Olympiad problems

2009 Middle European Mathematical Olympiad, 6

Let $ a$, $ b$, $ c$ be real numbers such that for every two of the equations \[ x^2\plus{}ax\plus{}b\equal{}0, \quad x^2\plus{}bx\plus{}c\equal{}0, \quad x^2\plus{}cx\plus{}a\equal{}0\] there is exactly one real number satisfying both of them. Determine all possible values of $ a^2\plus{}b^2\plus{}c^2$.

2011 Brazil National Olympiad, 5

Tags: geometry , circumcircle , trigonometry , parallelogram , harmonic quadrilateral

Let $ABC$ be an acute triangle and $H$ is orthocenter. Let $D$ be the intersection of $BH$ and $AC$ and $E$ be the intersection of $CH$ and $AB$. The circumcircle of $ADE$ cuts the circumcircle of $ABC$ at $F \neq A$. Prove that the angle bisectors of $\angle BFC$ and $\angle BHC$ concur at a point on $BC.$

1995 Turkey MO (2nd round), 4

Tags: ratio , trigonometry , geometry

In a triangle $ABC$ with $AB\neq AC$, the internal and external bisectors of angle $A$ meet the line $BC$ at $D$ and $E$ respectively. If the feet of the perpendiculars from a point $F$ on the circle with diameter $DE$ to $BC,CA,AB$ are $K,L,M$, respectively, show that $KL=KM$.

2002 France Team Selection Test, 2

Tags: incenter , circumcircle , trigonometry , geometry

Let $ ABC$ be a non-equilateral triangle. Denote by $ I$ the incenter and by $ O$ the circumcenter of the triangle $ ABC$. Prove that $ \angle AIO\leq\frac{\pi}{2}$ holds if and only if $ 2\cdot BC\leq AB\plus{}AC$.

2008 Moldova National Olympiad, 12.8

Tags: integration , trigonometry , logarithm , real analysis

Evaluate $ \displaystyle I \equal{} \int_0^{\frac\pi4}\left(\sin^62x \plus{} \cos^62x\right)\cdot \ln(1 \plus{} \tan x)\text{d}x$.

2008 Bosnia And Herzegovina - Regional Olympiad, 1

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

Given is an acute angled triangle $ \triangle ABC$ with side lengths $ a$, $ b$ and $ c$ (in an usual way) and circumcenter $ O$. Angle bisector of angle $ \angle BAC$ intersects circumcircle at points $ A$ and $ A_{1}$. Let $ D$ be projection of point $ A_{1}$ onto line $ AB$, $ L$ and $ M$ be midpoints of $ AC$ and $ AB$ , respectively. (i) Prove that $ AD\equal{}\frac{1}{2}(b\plus{}c)$ (ii) If triangle $ \triangle ABC$ is an acute angled prove that $ A_{1}D\equal{}OM\plus{}OL$

1992 IMO Longlists, 4

Tags: trigonometry , geometry , area of a triangle , geometric inequality

Let $p, q$, and $r$ be the angles of a triangle, and let $a = \sin2p, b = \sin2q$, and $c = \sin2r$. If $s = \frac{(a + b + c)}2$, show that \[s(s - a)(s - b)(s -c) \geq 0.\] When does equality hold?

2007 Bulgaria Team Selection Test, 1

Tags: trigonometry , geometric transformation , homothety , power of a point , radical axis , geometry

In isosceles triangle $ABC(AC=BC)$ the point $M$ is in the segment $AB$ such that $AM=2MB,$ $F$ is the midpoint of $BC$ and $H$ is the orthogonal projection of $M$ in $AF.$ Prove that $\angle BHF=\angle ABC.$

2007 ISI B.Stat Entrance Exam, 5

Tags: trigonometry , inequalities

Show that \[-2 \le \cos \theta\left(\sin \theta + \sqrt{\sin ^2 \theta +3}\right) \le 2\] for all values of $\theta$.

2007 China Western Mathematical Olympiad, 1

Tags: trigonometry , geometry

Is there a triangle with sides of integer lengths such that the length of the shortest side is $ 2007$ and that the largest angle is twice the smallest?

1975 IMO Shortlist, 15

Tags: trigonometry , algebra , point set , rational , roots of unity

Can there be drawn on a circle of radius $1$ a number of $1975$ distinct points, so that the distance (measured on the chord) between any two points (from the considered points) is a rational number?

1995 India National Olympiad, 4

Tags: ratio , geometry , trigonometry , incenter , homothety , angle bisector

Let $ABC$ be a triangle and a circle $\Gamma'$ be drawn lying outside the triangle, touching its incircle $\Gamma$ externally, and also the two sides $AB$ and $AC$. Show that the ratio of the radii of the circles $\Gamma'$ and $\Gamma$ is equal to $\tan^ 2 { \left( \dfrac{ \pi - A }{4} \right) }.$

VI Soros Olympiad 1999 - 2000 (Russia), 10.1

Tags: algebra , number theory , trigonometry

For a real number $a$, denote by $(a]$ the smallest integer that is not less than $a$. Find all real values of $x$ for which holds the equality $$(\sin x]^2 + (\cos x]^2 =|tg x| +|ctg x|.$$

VI Soros Olympiad 1999 - 2000 (Russia), 10.2

Tags: algebra , trigonometry

Solve the equation $$\frac{\pi-2}{2} + \frac{2}{1+\sin (2\sqrt{x})}+arccos(x^3-8x-1)=tg^2\sqrt{x}- \sqrt{x^4+x^3-5x^2-8x-24}$$

2013 Sharygin Geometry Olympiad, 3

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

Let $ABC$ be a right-angled triangle ($\angle B = 90^\circ$). The excircle inscribed into the angle $A$ touches the extensions of the sides $AB$, $AC$ at points $A_1, A_2$ respectively; points $C_1, C_2$ are defined similarly. Prove that the perpendiculars from $A, B, C$ to $C_1C_2, A_1C_1, A_1A_2$ respectively, concur.

2013 ELMO Shortlist, 7

Tags: ratio , geometry , parallelogram , circumcircle , trigonometry , collinear , complex

Let $ABC$ be a triangle inscribed in circle $\omega$, and let the medians from $B$ and $C$ intersect $\omega$ at $D$ and $E$ respectively. Let $O_1$ be the center of the circle through $D$ tangent to $AC$ at $C$, and let $O_2$ be the center of the circle through $E$ tangent to $AB$ at $B$. Prove that $O_1$, $O_2$, and the nine-point center of $ABC$ are collinear. [i]Proposed by Michael Kural[/i]

2016 German National Olympiad, 5

Tags: geometry , cyclic quadrilateral , trigonometry , angle

Let $A,B,C,D$ be points on a circle with radius $r$ in this order such that $|AB|=|BC|=|CD|=s$ and $|AD|=s+r$. Find all possible values of the interior angles of the quadrilateral $ABCD$.

2002 India IMO Training Camp, 15

Tags: inequalities , trigonometry , algebra

Let $x_1,x_2,\ldots,x_n$ be arbitrary real numbers. Prove the inequality \[ \frac{x_1}{1+x_1^2} + \frac{x_2}{1+x_1^2 + x_2^2} + \cdots + \frac{x_n}{1 + x_1^2 + \cdots + x_n^2} < \sqrt{n}. \]

2008 AIME Problems, 8

Tags: trigonometry , modular arithmetic

Let $ a\equal{}\pi/2008$. Find the smallest positive integer $ n$ such that \[ 2[\cos(a)\sin(a)\plus{}\cos(4a)\sin(2a)\plus{}\cos(9a)\sin(3a)\plus{}\cdots\plus{}\cos(n^2a)\sin(na)]\] is an integer.

2000 Estonia National Olympiad, 4

Tags: trigonometry , sidelenghts , angle

Prove that for any triangle the equation holds $a \cdot \cos (\beta + \gamma ) + b \cdot \cos (\gamma +\alpha) + c\cdot \cos (\alpha -\beta) = 0$, where $a, b, c$ are the sides of the triangle and $\alpha, \beta,\gamma$ according to their angles sizes of opposite angles.

1981 AMC 12/AHSME, 21

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

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 $\displaystyle \text{(A)} \ 15^\circ \qquad \text{(B)} \ 30^\circ \qquad \text{(C)} \ 45^\circ \qquad \text{(D)} \ 60^\circ \qquad \text{(E)} \ 150^\circ$

1999 Vietnam National Olympiad, 2

Tags: trigonometry , vector , geometry

$ OA, OB, OC, OD$ are 4 rays in space such that the angle between any two is the same. Show that for a variable ray $ OX,$ the sum of the cosines of the angles $ XOA, XOB, XOC, XOD$ is constant and the sum of the squares of the cosines is also constant.