Olympiad problems

1982 Vietnam National Olympiad, 3

Tags: geometry

Let be given a triangle $ABC$. Equilateral triangles $BCA_1$ and $BCA_2$ are drawn so that $A$ and $A_1$ are on one side of $BC$, whereas $A_2$ is on the other side. Points $B_1,B_2,C_1,C_2$ are analogously defined. Prove that \[S_{ABC} + S_{A_1B_1C_1} = S_{A_2B_2C_2}.\]

2011 Federal Competition For Advanced Students, Part 2, 3

Tags: circumcircle , geometry , trapezoid , incenter

We are given a non-isosceles triangle $ABC$ with incenter $I$. Show that the circumcircle $k$ of the triangle $AIB$ does not touch the lines $CA$ and $CB$. Let $P$ be the second point of intersection of $k$ with $CA$ and let $Q$ be the second point of intersection of $k$ with $CB$. Show that the four points $A$, $B$, $P$ and $Q$ (not necessarily in this order) are the vertices of a trapezoid.

2014 India Regional Mathematical Olympiad, 3

Tags: geometry , trigonometry , angle bisector

Let $ABC$ be an acute-angled triangle in which $\angle ABC$ is the largest angle. Let $O$ be its circumcentre. The perpendicular bisectors of $BC$ and $AB$ meet $AC$ at $X$ and $Y$ respectively. The internal angle bisectors of $\angle AXB$ and $\angle BYC$ meet $AB$ and $BC$ at $D$ and $E$ respectively. Prove that $BO$ is perpendicular to $AC$ if $DE$ is parallel to $AC$.

2016 Argentina National Olympiad, 4

Tags: angle , geometry

Find the angles of a convex quadrilateral $ABCD$ such that $\angle ABD = 29^o$, $\angle ADB = 41^o$, $\angle ACB = 82^o$ and $\angle ACD = 58^o$

1990 Tournament Of Towns, (260) 4

Tags: trapezoid , angle , geometry

Let $ABCD$ be a trapezium with $AC = BC$. Let $H$ be the midpoint of the base $AB$ and let $\ell$ be a line passing through $H$. Let $\ell$ meet $AD$ at $P$ and $BD$ at $Q$. Prove that the angles $ACP$ and $QCB$ are either equal or have a sum of $180^o$. (I. Sharygin, Moscow)

2014 AMC 12/AHSME, 10

Tags: reflection , circumcircle , trig identities , law of sines , geometry , trigonometry , geometric transformation

Three congruent isosceles triangles are constructed with their bases on the sides of an equilateral triangle of side length $1$. The sum of the areas of the three isosceles triangles is the same as the area of the equilateral triangle. What is the length of one of the two congruent sides of one of the isosceles triangles? $\textbf{(A) }\dfrac{\sqrt3}4\qquad \textbf{(B) }\dfrac{\sqrt3}3\qquad \textbf{(C) }\dfrac23\qquad \textbf{(D) }\dfrac{\sqrt2}2\qquad \textbf{(E) }\dfrac{\sqrt3}2$

2014 Harvard-MIT Mathematics Tournament, 7

Tags: geometric transformation , reflection , geometry

The Evil League of Evil is plotting to poison the city's water supply. They plan to set out from their headquarters at $(5, 1)$ and put poison in two pipes, one along the line $y=x$ and one along the line $x=7$. However, they need to get the job done quickly before Captain Hammer catches them. What's the shortest distance they can travel to visit both pipes and then return to their headquarters?

1964 Vietnam National Olympiad, 1

Tags: trigonometry , sum , geometry

Given an arbitrary angle $\alpha$, compute $cos \alpha + cos \big( \alpha +\frac{2\pi }{3 }\big) + cos \big( \alpha +\frac{4\pi }{3 }\big)$ and $sin \alpha + sin \big( \alpha +\frac{2\pi }{3 } \big) + sin \big( \alpha +\frac{4\pi }{3 } \big)$ . Generalize this result and justify your answer.

1995 Moldova Team Selection Test, 8

Tags: geometry

Each pair of three circles have the common chords $AA_1, BB_1$ and $CC_1{}$ such that lines $AB{}$ and $A_1B_1$ intersect in point $M{}$, $BC$ and $B_1C_1$ intersect in point $N{}$, $CA{}$ and $C_1A_1$ intersect in point $P{}$. Prove that points $M, N$ and $P$ are collinear.

1974 IMO Longlists, 51

Tags: geometry , geometric transformation , combinatorics , vector

There are $n$ points on a flat piece of paper, any two of them at a distance of at least $2$ from each other. An inattentive pupil spills ink on a part of the paper such that the total area of the damaged part equals $\frac 32$. Prove that there exist two vectors of equal length less than $1$ and with their sum having a given direction, such that after a translation by either of these two vectors no points of the given set remain in the damaged area.

2002 AIME Problems, 2

Tags: rectangle , rhombus , ratio , geometry

The diagram shows twenty congruent circles arranged in three rows and enclosed in a rectangle. The circles are tangent to one another and to the sides of the rectangle as shown in the diagram. The ratio of the longer dimension of the rectangle to the shorter dimension can be written as $\frac{1}{2}\left(\sqrt{p}-q\right),$ where $p$ and $q$ are positive integers. Find $p+q.$ [asy] size(250);real x=sqrt(3); int i; draw(origin--(14,0)--(14,2+2x)--(0,2+2x)--cycle); for(i=0; i<7; i=i+1) { draw(Circle((2*i+1,1), 1)^^Circle((2*i+1,1+2x), 1)); } for(i=0; i<6; i=i+1) { draw(Circle((2*i+2,1+x), 1)); }[/asy]

2016 Israel Team Selection Test, 3

Tags: conic , geometry , 3d geometry , octahedron

Prove that there exists an ellipsoid touching all edges of an octahedron if and only if the octahedron's diagonals intersect. (Here an octahedron is a polyhedron consisting of eight triangular faces, twelve edges, and six vertices such that four faces meat at each vertex. The diagonals of an octahedron are the lines connecting pairs of vertices not connected by an edge).

2012 IMO Shortlist, G7

Tags: geometry , incenter , homothety

Let $ABCD$ be a convex quadrilateral with non-parallel sides $BC$ and $AD$. Assume that there is a point $E$ on the side $BC$ such that the quadrilaterals $ABED$ and $AECD$ are circumscribed. Prove that there is a point $F$ on the side $AD$ such that the quadrilaterals $ABCF$ and $BCDF$ are circumscribed if and only if $AB$ is parallel to $CD$.

2024 European Mathematical Cup, 3

Tags: parallelogram , incenter , geometry

Let $\omega$ be a semicircle with diamater $AB$. Let $M$ be the midpoint of $AB$. Let $X,Y$ be points on the same semiplane with $\omega$ with respect to the line $AB$ such that $AMXY$ is a parallelogram. Let $XM\cap \omega = C$ and $YM \cap \omega = D$. Let $I$ be the incenter of $\triangle XYM$. Let $AC \cap BD= E$ and $ME$ intersects $XY$ at $T$. Let the intersection point of $TI$ and $AB$ be $Q$ and let the perpendicular projection of $T$ onto $AB$ be $P$. Prove that $M$ is midpoint of $PQ$

2015 India IMO Training Camp, 1

Tags: geometry

Consider a fixed circle $\Gamma$ with three fixed points $A, B,$ and $C$ on it. Also, let us fix a real number $\lambda \in(0,1)$. For a variable point $P \not\in\{A, B, C\}$ on $\Gamma$, let $M$ be the point on the segment $CP$ such that $CM =\lambda\cdot CP$ . Let $Q$ be the second point of intersection of the circumcircles of the triangles $AMP$ and $BMC$. Prove that as $P$ varies, the point $Q$ lies on a fixed circle. [i]Proposed by Jack Edward Smith, UK[/i]

2023 Stanford Mathematics Tournament, 9

Tags: geometry

Triangle $\vartriangle ABC$ is isosceles with $AC = AB$, $BC = 1$, and $\angle BAC = 36^o$. Let $\omega$ be a circle with center B and radius $r_{\omega}= \frac{P_{ABC}}{4}$, where $P_{ABC}$ denotes the perimeter of $\vartriangle ABC$. Let $\omega$ intersect line $AB$ at $P$ and line $BC$ at $Q$. Let $I_B$ be the center of the excircle with of $\vartriangle ABC$ with respect to point $B$, and let $BI_B$ intersect $P Q$ at $S$. We draw a tangent line from $S$ to $\odot I_B$ that intersects $\odot I_B$ at point $T$. Compute the length of ST.

2014 Online Math Open Problems, 2

Tags: trigonometry , geometry

Consider two circles of radius one, and let $O$ and $O'$ denote their centers. Point $M$ is selected on either circle. If $OO' = 2014$, what is the largest possible area of triangle $OMO'$? [i]Proposed by Evan Chen[/i]

Russian TST 2022, P1

Tags: incenter , geometry

In triangle $ABC$, a point $M$ is the midpoint of $AB$, and a point $I$ is the incentre. Point $A_1$ is the reflection of $A$ in $BI$, and $B_1$ is the reflection of $B$ in $AI$. Let $N$ be the midpoint of $A_1B_1$. Prove that $IN > IM$.

2008 Princeton University Math Competition, A9

Tags: circumcircle , tetrahedron , 3d geometry , geometry

In tetrahedron $ABCD$ with circumradius $2$, $AB = 2$, $CD = \sqrt{7}$, and $\angle ABC = \angle BAD = \frac{\pi}{2}$. Find all possible angles between the planes containing $ABC$ and $ABD$.

2010 Czech-Polish-Slovak Match, 2

Tags: geometry

Given any $60$ points on a circle of radius $1$, prove that there is a point on the circle the sum of whose distances to these $60$ points is at most $80$.

2012 Balkan MO Shortlist, G3

Tags: circumcircle , congruent triangles , geometry

Let $ABC$ be a triangle with circumcircle $c$ and circumcenter $O$, and let $D$ be a point on the side $BC$ different from the vertices and the midpoint of $BC$. Let $K$ be the point where the circumcircle $c_1$ of the triangle $BOD$ intersects $c$ for the second time and let $Z$ be the point where $c_1$ meets the line $AB$. Let $M$ be the point where the circumcircle $c_2$ of the triangle $COD$ intersects $c$ for the second time and let $E$ be the point where $c_2$ meets the line $AC$. Finally let $N$ be the point where the circumcircle $c_3$ of the triangle $AEZ$ meets $c$ again. Prove that the triangles $ABC$ and $NKM$ are congruent.

1965 Spain Mathematical Olympiad, 3

Tags: geometry , length

A disk in a record turntable makes $100$ revolutions per minute and it plays during $24$ minutes and $30$ seconds. The recorded line over the disk is a spiral with a diameter that decreases uniformly from $29$cm to $11.5$cm. Compute the length of the recorded line.

2024 Thailand October Camp, 2

Tags: geometry

Let $ABC$ be a triangle with $AC > BC,$ let $\omega$ be the circumcircle of $\triangle ABC,$ and let $r$ be its radius. Point $P$ is chosen on $\overline{AC}$ such taht $BC=CP,$ and point $S$ is the foot of the perpendicular from $P$ to $\overline{AB}$. Ray $BP$ mets $\omega$ again at $D$. Point $Q$ is chosen on line $SP$ such that $PQ = r$ and $S,P,Q$ lie on a line in that order. Finally, let $E$ be a point satisfying $\overline{AE} \perp \overline{CQ}$ and $\overline{BE} \perp \overline{DQ}$. Prove that $E$ lies on $\omega$.

2005 China Team Selection Test, 2

Tags: geometry , perimeter , cyclic quadrilateral , pythagorean theorem

Cyclic quadrilateral $ABCD$ has positive integer side lengths $AB$, $BC$, $CA$, $AD$. It is known that $AD=2005$, $\angle{ABC}=\angle{ADC} = 90^o$, and $\max \{ AB,BC,CD \} < 2005$. Determine the maximum and minimum possible values for the perimeter of $ABCD$.

2017 Canada National Olympiad, 4

Tags: geometry , parallelogram

Let $ABCD$ be a parallelogram. Points $P$ and $Q$ lie inside $ABCD$ such that $\bigtriangleup ABP$ and $\bigtriangleup{BCQ}$ are equilateral. Prove that the intersection of the line through $P$ perpendicular to $PD$ and the line through $Q$ perpendicular to $DQ$ lies on the altitude from $B$ in $\bigtriangleup{ABC}$.