Olympiad problems

1991 APMO, 1

Tags: trigonometry , geometry

Let $G$ be the centroid of a triangle $ABC$, and $M$ be the midpoint of $BC$. Let $X$ be on $AB$ and $Y$ on $AC$ such that the points $X$, $Y$, and $G$ are collinear and $XY$ and $BC$ are parallel. Suppose that $XC$ and $GB$ intersect at $Q$ and $YB$ and $GC$ intersect at $P$. Show that triangle $MPQ$ is similar to triangle $ABC$.

2010 Today's Calculation Of Integral, 658

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

Consider a parameterized curve $C: x=e^{-t}\cos t,\ y=e^{-t}\sin t\left (0\leq t\leq \frac{\pi}{2}\right).$ (1) Find the length $L$ of $C$. (2) Find the area $S$ of the region enclosed by the $x,\ y$ axis and $C$. Please solve the problem without using the formula of area for polar coordinate for Japanese High School Students who don't study it in High School. [i]1997 Kyoto University entrance exam/Science[/i]

1990 AIME Problems, 10

Tags: trigonometry , number theory , least common multiple , greatest common divisor , modular arithmetic , complex number , relatively prime

The sets $A = \{z : z^{18} = 1\}$ and $B = \{w : w^{48} = 1\}$ are both sets of complex roots of unity. The set $C = \{zw : z \in A \ \text{and} \ w \in B\}$ is also a set of complex roots of unity. How many distinct elements are in $C$?

2014 Taiwan TST Round 1, 1

Tags: geometry , trigonometry , geometric transformation

Let $O_1$, $O_2$ be two circles with radii $R_1$ and $R_2$, and suppose the circles meet at $A$ and $D$. Draw a line $L$ through $D$ intersecting $O_1$, $O_2$ at $B$ and $C$. Now allow the distance between the centers as well as the choice of $L$ to vary. Find the length of $AD$ when the area of $ABC$ is maximized.

1979 Chisinau City MO, 176

Tags: trigonometry , algebra

Indicate all the roots of the equation $x^2+1 = \cos x$.

III Soros Olympiad 1996 - 97 (Russia), 11.4

Tags: inequalities , algebra , trigonometry

Find the smallest value of a function $$y = \cos 8x + 3\cos 4x +3\cos2x + 2\cos x.$$

2006 MOP Homework, 1

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

$ ABC$ is an acute triangle. The points $ B'$ and $ C'$are the reflections of $ B$ and $ C$ across the lines $ AC$ and $ AB$ respectively. Suppose that the circumcircles of triangles$ ABB$' and $ ACC'$ meet at $ A$ and $ P$. Prove that the line $ PA$ passes through the circumcenter of triangle$ ABC.$

1990 APMO, 1

Tags: trigonometry , similar triangles , cyclic quadrilateral , perpendicular bisector , geometry

Given triangle $ABC$, let $D$, $E$, $F$ be the midpoints of $BC$, $AC$, $AB$ respectively and let $G$ be the centroid of the triangle. For each value of $\angle BAC$, how many non-similar triangles are there in which $AEGF$ is a cyclic quadrilateral?

Today's calculation of integrals, 886

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

Find the functions $f(x),\ g(x)$ such that $f(x)=e^{x}\sin x+\int_0^{\pi} ug(u)\ du$ $g(x)=e^{x}\cos x+\int_0^{\pi} uf(u)\ du$

2008 Moldova MO 11-12, 2

Tags: integration , trigonometry , real analysis

Find the exact value of $ E\equal{}\displaystyle\int_0^{\frac\pi2}\cos^{1003}x\text{d}x\cdot\int_0^{\frac\pi2}\cos^{1004}x\text{d}x\cdot$.

Indonesia MO Shortlist - geometry, g11.8

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.

2016 BMT Spring, 20

Tags: trigonometry , algebra

Find $$\prod^{2017}_{k=1} e^{\pi ik/2017}2 cos \left( \frac{\pi k}{2017} \right)$$

1990 Poland - Second Round, 4

Tags: algebra , trigonometry

For each pair of even natural numbers $ k $, $ m $determine all real numbers $ x $that satisfy the equation $$ (\sin x)^k + (\cos x)^{-m} = (\cos x)^k + (\sin x)^{-m}$$

1996 India National Olympiad, 2

Tags: inequalities , analytic geometry , vector , circumcircle , function , trigonometry , geometry

Let $C_1$ and $C_2$ be two concentric circles in the plane with radii $R$ and $3R$ respectively. Show that the orthocenter of any triangle inscribed in circle $C_1$ lies in the interior of circle $C_2$. Conversely, show that every point in the interior of $C_2$ is the orthocenter of some triangle inscribed in $C_1$.

1996 AIME Problems, 10

Tags: trigonometry , modular arithmetic , algorithm , function

Find the smallest positive integer solution to $\tan 19x^\circ=\frac{\cos 96^\circ+\sin 96^\circ}{\cos 96^\circ-\sin 96^\circ}.$

2002 Czech and Slovak Olympiad III A, 5

Tags: geometry , rectangle , analytic geometry , trigonometry , parameterization , graphing lines , conic

A triangle $KLM$ is given in the plane together with a point $A$ lying on the half-line opposite to $KL$. Construct a rectangle $ABCD$ whose vertices $B, C$ and $D$ lie on the lines $KM, KL$ and $LM$, respectively. (We allow the rectangle to be a square.)

1991 AIME Problems, 12

Tags: geometry , rhombus , rectangle , perimeter , geometric transformation , reflection , trigonometry

Rhombus $PQRS$ is inscribed in rectangle $ABCD$ so that vertices $P$, $Q$, $R$, and $S$ are interior points on sides $\overline{AB}$, $\overline{BC}$, $\overline{CD}$, and $\overline{DA}$, respectively. It is given that $PB=15$, $BQ=20$, $PR=30$, and $QS=40$. Let $m/n$, in lowest terms, denote the perimeter of $ABCD$. Find $m+n$.

2008 Bosnia Herzegovina Team Selection Test, 1

Tags: inequalities , trigonometry , calculus , derivative , geometry

Prove that in an isosceles triangle $ \triangle ABC$ with $ AC\equal{}BC\equal{}b$ following inequality holds $ b> \pi r$, where $ r$ is inradius.

1977 Polish MO Finals, 1

Tags: 3d geometry , tetrahedron , trigonometry , trig identities , law of sines , geometry

Let $ABCD$ be a tetrahedron with $\angle BAD = 60^{\cdot}$, $\angle BAC = 40^{\cdot}$, $\angle ABD = 80^{\cdot}$, $\angle ABC = 70^{\cdot}$. Prove that the lines $AB$ and $CD$ are perpendicular.

2000 AIME Problems, 14

Tags: geometry , rhombus , trapezoid , floor function , trigonometry

In triangle $ABC,$ it is given that angles $B$ and $C$ are congruent. Points $P$ and $Q$ lie on $\overline{AC}$ and $\overline{AB},$ respectively, so that $AP=PQ=QB=BC.$ Angle $ACB$ is $r$ times as large as angle $APQ,$ where $r$ is a positive real number. Find the greatest integer that does not exceed $1000r.$

2001 Turkey MO (2nd round), 2

Tags: trigonometry , circumcircle , perpendicular bisector , geometric transformation , geometry

Two nonperpendicular lines throught the point $A$ and a point $F$ on one of these lines different from $A$ are given. Let $P_{G}$ be the intersection point of tangent lines at $G$ and $F$ to the circle through the point $A$, $F$ and $G$ where $G$ is a point on the given line different from the line $FA$. What is the locus of $P_{G}$ as $G$ varies.

2010 China Team Selection Test, 1

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

Let $\omega$ be a semicircle and $AB$ its diameter. $\omega_1$ and $\omega_2$ are two different circles, both tangent to $\omega$ and to $AB$, and $\omega_1$ is also tangent to $\omega_2$. Let $P,Q$ be the tangent points of $\omega_1$ and $\omega_2$ to $AB$ respectively, and $P$ is between $A$ and $Q$. Let $C$ be the tangent point of $\omega_1$ and $\omega$. Find $\tan\angle ACQ$.

2014 Baltic Way, 5

Tags: geometry , trigonometry , cyclic quadrilateral , algebra

Given positive real numbers $a, b, c, d$ that satisfy equalities \[a^2 + d^2 - ad = b^2 + c^2 + bc \ \ \text{and} \ \ a^2 + b^2 = c^2 + d^2\] find all possible values of the expression $\frac{ab+cd}{ad+bc}.$

2000 Brazil Team Selection Test, Problem 1

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

Consider a triangle $ABC$ and $I$ its incenter. The line $(AI)$ meets the circumcircle of $ABC$ in $D$. Let $E$ and $F$ be the orthogonal projections of $I$ on $(BD)$ and $(CD)$ respectively. Assume that $IE+IF=\frac{1}{2}AD$. Calculate $\angle{BAC}$. [color=red][Moderator edited: Also discussed at http://www.mathlinks.ro/Forum/viewtopic.php?t=5088 .][/color]

1999 Putnam, 5

Tags: linear algebra , matrix , vector , trigonometry

For an integer $n\geq 3$, let $\theta=2\pi/n$. Evaluate the determinant of the $n\times n$ matrix $I+A$, where $I$ is the $n\times n$ identity matrix and $A=(a_{jk})$ has entries $a_{jk}=\cos(j\theta+k\theta)$ for all $j,k$.