Olympiad problems

2010 Today's Calculation Of Integral, 533

Tags: trigonometry , algebra , integration , analytic geometry , calculus , parameterization , function

Let $ C$ be the circle with radius 1 centered on the origin. Fix the endpoint of the string with length $ 2\pi$ on the point $ A(1,\ 0)$ and put the other end point $ P$ on the point $ P_0(1,\ 2\pi)$. From this situation, when we twist the string around $ C$ by moving the point $ P$ in anti clockwise with the string streched tightly, find the length of the curve that the point $ P$ draws from sarting point $ P_0$ to reaching point $ A$.

2010 JBMO Shortlist, 4

Tags: trigonometry , geometry , angle bisector , circumcircle , perpendicular bisector

Let $AL$ and $BK$ be angle bisectors in the non-isosceles triangle $ABC$ ($L$ lies on the side $BC$, $K$ lies on the side $AC$). The perpendicular bisector of $BK$ intersects the line $AL$ at point $M$. Point $N$ lies on the line $BK$ such that $LN$ is parallel to $MK$. Prove that $LN = NA$.

2000 Pan African, 2

Tags: trigonometry , geometry

Let $\gamma$ be circle and let $P$ be a point outside $\gamma$. Let $PA$ and $PB$ be the tangents from $P$ to $\gamma$ (where $A, B \in \gamma$). A line passing through $P$ intersects $\gamma$ at points $Q$ and $R$. Let $S$ be a point on $\gamma$ such that $BS \parallel QR$. Prove that $SA$ bisects $QR$.

1973 IMO Shortlist, 16

Tags: trigonometry , algebra , polynomial , factorization , geometry

Given $a, \theta \in \mathbb R, m \in \mathbb N$, and $P(x) = x^{2m}- 2|a|^mx^m \cos \theta +a^{2m}$, factorize $P(x)$ as a product of $m$ real quadratic polynomials.

2005 Estonia National Olympiad, 1

Tags: trigonometry , algebra , system of equations

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 $\cos 2x = \cos 2y$.

2018 Flanders Math Olympiad, 2

Tags: trigonometry , algebra , inequalities

Prove that for every acute angle $\alpha$, $\sin (\cos \alpha) < \cos(\sin \alpha)$.

1998 Niels Henrik Abels Math Contest (Norwegian Math Olympiad) Round 2, 9

Tags: trigonometry , geometry

In the triangle $ ABC$ we have $ AB \equal{} 5$ and $ AC \equal{} 6$. The area of the triangle when the $ \angle ACB$ is as large as possible is $ \text{(A)}\ 15 \qquad \text{(B)}\ 5 \sqrt{7} \qquad \text{(C)}\ \frac{7}{2} \sqrt{7} \qquad \text{(D)}\ 3 \sqrt{11} \qquad \text{(E)}\ \frac{5}{2} \sqrt{11}$

2009 Belarus Team Selection Test, 3

Tags: trigonometry , angle bisector , altitude , geometry

Points $T,P,H$ lie on the side $BC,AC,AB$ respectively of triangle $ABC$, so that $BP$ and $AT$ are angle bisectors and $CH$ is an altitude of $ABC$. Given that the midpoint of $CH$ belongs to the segment $PT,$ find the value of $\cos A + \cos B$ I. Voronovich

2004 India IMO Training Camp, 1

Tags: trigonometry , circumcircle , geometry

Let $ABC$ be an acute-angled triangle and $\Gamma$ be a circle with $AB$ as diameter intersecting $BC$ and $CA$ at $F ( \not= B)$ and $E (\not= A)$ respectively. Tangents are drawn at $E$ and $F$ to $\Gamma$ intersect at $P$. Show that the ratio of the circumcentre of triangle $ABC$ to that if $EFP$ is a rational number.

1998 China Team Selection Test, 1

Tags: circumcircle , incenter , trigonometry , geometry

In acute-angled $\bigtriangleup ABC$, $H$ is the orthocenter, $O$ is the circumcenter and $I$ is the incenter. Given that $\angle C > \angle B > \angle A$, prove that $I$ lies within $\bigtriangleup BOH$.

2019 Romania Team Selection Test, 2

Tags: trigonometry , geometry , circumcircle , triangle , easy geometry

The altitudes through the vertices $ A,B,C$ of an acute-angled triangle $ ABC$ meet the opposite sides at $ D,E, F,$ respectively. The line through $ D$ parallel to $ EF$ meets the lines $ AC$ and $ AB$ at $ Q$ and $ R,$ respectively. The line $ EF$ meets $ BC$ at $ P.$ Prove that the circumcircle of the triangle $ PQR$ passes through the midpoint of $ BC.$

Today's calculation of integrals, 866

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

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 Romania National Olympiad, 2

Tags: trigonometry , rotation , conic , 3d geometry , geometric transformation , pyramid , geometry

The base $A_{1}A_{2}\ldots A_{n}$ of the pyramid $VA_{1}A_{2}\ldots A_{n}$ is a regular polygon. Prove that if \[\angle VA_{1}A_{2}\equiv \angle VA_{2}A_{3}\equiv \cdots \equiv \angle VA_{n-1}A_{n}\equiv \angle VA_{n}A_{1},\] then the pyramid is regular.

2013 Online Math Open Problems, 32

Tags: geometry , trigonometry , inradius , trapezoid , incenter , circumcircle

In $\triangle ABC$ with incenter $I$, $AB = 61$, $AC = 51$, and $BC=71$. The circumcircles of triangles $AIB$ and $AIC$ meet line $BC$ at points $D$ ($D \neq B$) and $E$ ($E \neq C$), respectively. Determine the length of segment $DE$. [i]James Tao[/i]

1994 All-Russian Olympiad Regional Round, 11.1

Tags: trigonometry , inequalities , inequality , trigonometric inequality

Prove that for all $x \in \left( 0, \frac{\pi}{3} \right)$ inequality $sin2x+cosx>1$ holds.

1988 IMO Longlists, 48

Tags: geometry , trigonometry , perimeter

Consider 2 concentric circle radii $ R$ and $ r$ ($ R > r$) with centre $ O.$ Fix $ P$ on the small circle and consider the variable chord $ PA$ of the small circle. Points $ B$ and $ C$ lie on the large circle; $ B,P,C$ are collinear and $ BC$ is perpendicular to $ AP.$ [b]i.)[/b] For which values of $ \angle OPA$ is the sum $ BC^2 \plus{} CA^2 \plus{} AB^2$ extremal? [b]ii.)[/b] What are the possible positions of the midpoints $ U$ of $ BA$ and $ V$ of $ AC$ as $ \angle OPA$ varies?

2010 Balkan MO Shortlist, C3

Tags: geometry , extremal combinatorics , marvio , trigonometry , induction , combinatorics

A strip of width $w$ is the set of all points which lie on, or between, two parallel lines distance $w$ apart. Let $S$ be a set of $n$ ($n \ge 3$) points on the plane such that any three different points of $S$ can be covered by a strip of width $1$. Prove that $S$ can be covered by a strip of width $2$.

2005 Germany Team Selection Test, 3

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

Let ABC be a triangle and let $r, r_a, r_b, r_c$ denote the inradius and ex-radii opposite to the vertices $A, B, C$, respectively. Suppose that $a>r_a, b>r_b, c>r_c$. Prove that [b](a)[/b] $\triangle ABC$ is acute. [b](b)[/b] $a+b+c > r+r_a+r_b+r_c$.

1952 Putnam, B7

Tags: trigonometry , cosine , convergence

Given any real number $N_0,$ if $N_{j+1}= \cos N_j ,$ prove that $\lim_{j\to \infty} N_j$ exists and is independent of $N_0.$

1999 Putnam, 1

Tags: trigonometry , limit

Right triangle $ABC$ has right angle at $C$ and $\angle BAC=\theta$; the point $D$ is chosen on $AB$ so that $|AC|=|AD|=1$; the point $E$ is chosen on $BC$ so that $\angle CDE=\theta$. The perpendicular to $BC$ at $E$ meets $AB$ at $F$. Evaluate $\lim_{\theta\to 0}|EF|$.

2009 Moldova Team Selection Test, 3

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

[color=darkred]Quadrilateral $ ABCD$ is inscribed in the circle of diameter $ BD$. Point $ A_1$ is reflection of point $ A$ wrt $ BD$ and $ B_1$ is reflection of $ B$ wrt $ AC$. Denote $ \{P\}\equal{}CA_1 \cap BD$ and $ \{Q\}\equal{}DB_1\cap AC$. Prove that $ AC\perp PQ$.[/color]

1970 IMO Longlists, 14

Tags: trigonometry

Let $\alpha + \beta +\gamma = \pi$. Prove that $\sum_{cyc}{\sin 2\alpha} = 2\cdot \left(\sum_{cyc}{\sin \alpha}\right)\cdot\left(\sum_{cyc}{\cos \alpha}\right)- 2\sum_{cyc}{\sin \alpha}$.

2008 AMC 10, 24

Tags: trigonometry , parallelogram , geometry , geometric transformation , rhombus

Quadrilateral $ABCD$ has $AB=BC=CD$, $\angle ABC=70^\circ$, and $\angle BCD=170^\circ$. What is the degree measure of $\angle BAD$? $ \textbf{(A)}\ 75\qquad \textbf{(B)}\ 80\qquad \textbf{(C)}\ 85\qquad \textbf{(D)}\ 90\qquad \textbf{(E)}\ 95$

2014 AMC 10, 21

Tags: trigonometry , geometry , trapezoid

Trapezoid $ABCD$ has parallel sides $\overline{AB}$ or length $33$ and $\overline{CD}$ of length $21$. The other two sides are of lengths $10$ and $14$. The angles at $A$ and $B$ are acute. What is the length of the shorter diagonal of $ABCD$? $ \textbf {(A) } 10\sqrt{6} \qquad \textbf {(B) } 25 \qquad \textbf {(C) } 8\sqrt{10} \qquad \textbf {(D) } 18\sqrt{2} \qquad \textbf {(E) } 26 $

2005 Today's Calculation Of Integral, 2

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

Calculate the following indefinite integrals. [1] $\int \cos \left(2x-\frac{\pi}{3}\right)dx$ [2]$\int \frac{dx}{\cos ^2 (3x+4)}$ [3]$\int (x-1)\sqrt[3]{x-2}dx$ [4]$\int x\cdot 3^{x^2+1}dx$ [5]$\int \frac{dx}{\sqrt{1-x}}dx$