Olympiad problems

2012 Today's Calculation Of Integral, 792

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

Answer the following questions: (1) Let $a$ be positive real number. Find $\lim_{n\to\infty} (1+a^{n})^{\frac{1}{n}}.$ (2) Evaluate $\int_1^{\sqrt{3}} \frac{1}{x^2}\ln \sqrt{1+x^2}dx.$ 35 points

2022 JHMT HS, 10

Tags: calculus , trigonometry

The maximum value of \[ 2\sum_{n=1}^{\infty} \frac{\sin(n\theta)}{44^n} \] over all real numbers $\theta$ can be expressed as a common fraction $\tfrac{p}{q}$. Compute $p + q$.

II Soros Olympiad 1995 - 96 (Russia), 11.3

Tags: algebra , system of equations , trigonometry

Solve the system of equations $$\begin{cases} \sin \frac{\pi}{2}xy =z \\ \sin \frac{\pi}{2}yz =x \\ \sin \frac{\pi}{2}zx =y \end{cases} \,\,\, ?$$

2003 All-Russian Olympiad Regional Round, 10.1

Tags: trigonometry , algebra

Find all angles a for which the set of numbers $\sin a$, $\sin 2a$, $\sin 3a$ coincides with the set $cos a$, $cos 2a$, $cos 3a$.

2008 China Team Selection Test, 1

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

Let $ ABC$ be an acute triangle, let $ M,N$ be the midpoints of minor arcs $ \widehat{CA},\widehat{AB}$ of the circumcircle of triangle $ ABC,$ point $ D$ is the midpoint of segment $ MN,$ point $ G$ lies on minor arc $ \widehat{BC}.$ Denote by $ I,I_{1},I_{2}$ the incenters of triangle $ ABC,ABG,ACG$ respectively.Let $ P$ be the second intersection of the circumcircle of triangle $ GI_{1}I_{2}$ with the circumcircle of triangle $ ABC.$ Prove that three points $ D,I,P$ are collinear.

1967 IMO Shortlist, 4

Tags: trigonometric equation , trigonometry , algebra , geometry

[b](i)[/b] Solve the equation: \[ \sin^3(x) + \sin^3\left( \frac{2 \pi}{3} + x\right) + \sin^3\left( \frac{4 \pi}{3} + x\right) + \frac{3}{4} \cos {2x} = 0.\] [b](ii)[/b] Supposing the solutions are in the form of arcs $AB$ with one end at the point $A$, the beginning of the arcs of the trigonometric circle, and $P$ a regular polygon inscribed in the circle with one vertex in $A$, find: 1) The subsets of arcs having the other end in $B$ in one of the vertices of the regular dodecagon. 2) Prove that no solution can have the end $B$ in one of the vertices of polygon $P$ whose number of sides is prime or having factors other than 2 or 3.

1997 AIME Problems, 11

Tags: ratio , trigonometry , floor function , integration , calculus , complex number , geometric series

Let $x=\frac{\displaystyle\sum_{n=1}^{44} \cos n^\circ}{\displaystyle \sum_{n=1}^{44} \sin n^\circ}.$ What is the greatest integer that does not exceed $100x$?

2006 AMC 12/AHSME, 15

Tags: trigonometry

Suppose $ \cos x \equal{} 0$ and $ \cos (x \plus{} z) \equal{} 1/2$. What is the smallest possible positive value of $ z$? $ \textbf{(A) } \frac {\pi}{6}\qquad \textbf{(B) } \frac {\pi}{3}\qquad \textbf{(C) } \frac {\pi}{2}\qquad \textbf{(D) } \frac {5\pi}{6}\qquad \textbf{(E) } \frac {7\pi}{6}$

2006 Iran MO (3rd Round), 6

Tags: geometry , geometric transformation , vector , trigonometry , dilation , ratio , real analysis

Assume that $C$ is a convex subset of $\mathbb R^{d}$. Suppose that $C_{1},C_{2},\dots,C_{n}$ are translations of $C$ that $C_{i}\cap C\neq\emptyset$ but $C_{i}\cap C_{j}=\emptyset$. Prove that \[n\leq 3^{d}-1\] Prove that $3^{d}-1$ is the best bound. P.S. In the exam problem was given for $n=3$.

2009 Today's Calculation Of Integral, 508

Tags: calculus , integration , trigonometry , calculus computations

Compare the size of the definite integrals? \[ \int_0^{\frac {\pi}{4}} x^{2008}\tan ^{2008}x\ dx,\ \int_0^{\frac {\pi}{4}} x^{2009}\tan ^{2009}x\ dx,\ \int_0^{\frac {\pi}{4}} x^{2010}\tan ^{2010}x\ dx\]

1949-56 Chisinau City MO, 51

Tags: algebra , logarithm , trigonometry

Determine graphically the number of roots of the equation $\sin x = \lg x$.

VI Soros Olympiad 1999 - 2000 (Russia), 11.9

Tags: algebra , inequalities , trigonometry

Find the largest $c$ such that for any $\lambda \ge 1$ there is an a that satisfies the inequality $$\sin a + \sin (a\lambda ) \ge c.$$

2008 Harvard-MIT Mathematics Tournament, 32

Tags: trigonometry , circumcircle , symmetry , geometry

Cyclic pentagon $ ABCDE$ has side lengths $ AB\equal{}BC\equal{}5$, $ CD\equal{}DE\equal{}12$, and $ AE \equal{} 14$. Determine the radius of its circumcircle.

2013 Today's Calculation Of Integral, 888

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

In the coordinate plane, given a circle $K: x^2+y^2=1,\ C: y=x^2-2$. Let $l$ be the tangent line of $K$ at $P(\cos \theta,\ \sin \theta)\ (\pi<\theta <2\pi).$ Find the minimum area of the part enclosed by $l$ and $C$.

1996 IMO Shortlist, 8

Tags: inequalities , trigonometry , geometry , circumcircle

Let $ ABCD$ be a convex quadrilateral, and let $ R_A, R_B, R_C, R_D$ denote the circumradii of the triangles $ DAB, ABC, BCD, CDA,$ respectively. Prove that $ R_A \plus{} R_C > R_B \plus{} R_D$ if and only if $ \angle A \plus{} \angle C > \angle B \plus{} \angle D.$

2013 Math Prize For Girls Problems, 20

Tags: inequalities , trigonometry

Let $a_0$, $a_1$, $a_2$, $\dots$ be an infinite sequence of real numbers such that $a_0 = \frac{4}{5}$ and \[ a_{n} = 2 a_{n-1}^2 - 1 \] for every positive integer $n$. Let $c$ be the smallest number such that for every positive integer $n$, the product of the first $n$ terms satisfies the inequality \[ a_0 a_1 \dots a_{n - 1} \le \frac{c}{2^n}. \] What is the value of $100c$, rounded to the nearest integer?

2012 National Olympiad First Round, 27

Tags: trigonometry

What is the least real number $C$ that satisfies $\sin x \cos x \leq C(\sin^6x+\cos^6x)$ for every real number $x$? $ \textbf{(A)}\ \sqrt3 \qquad \textbf{(B)}\ 2\sqrt2 \qquad \textbf{(C)}\ \sqrt 2 \qquad \textbf{(D)}\ 2 \qquad \textbf{(E)}\ \text{None}$

2005 Harvard-MIT Mathematics Tournament, 3

Tags: geometry , rectangle , trigonometry

Let $ABCD$ be a rectangle with area $1$, and let $E$ lie on side $CD$. What is the area of the triangle formed by the centroids of triangles $ABE$, $BCE$, and $ADE$?

2009 Indonesia MO, 4

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

Given an acute triangle $ ABC$. The incircle of triangle $ ABC$ touches $ BC,CA,AB$ respectively at $ D,E,F$. The angle bisector of $ \angle A$ cuts $ DE$ and $ DF$ respectively at $ K$ and $ L$. Suppose $ AA_1$ is one of the altitudes of triangle $ ABC$, and $ M$ be the midpoint of $ BC$. (a) Prove that $ BK$ and $ CL$ are perpendicular with the angle bisector of $ \angle BAC$. (b) Show that $ A_1KML$ is a cyclic quadrilateral.