Olympiad problems

2006 Kyiv Mathematical Festival, 4

Tags: geometry , circumcircle , incenter , trigonometry , trig identities , Law of Sines , geometry proposed

See all the problems from 5-th Kyiv math festival [url=http://www.mathlinks.ro/Forum/viewtopic.php?p=506789#p506789]here[/url] Let $O$ be the circumcenter and $H$ be the intersection point of the altitudes of acute triangle $ABC.$ The straight lines $BH$ and $CH$ intersect the segments $CO$ and $BO$ at points $D$ and $E$ respectively. Prove that if triangles $ODH$ and $OEH$ are isosceles then triangle $ABC$ is isosceles too.

2010 Turkey Team Selection Test, 1

Tags: geometry , incenter , circumcircle , trigonometry , trig identities , Law of Sines , geometry proposed

$D, \: E , \: F$ are points on the sides $AB, \: BC, \: CA,$ respectively, of a triangle $ABC$ such that $AD=AF, \: BD=BE,$ and $DE=DF.$ Let $I$ be the incenter of the triangle $ABC,$ and let $K$ be the point of intersection of the line $BI$ and the tangent line through $A$ to the circumcircle of the triangle $ABI.$ Show that $AK=EK$ if $AK=AD.$

1953 AMC 12/AHSME, 37

Tags: geometry , circumcircle , trigonometry , perpendicular bisector , trig identities , Law of Sines

The base of an isosceles triangle is $ 6$ inches and one of the equal sides is $ 12$ inches. The radius of the circle through the vertices of the triangle is: $ \textbf{(A)}\ \frac{7\sqrt{15}}{5} \qquad\textbf{(B)}\ 4\sqrt{3} \qquad\textbf{(C)}\ 3\sqrt{5} \qquad\textbf{(D)}\ 6\sqrt{3} \qquad\textbf{(E)}\ \text{none of these}$

1992 IMO Shortlist, 11

Tags: trigonometry , geometry , Law of Sines , Triangle , IMO Shortlist , IMO Longlist

In a triangle $ ABC,$ let $ D$ and $ E$ be the intersections of the bisectors of $ \angle ABC$ and $ \angle ACB$ with the sides $ AC,AB,$ respectively. Determine the angles $ \angle A,\angle B, \angle C$ if $ \angle BDE \equal{} 24 ^{\circ},$ $ \angle CED \equal{} 18 ^{\circ}.$

2010 Contests, 2

Tags: geometry , trigonometry , symmetry , trig identities , Law of Sines

Let $ABC$ be a triangle with $AB = AC$. The incircle touches $BC$, $AC$ and $AB$ at $D$, $E$ and $F$ respectively. Let $P$ be a point on the arc $\overarc{EF}$ that does not contain $D$. Let $Q$ be the second point of intersection of $BP$ and the incircle of $ABC$. The lines $EP$ and $EQ$ meet the line $BC$ at $M$ and $N$, respectively. Prove that the four points $P, F, B, M$ lie on a circle and $\frac{EM}{EN} = \frac{BF}{BP}$.

1991 IMO Shortlist, 5

Tags: geometry , incenter , trigonometry , trig identities , Law of Sines , IMO Shortlist

In the triangle $ ABC,$ with $ \angle A \equal{} 60 ^{\circ},$ a parallel $ IF$ to $ AC$ is drawn through the incenter $ I$ of the triangle, where $ F$ lies on the side $ AB.$ The point $ P$ on the side $ BC$ is such that $ 3BP \equal{} BC.$ Show that $ \angle BFP \equal{} \frac{\angle B}{2}.$

1973 AMC 12/AHSME, 4

Tags: geometry , trigonometry , trig identities , Law of Sines , Law of Cosines

Two congruent $ 30^{\circ}$-$ 60^{\circ}$-$ 90^{\circ}$ are placed so that they overlap partly and their hypotenuses coincide. If the hypotenuse of each triangle is 12, the area common to both triangles is $ \textbf{(A)}\ 6\sqrt3 \qquad \textbf{(B)}\ 8\sqrt3 \qquad \textbf{(C)}\ 9\sqrt3 \qquad \textbf{(D)}\ 12\sqrt3 \qquad \textbf{(E)}\ 24$

2025 Sharygin Geometry Olympiad, 3

Tags: geometry , Law of Sines , excircle , pedal circles , Sharygin Geometry Olympiad

An excircle centered at $I_{A}$ touches the side $BC$ of a triangle $ABC$ at point $D$. Prove that the pedal circles of $D$ with respect to the triangles $ABI_{A}$ and $ACI_{A}$ are congruent. Proposed by:K.Belsky

1994 Brazil National Olympiad, 2

Tags: geometry , circumcircle , trigonometry , combinatorial geometry , trig identities , Law of Sines , geometry proposed

Given any convex polygon, show that there are three consecutive vertices such that the polygon lies inside the circle through them.

1993 Brazil National Olympiad, 4

Tags: trigonometry , geometry , circumcircle , Law of Sines , perpendicular bisector , Brazilian Math Olympiad

$ABCD$ is a convex quadrilateral with \[\angle BAC = 30^\circ \]\[\angle CAD = 20^\circ\]\[\angle ABD = 50^\circ\]\[\angle DBC = 30^\circ\] If the diagonals intersect at $P$, show that $PC = PD$.

2012 AMC 10, 14

Tags: geometry , rhombus , trigonometry , trig identities , Law of Sines , AMC

Two equilateral triangles are contained in a square whose side length is $2\sqrt3$. The bases of these triangles are the opposite sides of the square, and their intersection is a rhombus. What is the area of the rhombus? $ \textbf{(A)}\ \frac{3}{2}\qquad\textbf{(B)}\ \sqrt3\qquad\textbf{(C)}\ 2\sqrt2-1\qquad\textbf{(D)}\ 8\sqrt3-12\qquad\textbf{(E)}\ \frac{4\sqrt3}{3}$

2003 AIME Problems, 10

Tags: trigonometry , AMC , AIME , geometry , trig identities , Law of Sines , Law of Cosines

Triangle $ABC$ is isosceles with $AC = BC$ and $\angle ACB = 106^\circ$. Point $M$ is in the interior of the triangle so that $\angle MAC = 7^\circ$ and $\angle MCA = 23^\circ$. Find the number of degrees in $\angle CMB$.

2012 Sharygin Geometry Olympiad, 7

Tags: trigonometry , geometry , trig identities , Law of Sines , geometry unsolved

In a non-isosceles triangle $ABC$ the bisectors of angles $A$ and $B$ are inversely proportional to the respective sidelengths. Find angle $C$.

2006 Kyiv Mathematical Festival, 3

Tags: geometry , circumcircle , incenter , trigonometry , trig identities , Law of Sines , geometry proposed

See all the problems from 5-th Kyiv math festival [url=http://www.mathlinks.ro/Forum/viewtopic.php?p=506789#p506789]here[/url] Let $O$ be the circumcenter and $H$ be the intersection point of the altitudes of acute triangle $ABC.$ The straight lines $BH$ and $CH$ intersect the segments $CO$ and $BO$ at points $D$ and $E$ respectively. Prove that if triangles $ODH$ and $OEH$ are isosceles then triangle $ABC$ is isosceles too.

2003 France Team Selection Test, 3

Tags: geometry , circumcircle , trigonometry , incenter , trig identities , Law of Sines , geometry unsolved

$M$ is an arbitrary point inside $\triangle ABC$. $AM$ intersects the circumcircle of the triangle again at $A_1$. Find the points $M$ that minimise $\frac{MB\cdot MC}{MA_1}$.

2005 China Girls Math Olympiad, 2

Tags: trigonometry , geometry , trig identities , Law of Sines , algebra unsolved , algebra

Find all ordered triples $ (x, y, z)$ of real numbers such that \[ 5 \left(x \plus{} \frac{1}{x} \right) \equal{} 12 \left(y \plus{} \frac{1}{y} \right) \equal{} 13 \left(z \plus{} \frac{1}{z} \right),\] and \[ xy \plus{} yz \plus{} zy \equal{} 1.\]

2003 Bulgaria Team Selection Test, 5

Tags: Asymptote , geometry , trigonometry , trig identities , Law of Sines , geometry proposed

Let $ABCD$ be a circumscribed quadrilateral and let $P$ be the orthogonal projection of its in center on $AC$. Prove that $\angle {APB}=\angle {APD}$

2013 Canadian Mathematical Olympiad Qualification Repechage, 2

Tags: trigonometry , geometry , trig identities , Law of Sines , geometry proposed

In triangle $ABC$, $\angle A = 90^\circ$ and $\angle C = 70^\circ$. $F$ is point on $AB$ such that $\angle ACF = 30^\circ$, and $E$ is a point on $CA$ such that $\angle CF E = 20^\circ$. Prove that $BE$ bisects $\angle B$.

2013 India IMO Training Camp, 2

Tags: trigonometry , LaTeX , geometry , geometric transformation , reflection , trig identities , Law of Sines

In a triangle $ABC$ with $B = 90^\circ$, $D$ is a point on the segment $BC$ such that the inradii of triangles $ABD$ and $ADC$ are equal. If $\widehat{ADB} = \varphi$ then prove that $\tan^2 (\varphi/2) = \tan (C/2)$.

2020 Candian MO, 2#

Tags: USAMO , trigonometry , iscosceles triangle , Law of Cosines , Law of Sines , reflection , geometry

Let $ABC$ be a triangle, and $M$ an interior point such that $\angle MAB=10^\circ$, $\angle MBA=20^\circ$, $\angle MAC=40^\circ$ and $\angle MCA=30^\circ$. Prove that the triangle is isosceles.

2013 Stanford Mathematics Tournament, 2

Tags: geometry , quadratics , trigonometry , area of a triangle , trig identities , Law of Sines

Points $A$, $B$, and $C$ lie on a circle of radius $5$ such that $AB=6$ and $AC=8$. Find the smaller of the two possible values of $BC$.

2007 Purple Comet Problems, 23

Tags: geometry , trigonometry , trig identities , Law of Sines

Two circles with radius $2$ and radius $4$ have a common center at P. Points $A, B,$ and $C$ on the larger circle are the vertices of an equilateral triangle. Point $D$ is the intersection of the smaller circle and the line segment $PB$. Find the square of the area of triangle $ADC$.

2012 Turkey Team Selection Test, 1

Tags: geometry , circumcircle , trigonometry , geometric transformation , angle bisector , trig identities , Law of Sines

In a triangle $ABC,$ incircle touches the sides $BC, CA, AB$ at $D, E, F,$ respectively. A circle $\omega$ passing through $A$ and tangent to line $BC$ at $D$ intersects the line segments $BF$ and $CE$ at $K$ and $L,$ respectively. The line passing through $E$ and parallel to $DL$ intersects the line passing through $F$ and parallel to $DK$ at $P.$ If $R_1, R_2, R_3, R_4$ denotes the circumradius of the triangles $AFD, AED, FPD, EPD,$ respectively, prove that $R_1R_4=R_2R_3.$

2011 Indonesia MO, 3

Tags: geometry , circumcircle , trigonometry , inequalities , trig identities , Law of Sines , Law of Cosines

Given an acute triangle $ABC$, let $l_a$ be the line passing $A$ and perpendicular to $AB$, $l_b$ be the line passing $B$ and perpendicular to $BC$, and $l_c$ be the line passing $C$ and perpendicular to $CA$. Let $D$ be the intersection of $l_b$ and $l_c$, $E$ be the intersection of $l_c$ and $l_a$, and $F$ be the intersection of $l_a$ and $l_b$. Prove that the area of the triangle $DEF$ is at least three times of the area of $ABC$.