Olympiad problems

2013 Brazil Team Selection Test, 3

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.

2003 Purple Comet Problems, 8

Tags: trigonometry

Let $ABCDEFGHIJKL$ be a regular dodecagon. Find $\frac{AB}{AF} + \frac{AF}{AB}$.

2005 ISI B.Stat Entrance Exam, 4

Tags: trigonometry

Find all real solutions of the equation $\sin^{5}x+\cos^{3}x=1$.

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 $q$.

2009 AMC 10, 23

Tags: trigonometry , geometry , ratio , trapezoid , parallelogram

Convex quadrilateral $ ABCD$ has $ AB\equal{}9$ and $ CD\equal{}12$. Diagonals $ AC$ and $ BD$ intersect at $ E$, $ AC\equal{}14$, and $ \triangle AED$ and $ \triangle BEC$ have equal areas. What is $ AE$? $ \textbf{(A)}\ \frac{9}{2}\qquad \textbf{(B)}\ \frac{50}{11}\qquad \textbf{(C)}\ \frac{21}{4}\qquad \textbf{(D)}\ \frac{17}{3}\qquad \textbf{(E)}\ 6$

1998 Harvard-MIT Mathematics Tournament, 5

Tags: calculus , limit , trigonometry , logarithm

Evaluate $\displaystyle\lim_{x\to 1}x^{\dfrac{x}{\sin(1-x)}}$.

1995 Greece National Olympiad, 2

Tags: trigonometry , geometry

Let $ABC$ be a triangle with $AB = AC$ and let $D$ be a point on $BC$ such that the incircle of $ABD$ and the excircle of $ADC$ corresponding to $A$ have the same radius. Prove that this radius is equal to one quarter of the altitude from $B$ of triangle $ABC$.

1998 Turkey Team Selection Test, 2

Tags: induction , trigonometry , algebra

Let the sequence $(a_{n})$ be deﬁned by $a_{1} = t$ and $a_{n+1} = 4a_{n}(1 - a_{n})$ for $n \geq 1$. How many possible values of t are there, if $a_{1998} = 0$?

1987 IMO Longlists, 38

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

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]

1971 Swedish Mathematical Competition, 5

Tags: trigonometry , inequalities , max , algebra

Show that \[ \max\limits_{|x|\leq t} |1 - a \cos x| \geq \tan^2 \frac{t}{2} \] for $a$ positive and $t \in (0, \frac{\pi}{2})$.

2008 AMC 12/AHSME, 23

Tags: geometry , symmetry , geometric transformation , rotation , analytic geometry , trigonometry

The solutions of the equation $ z^4 \plus{} 4z^3i \minus{} 6z^2 \minus{} 4zi \minus{} i \equal{} 0$ are the vertices of a convex polygon in the complex plane. What is the area of the polygon? $ \textbf{(A)}\ 2^{5/8} \qquad \textbf{(B)}\ 2^{3/4} \qquad \textbf{(C)}\ 2 \qquad \textbf{(D)}\ 2^{5/4} \qquad \textbf{(E)}\ 2^{3/2}$

1979 Spain Mathematical Olympiad, 5

Tags: trigonometry , integration , integral , calculus

Calculate the definite integral $$\int_2^4 \sin ((x-3)^3) dx$$

2010 Contests, 2

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

Let $ABC$ be a triangle with $AB = AC$. The incircle touches $BC$, $AC$ and $AB$ at $D$, $E$ and $F$ respectively. Let $P$ be a point on the arc $\overarc{EF}$ that does not contain $D$. Let $Q$ be the second point of intersection of $BP$ and the incircle of $ABC$. The lines $EP$ and $EQ$ meet the line $BC$ at $M$ and $N$, respectively. Prove that the four points $P, F, B, M$ lie on a circle and $\frac{EM}{EN} = \frac{BF}{BP}$.

PEN E Problems, 41

Tags: limit , trigonometry

Show that $n$ is prime iff $\lim_{r \rightarrow\infty}\,\lim_{s \rightarrow\infty}\,\lim_{t \rightarrow \infty}\,\sum_{u=0}^{s}\left(1-\left(\cos\,\frac{(u!)^{r} \pi}{n} \right)^{2t} \right)=n$ PS : I posted it because it's in the PDF file but not here ...

1994 Vietnam National Olympiad, 2

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

$ABC$ is a triangle. Reflect each vertex in the opposite side to get the triangle $A'B'C'$. Find a necessary and sufficient condition on $ABC$ for $A'B'C'$ to be equilateral.

2000 Spain Mathematical Olympiad, 2

Tags: trigonometry , combinatorics

Four points are given inside or on the boundary of a unit square. Prove that at least two of these points are on a mutual distance at most $1.$

2011 Uzbekistan National Olympiad, 3

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

Given an acute triangle $ABC$ with altituties AD and BE. O circumcinter of $ABC$.If o lies on the segment DE then find the value of $sinAsinBcosC$

Today's calculation of integrals, 877

Tags: limit , calculus , integration , trigonometry , calculus computations , definite integral

Let $f(x)=\lim_{n\to\infty} \frac{\sin^{n+2}x+\cos^{n+2}x}{\sin^n x+\cos^n x}$ for $0\leq x\leq \frac{\pi}2.$ Evaluate $\int_0^{\frac{\pi}2} f(x)\ dx.$

2019 AMC 12/AHSME, 19

Tags: trigonometry , perimeter , geometry

In $\triangle ABC$ with integer side lengths, \[ \cos A=\frac{11}{16}, \qquad \cos B= \frac{7}{8}, \qquad \text{and} \qquad\cos C=-\frac{1}{4}. \] What is the least possible perimeter for $\triangle ABC$? $\textbf{(A) } 9 \qquad \textbf{(B) } 12 \qquad \textbf{(C) } 23 \qquad \textbf{(D) } 27 \qquad \textbf{(E) } 44$

1994 APMO, 4

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

Is there an infinite set of points in the plane such that no three points are collinear, and the distance between any two points is rational?

1984 Vietnam National Olympiad, 3

Tags: trigonometry , incenter , geometry

Consider a trihedral angle $Sxyz$ with $\angle xSz = \alpha, \angle xSy = \beta$ and $\angle ySz =\gamma$. Let $A,B,C$ denote the dihedral angles at edges $y, z, x$ respectively. $(a)$ Prove that $\frac{\sin\alpha}{\sin A}=\frac{\sin\beta}{\sin B}=\frac{\sin\gamma}{\sin C}$ $(b)$ Show that $\alpha + \beta = 180^{\circ}$ if and only if $\angle A + \angle B = 180^{\circ}.$ $(c)$ Assume that $\alpha=\beta =\gamma = 90^{\circ}$. Let $O \in Sz$ be a fixed point such that $SO = a$ and let $M,N$ be variable points on $x, y$ respectively. Prove that $\angle SOM +\angle SON +\angle MON$ is constant and find the locus of the incenter of $OSMN$.

1979 IMO Shortlist, 10

Tags: vector , trigonometry , geometry , geometric inequality , euclidean geometry

Show that for any vectors $a, b$ in Euclidean space, \[|a \times b|^3 \leq \frac{3 \sqrt 3}{8} |a|^2 |b|^2 |a-b|^2\] Remark. Here $\times$ denotes the vector product.

2013 AIME Problems, 8

Tags: algebra , function , domain , modular arithmetic , trigonometry , logarithm

The domain of the function $f(x) = \text{arcsin}(\log_{m}(nx))$ is a closed interval of length $\frac{1}{2013}$, where $m$ and $n$ are positive integers and $m > 1$. Find the remainder when the smallest possible sum $m+n$ is divided by $1000$.

2008 Alexandru Myller, 2

Tags: binomial coefficient , complex number , number theory , trigonometry , binomial theorem

There are no integers $ a,b,c $ that satisfy $ \left( a+b\sqrt{-3}\right)^{17}=c+\sqrt{-3} . $ [i]Dorin Andrica, Mihai Piticari[/i]

2014 NIMO Problems, 4

Tags: trigonometry , geometry , trapezoid , ratio , trig identities , law of sines , power of a point

Points $A$, $B$, $C$, and $D$ lie on a circle such that chords $\overline{AC}$ and $\overline{BD}$ intersect at a point $E$ inside the circle. Suppose that $\angle ADE =\angle CBE = 75^\circ$, $BE=4$, and $DE=8$. The value of $AB^2$ can be written in the form $a+b\sqrt{c}$ for positive integers $a$, $b$, and $c$ such that $c$ is not divisible by the square of any prime. Find $a+b+c$. [i]Proposed by Tony Kim[/i]