Olympiad problems

2007 Bundeswettbewerb Mathematik, 3

In triangle $ ABC$ points $ E$ and $ F$ lie on sides $ AC$ and $ BC$ such that segments $ AE$ and $ BF$ have equal length, and circles formed by $ A,C,F$ and by $ B,C,E,$ respectively, intersect at point $ C$ and another point $ D.$ Prove that that the line $ CD$ bisects $ \angle ACB.$

2014 Indonesia MO Shortlist, G3

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

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 $KL$.

1985 IberoAmerican, 3

Tags: circumcircle , geometry , trigonometry

Given an acute triangle $ABC$, let $D$, $E$ and $F$ be points in the lines $BC$, $AC$ and $AB$ respectively. If the lines $AD$, $BE$ and $CF$ pass through $O$ the centre of the circumcircle of the triangle $ABC$, whose radius is $R$, show that: \[\frac{1}{AD}\plus{}\frac{1}{BE}\plus{}\frac{1}{CF}\equal{}\frac{2}{R}\]

2013 China Team Selection Test, 2

Tags: geometric transformation , trigonometry , euler , circumcircle , homothety , geometry

Let $P$ be a given point inside the triangle $ABC$. Suppose $L,M,N$ are the midpoints of $BC, CA, AB$ respectively and \[PL: PM: PN= BC: CA: AB.\] The extensions of $AP, BP, CP$ meet the circumcircle of $ABC$ at $D,E,F$ respectively. Prove that the circumcentres of $APF, APE, BPF, BPD, CPD, CPE$ are concyclic.

2009 Vietnam National Olympiad, 3

Tags: trigonometry , incenter , geometry , euler , circumcircle

Let $ A$, $ B$ be two fixed points and $ C$ is a variable point on the plane such that $ \angle ACB\equal{}\alpha$ (constant) ($ 0^{\circ}\le \alpha\le 180^{\circ}$). Let $ D$, $ E$, $ F$ be the projections of the incenter $ I$ of triangle $ ABC$ to its sides $ BC$, $ CA$, $ AB$, respectively. Denoted by $ M$, $ N$ the intersections of $ AI$, $ BI$ with $ EF$, respectively. Prove that the length of the segment $ MN$ is constant and the circumcircle of triangle $ DMN$ always passes through a fixed point.

2005 Harvard-MIT Mathematics Tournament, 2

Tags: trigonometry , logarithm , calculus , integration

A plane curve is parameterized by $x(t)=\displaystyle\int_{t}^{\infty} \dfrac {\cos u}{u} \, \mathrm{d}u $ and $ y(t) = \displaystyle\int_{t}^{\infty} \dfrac {\sin u}{u} \, \mathrm{d}u $ for $ 1 \le t \le 2 $. What is the length of the curve?

2008 IMAC Arhimede, 5

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

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$.

2020 Costa Rica - Final Round, 5

Tags: trigonometry , algebra

Determine the value of the expression $$ (1 +\tan(1^o))(1 + \tan(2^o))...(1 + \tan(45^o)).$$

2010 Today's Calculation Of Integral, 575

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

For a function $ f(x)\equal{}\int_x^{\frac{\pi}{4}\minus{}x} \log_4 (1\plus{}\tan t)dt\ \left(0\leq x\leq \frac{\pi}{8}\right)$, answer the following questions. (1) Find $ f'(x)$. (2) Find the $ n$ th term of the sequence $ a_n$ such that $ a_1\equal{}f(0),\ a_{n\plus{}1}\equal{}f(a_n)\ (n\equal{}1,\ 2,\ 3,\ \cdots)$.

IV Soros Olympiad 1997 - 98 (Russia), 11.2

Tags: trigonometry , analytic geometry , algebra , inequalities

Find the area of a figure consisting of points whose coordinates satisfy the inequality $$(y^3 - arcsin x)(x^3 + arcsin y) \ge 0.$$

1981 AMC 12/AHSME, 21

Tags: special factorizations , geometry , difference of squares , trigonometry , algebra

In a triangle with sides of lengths $a,b,$ and $c,$ $(a+b+c)(a+b-c) = 3ab.$ The measure of the angle opposite the side length $c$ is $\displaystyle \text{(A)} \ 15^\circ \qquad \text{(B)} \ 30^\circ \qquad \text{(C)} \ 45^\circ \qquad \text{(D)} \ 60^\circ \qquad \text{(E)} \ 150^\circ$

2009 ISI B.Stat Entrance Exam, 1

Tags: trigonometry

Two train lines intersect each other at a junction at an acute angle $\theta$. A train is passing along one of the two lines. When the front of the train is at the junction, the train subtends an angle $\alpha$ at a station on the other line. It subtends an angle $\beta (<\alpha)$ at the same station, when its rear is at the junction. Show that \[\tan\theta=\frac{2\sin\alpha\sin\beta}{\sin(\alpha-\beta)}\]

2011 AMC 12/AHSME, 14

Tags: trig identities , parabola , geometry , conic , pythagorean theorem , trigonometry , vector

A segment through the focus $F$ of a parabola with vertex $V$ is perpendicular to $\overline{FV}$ and intersects the parabola in points $A$ and $B$. What is $\cos(\angle AVB)$? $ \textbf{(A)}\ -\frac{3\sqrt{5}}{7} \qquad \textbf{(B)}\ -\frac{2\sqrt{5}}{5} \qquad \textbf{(C)}\ -\frac{4}{5} \qquad \textbf{(D)}\ -\frac{3}{5} \qquad \textbf{(E)}\ -\frac{1}{2} $

2012 Pan African, 3

Tags: trigonometry , geometry

(i) Find the angles of $\triangle ABC$ if the length of the altitude through $B$ is equal to the length of the median through $C$ and the length of the altitude through $C$ is equal to the length of the median through $B$. (ii) Find all possible values of $\angle ABC$ of $\triangle ABC$ if the length of the altitude through $A$ is equal to the length of the median through $C$ and the length of the altitude through $C$ is equal to the length of the median through $B$.

2001 Romania National Olympiad, 3

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

Let $f:[-1,1]\rightarrow\mathbb{R}$ be a continuous function. Show that: a) if $\int_0^1 f(\sin (x+\alpha ))\, dx=0$, for every $\alpha\in\mathbb{R}$, then $f(x)=0,\ \forall x\in [-1,1]$. b) if $\int_0^1 f(\sin (nx))\, dx=0$, for every $n\in\mathbb{Z}$, then $f(x)=0,\ \forall x\in [-1,1]$.

2005 Silk Road, 3

Tags: circumcircle , radical axis , power of a point , similar triangles , geometry , trigonometry

Assume $A,B,C$ are three collinear points that $B \in [AC]$. Suppose $AA'$ and $BB'$ are to parrallel lines that $A'$, $B'$ and $C$ are not collinear. Suppose $O_1$ is circumcenter of circle passing through $A$, $A'$ and $C$. Also $O_2$ is circumcenter of circle passing through $B$, $B'$ and $C$. If area of $A'CB'$ is equal to area of $O_1CO_2$, then find all possible values for $\angle CAA'$

2013 AIME Problems, 12

Tags: trigonometry , geometry , relatively prime , number theory , area of a triangle

Let $\triangle PQR$ be a triangle with $\angle P = 75^\circ$ and $\angle Q = 60^\circ$. A regular hexagon $ABCDEF$ with side length 1 is drawn inside $\triangle PQR$ so that side $\overline{AB}$ lies on $\overline{PQ}$, side $\overline{CD}$ lies on $\overline{QR}$, and one of the remaining vertices lies on $\overline{RP}$. There are positive integers $a$, $b$, $c$, and $d$ such that the area of $\triangle PQR$ can be expressed in the form $\tfrac{a+b\sqrt c}d$, where $a$ and $d$ are relatively prime and $c$ is not divisible by the square of any prime. Find $a+b+c+d$.

2009 India Regional Mathematical Olympiad, 1

Tags: trigonometry , angle bisector , perpendicular bisector , geometry

Let $ ABC$ be a triangle in which $ AB \equal{} AC$ and let $ I$ be its in-centre. Suppose $ BC \equal{} AB \plus{} AI$. Find $ \angle{BAC}$

1988 IMO Longlists, 91

Tags: trigonometry , geometry

A regular 14-gon with side $a$ is inscribed in a circle of radius one. Prove \[ \frac{2-a}{2 \cdot a} > \sqrt{3 \cdot \cos \left( \frac{\pi}{7} \right)}. \]

VI Soros Olympiad 1999 - 2000 (Russia), 10.8

Tags: periodic , trigonometry , algebra

Find the smallest positive period of the function $f(x)=\sin (1998x)+ \sin (2000x)$

2003 Tuymaada Olympiad, 1

Tags: inequalities , trigonometry

Prove that for every $\alpha_{1}, \alpha_{2}, \ldots, \alpha_{n}$ in the interval $(0,\pi/2)$ \[\left({1\over \sin \alpha_{1}}+{1\over \sin \alpha_{2}}+\ldots+{1\over \sin \alpha_{n}}\right) \left({1\over \cos \alpha_{1}}+{1\over \cos \alpha_{2}}+\ldots+{1\over \cos \alpha_{n}}\right) \leq\] \[\leq 2 \left({1\over \sin 2\alpha_{1}}+{1\over \sin 2\alpha_{2}}+\ldots+{1\over \sin 2\alpha_{n}}\right)^{2}.\] [i]Proposed by A. Khrabrov[/i]

2005 Today's Calculation Of Integral, 49

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

For $x\geq 0$, Prove that $\int_0^x (t-t^2)\sin ^{2002} t \,dt<\frac{1}{2004\cdot 2005}$

2007 All-Russian Olympiad, 1

Tags: inequalities , trigonometry , function

Prove that for $k>10$ Nazar may replace in the following product some one $\cos$ by $\sin$ so that the new function $f_{1}(x)$ would satisfy inequality $|f_{1}(x)|\le 3\cdot 2^{-1-k}$ for all real $x$. \[f(x) = \cos x \cos 2x \cos 3x \dots \cos 2^{k}x \] [i]N. Agakhanov[/i]

2004 Swedish Mathematical Competition, 4

Tags: algebra , trigonometry , inequalities

If $0 < v <\frac{\pi}{2}$ and $\tan v = 2v$, decide whether $sinv < \frac{20}{21}$.

2012 Online Math Open Problems, 31

Tags: trigonometry , congruent triangles , geometric transformation , reflection , analytic geometry , geometry

Let $ABC$ be a triangle inscribed in circle $\Gamma$, centered at $O$ with radius $333.$ Let $M$ be the midpoint of $AB$, $N$ be the midpoint of $AC$, and $D$ be the point where line $AO$ intersects $BC$. Given that lines $MN$ and $BO$ concur on $\Gamma$ and that $BC = 665$, find the length of segment $AD$. [i]Author: Alex Zhu[/i]