Olympiad problems

1999 Mongolian Mathematical Olympiad, Problem 3

I couldn't solve this problem and the only solution I was able to find was very unnatural (it was an official solution, I think) and I couldn't be satisfied with it, so I ask you if you can find some different solutions. The problem is really great one! If $M$ is the centroid of a triangle $ABC$, prove that the following inequality holds: \[\sin\angle CAM+\sin\angle CBM\leq\frac{2}{\sqrt3}.\] The equality occurs in a very strange case, I don't remember it.

2014 France Team Selection Test, 2

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

Two circles $O_1$ and $O_2$ intersect each other at $M$ and $N$. The common tangent to two circles nearer to $M$ touch $O_1$ and $O_2$ at $A$ and $B$ respectively. Let $C$ and $D$ be the reflection of $A$ and $B$ respectively with respect to $M$. The circumcircle of the triangle $DCM$ intersect circles $O_1$ and $O_2$ respectively at points $E$ and $F$ (both distinct from $M$). Show that the circumcircles of triangles $MEF$ and $NEF$ have same radius length.

2014 China Team Selection Test, 1

Tags: circumcircle , geometric transformation , homothety , trigonometry , cyclic quadrilateral , geometry

Let the circumcenter of triangle $ABC$ be $O$. $H_A$ is the projection of $A$ onto $BC$. The extension of $AO$ intersects the circumcircle of $BOC$ at $A'$. The projections of $A'$ onto $AB, AC$ are $D,E$, and $O_A$ is the circumcentre of triangle $DH_AE$. Define $H_B, O_B, H_C, O_C$ similarly. Prove: $H_AO_A, H_BO_B, H_CO_C$ are concurrent

2009 India IMO Training Camp, 4

Tags: trigonometry , geometry , circumcircle

Let $ \gamma$ be circumcircle of $ \triangle ABC$.Let $ R_a$ be radius of circle touching $ AB,AC$&$ \gamma$ internally.Define $ R_b,R_c$ similarly. Prove That $ \frac {1}{aR_a} \plus{} \frac {1}{bR_b} \plus{} \frac {1}{cR_c} \equal{} \frac {s^2}{rabc}$.

1978 AMC 12/AHSME, 15

Tags: trigonometry , function , quadratic

If $\sin x+\cos x=1/5$ and $0\le x<\pi$, then $\tan x$ is $\textbf{(A) }-\frac{4}{3}\qquad\textbf{(B) }-\frac{3}{4}\qquad\textbf{(C) }\frac{3}{4}\qquad\textbf{(D) }\frac{4}{3}\qquad$ $\textbf{(E) }\text{not completely determined by the given information}$

2005 Today's Calculation Of Integral, 44

Tags: calculus , integration , trigonometry , induction , abstract algebra , calculus computations

Evaluate \[{\int_0^\frac{\pi}{2}} \frac{\sin 2005x}{\sin x}dx\]

2013 AMC 12/AHSME, 20

Tags: trigonometry , geometry , trapezoid , parabola , analytic geometry , graphing lines , conic

For $135^\circ < x < 180^\circ$, points $P=(\cos x, \cos^2 x), Q=(\cot x, \cot^2 x), R=(\sin x, \sin^2 x)$ and $S =(\tan x, \tan^2 x)$ are the vertices of a trapezoid. What is $\sin(2x)$? $ \textbf{(A)}\ 2-2\sqrt{2}\qquad\textbf{(B)}\ 3\sqrt{3}-6\qquad\textbf{(C)}\ 3\sqrt{2}-5\qquad\textbf{(D)}\ -\frac{3}{4}\qquad\textbf{(E)}\ 1-\sqrt{3} $

2010 Today's Calculation Of Integral, 525

Tags: calculus , integration , geometry , vector , inequalities , quadratic , trigonometry

Let $ a,\ b$ be real numbers satisfying $ \int_0^1 (ax\plus{}b)^2dx\equal{}1$. Determine the values of $ a,\ b$ for which $ \int_0^1 3x(ax\plus{}b)\ dx$ is maximized.

2004 France Team Selection Test, 2

Tags: trigonometry , angle bisector , geometry

Let $ABCD$ be a parallelogram. Let $M$ be a point on the side $AB$ and $N$ be a point on the side $BC$ such that the segments $AM$ and $CN$ have equal lengths and are non-zero. The lines $AN$ and $CM$ meet at $Q$. Prove that the line $DQ$ is the bisector of the angle $\measuredangle ADC$. [i]Alternative formulation.[/i] Let $ABCD$ be a parallelogram. Let $M$ and $N$ be points on the sides $AB$ and $BC$, respectively, such that $AM=CN\neq 0$. The lines $AN$ and $CM$ intersect at a point $Q$. Prove that the point $Q$ lies on the bisector of the angle $\measuredangle ADC$.

1976 Polish MO Finals, 1

Tags: trigonometry

Is the number $$\sin \frac{\pi}{18} \sin \frac{3\pi}{18} \sin \frac{5\pi}{18} \sin \frac{7\pi}{18} \sin \frac{9\pi}{18}$$ rational?

2012 Brazil Team Selection Test, 4

Tags: trigonometry , triangle inequality , algebra , triangle

Prove that for every positive integer $n,$ the set $\{2,3,4,\ldots,3n+1\}$ can be partitioned into $n$ triples in such a way that the numbers from each triple are the lengths of the sides of some obtuse triangle. [i]Proposed by Canada[/i]

1999 AMC 12/AHSME, 15

Tags: trigonometry

Let $ x$ be a real number such that $ \sec x \minus{} \tan x \equal{} 2$. Then $ \sec x \plus{} \tan x \equal{}$ $ \textbf{(A)}\ 0.1 \qquad \textbf{(B)}\ 0.2 \qquad \textbf{(C)}\ 0.3 \qquad \textbf{(D)}\ 0.4 \qquad \textbf{(E)}\ 0.5$

Russian TST 2014, P3

Tags: algebra , minimum value , trigonometry

Let $n>1$ be an integer and $x_1,x_2,\ldots,x_n$ be $n{}$ arbitrary real numbers. Determine the minimum value of \[\sum_{i<j}|\cos(x_i-x_j)|.\]

2008 India Regional Mathematical Olympiad, 1

Tags: semicircle , geometry , straight lines , trigonometry , intersection

On a semicircle with diameter $AB$ and centre $S$, points $C$ and $D$ are given such that point $C$ belongs to arc $AD$. Suppose $\angle CSD = 120^\circ$. Let $E$ be the point of intersection of the straight lines $AC$ and $BD$ and $F$ the point of intersection of the straight lines $AD$ and $BC$. Prove that $EF=\sqrt{3}AB$.

2011 China Team Selection Test, 1

Tags: geometry , rectangle , analytic geometry , trapezoid , geometric transformation , homothety , trigonometry

In $\triangle ABC$ we have $BC>CA>AB$. The nine point circle is tangent to the incircle, $A$-excircle, $B$-excircle and $C$-excircle at the points $T,T_A,T_B,T_C$ respectively. Prove that the segments $TT_B$ and lines $T_AT_C$ intersect each other.

2010 International Zhautykov Olympiad, 3

Tags: incenter , trigonometry , inequalities , euler , geometry

Let $ABC$ arbitrary triangle ($AB \neq BC \neq AC \neq AB$) And O,I,H it's circum-center, incenter and ortocenter (point of intersection altitudes). Prove, that 1) $\angle OIH > 90^0$(2 points) 2)$\angle OIH >135^0$(7 points) balls for 1) and 2) not additive.

2007 JBMO Shortlist, 2

Tags: trigonometry , geometry , incenter , function , circumcircle , cyclic quadrilateral , angle bisector

Let $ABCD$ be a convex quadrilateral with $\angle{DAC}= \angle{BDC}= 36^\circ$ , $\angle{CBD}= 18^\circ$ and $\angle{BAC}= 72^\circ$. The diagonals and intersect at point $P$ . Determine the measure of $\angle{APD}$.

1999 Harvard-MIT Mathematics Tournament, 2

Tags: calculus , geometry , rectangle , trigonometry

A rectangle has sides of length $\sin x$ and $\cos x$ for some $x$. What is the largest possible area of such a rectangle?

1998 Iran MO (3rd Round), 2

Tags: reflection , trigonometry , circumcircle , isogonal conjugate , geometry

Let $ M$ and $ N$ be two points inside triangle $ ABC$ such that \[ \angle MAB \equal{} \angle NAC\quad \mbox{and}\quad \angle MBA \equal{} \angle NBC. \] Prove that \[ \frac {AM \cdot AN}{AB \cdot AC} \plus{} \frac {BM \cdot BN}{BA \cdot BC} \plus{} \frac {CM \cdot CN}{CA \cdot CB} \equal{} 1. \]

2009 Canadian Mathematical Olympiad Qualification Repechage, 2

Tags: pythagorean theorem , trigonometry , geometry

Triangle $ABC$ is right-angled at $C$ with $AC = b$ and $BC = a$. If $d$ is the length of the altitude from $C$ to $AB$, prove that $\dfrac{1}{a^2}+\dfrac{1}{b^2}=\dfrac{1}{d^2}$

2003 Romania Team Selection Test, 11

Tags: trigonometry , pigeonhole principle , area of a triangle , geometry

In a square of side 6 the points $A,B,C,D$ are given such that the distance between any two of the four points is at least 5. Prove that $A,B,C,D$ form a convex quadrilateral and its area is greater than 21. [i]Laurentiu Panaitopol[/i]

2005 Today's Calculation Of Integral, 33

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

Evaluate \[\int_{-\ln 2}^0\ \frac{dx}{\cos ^2 h x \cdot \sqrt{1-2a\tanh x +a^2}}\ (a>0)\]

1997 APMO, 3

Tags: trigonometry , geometry , circumcircle , inequalities , ratio , angle bisector , geometric inequality

Let $ABC$ be a triangle inscribed in a circle and let \[ l_a = \frac{m_a}{M_a} \ , \ \ l_b = \frac{m_b}{M_b} \ , \ \ l_c = \frac{m_c}{M_c} \ , \] where $m_a$,$m_b$, $m_c$ are the lengths of the angle bisectors (internal to the triangle) and $M_a$, $M_b$, $M_c$ are the lengths of the angle bisectors extended until they meet the circle. Prove that \[ \frac{l_a}{\sin^2 A} + \frac{l_b}{\sin^2 B} + \frac{l_c}{\sin^2 C} \geq 3 \] and that equality holds iff $ABC$ is an equilateral triangle.

2013 Canada National Olympiad, 5

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

Let $O$ denote the circumcentre of an acute-angled triangle $ABC$. Let point $P$ on side $AB$ be such that $\angle BOP = \angle ABC$, and let point $Q$ on side $AC$ be such that $\angle COQ = \angle ACB$. Prove that the reflection of $BC$ in the line $PQ$ is tangent to the circumcircle of triangle $APQ$.

1951 Poland - Second Round, 5

Tags: geometry , trigonometry

Prove that if the relationship between the sides and opposite angles $ A $ and $ B $ of the triangle $ ABC $ is $$ (a^2 + b^2) \sin (A - B) = (a^2 - b^2) \sin (A + B)$$ then such a triangle is right-angled or isosceles.