30^\circ$ satisfies $\sin\left(90^\circ-\theta\right)\sin\left(60^\circ-\theta\right)\sin\left(30^\circ-\theta\right)=\sin^3\left(\theta\right)$, compute $\sin\left(\theta\right)$. [i]2019 CCA Math Bonanza Lightning Round #4.4[/i]

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 $1+ \frac{\pi}{4}.$

1967 IMO Longlists, 4

Tags: inequalities , trigonometry , geometry

Suppose, medians $m_a$ and $m_b$ of a triangle are orthogonal. Prove that: (a) The medians of the triangle correspond to the sides of a right-angled triangle. (b) If $a,b,c$ are the side-lengths of the triangle, then, the following inequality holds:\[5(a^2+b^2-c^2)\geq 8ab\]

1999 AIME Problems, 15

Tags: geometry , 3d geometry , pyramid , tetrahedron , analytic geometry , trigonometry , vector

Consider the paper triangle whose vertices are $(0,0), (34,0),$ and $(16,24).$ The vertices of its midpoint triangle are the midpoints of its sides. A triangular pyramid is formed by folding the triangle along the sides of its midpoint triangle. What is the volume of this pyramid?

2006 Iran Team Selection Test, 5

Tags: circumcircle , trigonometry , inequalities , function , incenter , geometry

Let $ABC$ be an acute angle triangle. Suppose that $D,E,F$ are the feet of perpendicluar lines from $A,B,C$ to $BC,CA,AB$. Let $P,Q,R$ be the feet of perpendicular lines from $A,B,C$ to $EF,FD,DE$. Prove that \[ 2(PQ+QR+RP)\geq DE+EF+FD \]

2013 Moldova Team Selection Test, 2

Tags: trigonometry , algebra

We call a triangle $\triangle ABC$, $Q$-angled if $\tan\angle A,\tan\angle B,\tan\angle C \in \mathbb{Q}$, where $\angle A,\angle B ,\angle C$ are the interior angles of the triangle $\triangle ABC$. $a)$ Prove that $Q$-angled triangles exist; $b)$ Let triangle $\triangle ABC$ be $Q$-angled. Prove that for any non-negative integer $n$, numbers of the form $E_n=\sin^n\angle A \sin^n\angle B \sin^n\angle C + \cos^n\angle A\cos^n\angle B\cos^n\angle C$ are rational.

1983 AIME Problems, 9

Tags: trigonometry , inequalities , calculus

Find the minimum value of \[\frac{9x^2 \sin^2 x + 4}{x \sin x}\] for $0 < x < \pi$.

2001 IMO, 5

Tags: geometry , ratio , trigonometry , similar triangles , angle bisector

Let $ABC$ be a triangle with $\angle BAC = 60^{\circ}$. Let $AP$ bisect $\angle BAC$ and let $BQ$ bisect $\angle ABC$, with $P$ on $BC$ and $Q$ on $AC$. If $AB + BP = AQ + QB$, what are the angles of the triangle?

1969 IMO Shortlist, 38

Tags: series summation , trigonometry , trigonometric identities , algebra

$(HUN 5)$ Let $r$ and $m (r \le m)$ be natural numbers and $Ak =\frac{2k-1}{2m}\pi$. Evaluate $\frac{1}{m^2}\displaystyle\sum_{k=1}^{m}\displaystyle\sum_{l=1}^{m}\sin(rA_k)\sin(rA_l)\cos(rA_k-rA_l)$

2014 Miklós Schweitzer, 11

Tags: trigonometry , function , probability and stats

Let $U$ be a random variable that is uniformly distributed on the interval $[0,1]$, and let \[S_n= 2\sum_{k=1}^n \sin(2kU\pi).\] Show that, as $n\to \infty$, the limit distribution of $S_n$ is the Cauchy distribution with density function $f(x)=\frac1{\pi(1+x^2)}$.

2005 Thailand Mathematical Olympiad, 21

Tags: trigonometry , algebra , min , inequalities

Compute the minimum value of $cos(a-b) + cos(b-c) + cos(c-a)$ as $a,b,c$ ranges over the real numbers.

2006 Romania National Olympiad, 4

Tags: integration , function , calculus , derivative , trigonometry , real analysis

Let $f: [0,1]\to\mathbb{R}$ be a continuous function such that \[ \int_{0}^{1}f(x)dx=0. \] Prove that there is $c\in (0,1)$ such that \[ \int_{0}^{c}xf(x)dx=0. \] [i]Cezar Lupu, Tudorel Lupu[/i]

1998 Croatia National Olympiad, Problem 1

Tags: triangle , trigonometry , geometry

Let $a,b,c$ be the sides and $\alpha,\beta,\gamma$ be the corresponding angles of a triangle. Prove the equality $$\left(\frac bc+\frac cb\right)\cos\alpha+\left(\frac ca+\frac ac\right)\cos\beta+\left(\frac ab+\frac ba\right)\cos\gamma=3.$$

2009 Today's Calculation Of Integral, 405

Tags: calculus , integration , trigonometry , calculus computations

Calculate $ \displaystyle \left|\frac {\int_0^{\frac {\pi}{2}} (x\cos x + 1)e^{\sin x}\ dx}{\int_0^{\frac {\pi}{2}} (x\sin x - 1)e^{\cos x}\ dx}\right|$.

2010 Tournament Of Towns, 3

Tags: function , trigonometry , algebra

Consider a composition of functions $\sin, \cos, \tan, \cot, \arcsin, \arccos, \arctan, \arccos$, applied to the number $1$. Each function may be applied arbitrarily many times and in any order. (ex: $\sin \cos \arcsin \cos \sin\cdots 1$). Can one obtain the number $2010$ in this way?

2007 Swedish Mathematical Competition, 3

Tags: trigonometry , geometry , circumradius

Let $\alpha$, $\beta$, $\gamma$ be the angles of a triangle. If $a$, $b$, $c$ are the side length of the triangle and $R$ is the circumradius, show that \[ \cot \alpha + \cot \beta +\cot \gamma =\frac{R\left(a^2+b^2+c^2\right)}{abc} \]

2008 Moldova National Olympiad, 9.3

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

From the vertex $ A$ of the equilateral triangle $ ABC$ a line is drown that intercepts the segment $ [BC]$ in the point $ E$. The point $ M \in (AE$ is such that $ M$ external to $ ABC$, $ \angle AMB \equal{} 20 ^\circ$ and $ \angle AMC \equal{} 30 ^ \circ$. What is the measure of the angle $ \angle MAB$?

1998 AMC 12/AHSME, 19

Tags: trigonometry , analytic geometry , geometry

How many triangles have area $ 10$ and vertices at $ (\minus{}5,0)$, $ (5,0)$, and $ (5\cos \theta, 5\sin \theta)$ for some angle $ \theta$? $ \textbf{(A)}\ 0\qquad \textbf{(B)}\ 2\qquad \textbf{(C)}\ 4\qquad \textbf{(D)}\ 6\qquad \textbf{(E)}\ 8$

2010 Today's Calculation Of Integral, 557

Tags: calculus , integration , limit , trigonometry , floor function , function , analytic geometry

Find the folllowing limit. \[ \lim_{n\to\infty} \frac{(2n\plus{}1)\int_0^1 x^{n\minus{}1}\sin \left(\frac{\pi}{2}x\right)dx}{(n\plus{}1)^2\int_0^1 x^{n\minus{}1}\cos \left(\frac{\pi}{2}x\right)dx}\ \ (n\equal{}1,\ 2,\ \cdots).\]

2006 Poland - Second Round, 2

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

Point $C$ is a midpoint of $AB$. Circle $o_1$ which passes through $A$ and $C$ intersect circle $o_2$ which passes through $B$ and $C$ in two different points $C$ and $D$. Point $P$ is a midpoint of arc $AD$ of circle $o_1$ which doesn't contain $C$. Point $Q$ is a midpoint of arc $BD$ of circle $o_2$ which doesn't contain $C$. Prove that $PQ \perp CD$.

1991 Arnold's Trivium, 79

Tags: trigonometry

How many solutions has the boundary-value problem \[u_{xx}+\lambda u=\sin x,\;u(0)=u(\pi)=0\]