Olympiad problems

MathLinks Contest 6th, 3.2

Tags: geometry

Let $ABCD$ be a convex quadrilateral, and the points $A_1 \in (CD)$, $A_2 \in (BC)$, $C_1 \in (AB)$, $C_2 \in (AD)$. Let $M, N$ be the intersection points between the lines $AA_2, CC_1$ and $AA_1, CC_2$ respectively. Prove that if three of the quadrilaterals $ABCD$, $A_2BC_1M$, $AMCN$, $A_1NC_2D$ are circumscriptive (i.e. there exists an incircle tangent to all the sides of the quadrilateral) then the forth quadrilateral is also circumscriptive.

2013 Korea - Final Round, 4

Tags: circumcircle , geometry , geometric transformation , homothety

For a triangle $ ABC $, let $ B_1 ,C_1 $ be the excenters of $ B, C $. Line $B_1 C_1 $ meets with the circumcircle of $ \triangle ABC $ at point $ D (\ne A) $. $ E $ is the point which satisfies $ B_1 E \bot CA $ and $ C_1 E \bot AB $. Let $ w $ be the circumcircle of $ \triangle ADE $. The tangent to the circle $ w $ at $ D $ meets $ AE $ at $ F $. $ G , H $ are the points on $ AE, w $ such that $ DGH \bot AE $. The circumcircle of $ \triangle HGF $ meets $ w $ at point $ I ( \ne H ) $, and $ J $ be the foot of perpendicular from $ D $ to $ AH $. Prove that $ AI $ passes the midpoint of $ DJ $.

1991 AMC 12/AHSME, 14

Tags: prime factorization , 3d geometry , geometry , number theory

If $x$ is the cube of a positive integer and $d$ is the number of positive integers that are divisors of $x$, then $d$ could be $ \textbf{(A)}\ 200\qquad\textbf{(B)}\ 201\qquad\textbf{(C)}\ 202\qquad\textbf{(D)}\ 203\qquad\textbf{(E)}\ 204 $

1975 IMO, 3

Tags: trigonometry , angle , triangle , geometry

In the plane of a triangle $ABC,$ in its exterior$,$ we draw the triangles $ABR, BCP, CAQ$ so that $\angle PBC = \angle CAQ = 45^{\circ}$, $\angle BCP = \angle QCA = 30^{\circ}$, $\angle ABR = \angle RAB = 15^{\circ}$. Prove that [b]a.)[/b] $\angle QRP = 90\,^{\circ},$ and [b]b.)[/b] $QR = RP.$

1987 IMO Shortlist, 12

Tags: geometry , collinearity , circumcircle , incenter

Given a nonequilateral triangle $ABC$, the vertices listed counterclockwise, find the locus of the centroids of the equilateral triangles $A'B'C'$ (the vertices listed counterclockwise) for which the triples of points $A,B', C'; A',B, C';$ and $A',B', C$ are collinear. [i]Proposed by Poland.[/i]

2016 IFYM, Sozopol, 6

Tags: geometry , polygon

On the sides of a convex, non-regular $m$-gon are built externally regular heptagons. It is known that their centers are vertices of a regular $m$-gon. What’s the least possible value of $m$?

2006 Sharygin Geometry Olympiad, 6

Tags: circles , construction , tangent , geometry

a) Given a segment $AB$ with a point $C$ inside it, which is the chord of a circle of radius $R$. Inscribe in the formed segment a circle tangent to point $C$ and to the circle of radius $R$. b) Given a segment $AB$ with a point $C$ inside it, which is the point of tangency of a circle of radius $r$. Draw through $A$ and $B$ a circle tangent to a circle of radius $r$.

1950 AMC 12/AHSME, 46

Tags: geometry

In triangle $ABC$, $AB=12$, $AC=7$, and $BC=10$. If sides $AB$ and $AC$ are doubled while $BC$ remains the same, then: $\textbf{(A)}\ \text{The area is doubled} \qquad\\ \textbf{(B)}\ \text{The altitude is doubled} \qquad\\ \textbf{(C)}\ \text{The area is four times the original area} \qquad\\ \textbf{(D)}\ \text{The median is unchanged} \qquad\\ \textbf{(E)}\ \text{The area of the triangle is 0}$

2002 All-Russian Olympiad Regional Round, 9.3

Tags: geometry , concyclic

In an isosceles triangle $ABC$ ($AB = BC$), point $O$ is the center of the circumcircle. Point $M$ lies on the segment $BO$, point $M' $ is symmetric to $M$ wrt the midpoint of $AB$. Point K is the intersection point of of $M'O$ and $AB$. Point $L$ lies on side BC such that $\angle CLO = \angle BLM$. Prove that points $O, K,B,L$ lie on the same circle

1938 Eotvos Mathematical Competition, 3

Tags: 3d geometry , geometry

Prove that for any acute triangle, there is a point in space such that every line segment from a vertex of the triangle to a point on the line joining the other two vertices subtends a right angle at this point.

2001 Estonia National Olympiad, 1

Tags: geometry , angle , convex , polygon

A convex $n$-gon has exactly three obtuse interior angles. Find all possible values of $n$.

2011 Dutch BxMO TST, 5

Tags: trapezoid , angle bisector , geometry

A trapezoid $ABCD$ is given with $BC // AD$. Assume that the bisectors of the angles $BAD$ and $CDA$ intersect on the perpendicular bisector of the line segment $BC$. Prove that $|AB|= |CD|$ or $|AB| +|CD| =|AD|$.

2007 Estonia National Olympiad, 3

Tags: circumcircle , geometry

A circle passing through the endpoints of the leg AB of an isosceles triangle ABC intersects the base BC in point P. A line tangent to the circle in point B intersects the circumcircle of ABC in point Q. Prove that P lies on line AQ if and only if AQ and BC are perpendicular.

2012 IMO Shortlist, G5

Tags: right triangle , geometry , circumcircle

Let $ABC$ be a triangle with $\angle BCA=90^{\circ}$, and let $D$ be the foot of the altitude from $C$. Let $X$ be a point in the interior of the segment $CD$. Let $K$ be the point on the segment $AX$ such that $BK=BC$. Similarly, let $L$ be the point on the segment $BX$ such that $AL=AC$. Let $M$ be the point of intersection of $AL$ and $BK$. Show that $MK=ML$. [i]Proposed by Josef Tkadlec, Czech Republic[/i]

2024 Serbia Team Selection Test, 3

Tags: geometry

Let $ABC$ be a triangle with circumcenter $O$, angle bisector $AD$ with $D \in BC$ and altitude $AE$ with $E \in BC$. The lines $AO$ and $BC$ meet at $I$. The circumcircle of $\triangle ADE$ meets $AB, AC$ at $F, G$ and $FG$ meets $BC$ at $H$. The circumcircles of triangles $AHI$ and $ABC$ meet at $J$. Show that $AJ$ is a symmedian in $\triangle ABC$

2023 Portugal MO, 2

Tags: geometry , arc midpoint

Let $[AB]$ be a diameter of a circle with center $O$ and radius $1$. Consider $P$ a point on the circumference, different from $A$ and $B$ and let $Q$ be the midpoint of the arc $AP$. The line parallel to $PQ$ that passes through $O$ intersects the line $PB$ at point $S$. Determine $\overline{PS}$.

2016 Bosnia and Herzegovina Team Selection Test, 5

Tags: incenter , arc midpoint , angle bisector , geometry , cyclic , circumcircle

Let $k$ be a circumcircle of triangle $ABC$ $(AC<BC)$. Also, let $CL$ be an angle bisector of angle $ACB$ $(L \in AB)$, $M$ be a midpoint of arc $AB$ of circle $k$ containing the point $C$, and let $I$ be an incenter of a triangle $ABC$. Circle $k$ cuts line $MI$ at point $K$ and circle with diameter $CI$ at $H$. If the circumcircle of triangle $CLK$ intersects $AB$ again at $T$, prove that $T$, $H$ and $C$ are collinear. .

1999 BAMO, 5

Tags: angle bisector , angle chasing , geometry

Let $ABCD$ be a cyclic quadrilateral (a quadrilateral which can be inscribed in a circle). Let $E$ and $F$ be variable points on the sides $AB$ and $CD$, respectively, such that $\frac{AE}{EB} = \frac{C}{FD}$. Let $P$ be the point on the segment $EF$ such that $\frac{PE}{PF} = \frac{AB}{CD}$. Prove that the ratio between the areas of triangle $APD$ and $BPC$ does not depend on the choice of $E$ and $F$.

1971 IMO Shortlist, 12

Tags: congruent triangles , geometry , midpoint , collinear

Two congruent equilateral triangles $ABC$ and $A'B'C'$ in the plane are given. Show that the midpoints of the segments $AA',BB', CC'$ either are collinear or form an equilateral triangle.

1970 IMO Longlists, 39

Tags: geometry , incircle , radius , ratio

$M$ is any point on the side $AB$ of the triangle $ABC$. $r,r_1,r_2$ are the radii of the circles inscribed in $ABC,AMC,BMC$. $q$ is the radius of the circle on the opposite side of $AB$ to $C$, touching the three sides of $AB$ and the extensions of $CA$ and $CB$. Similarly, $q_1$ and $q_2$. Prove that $r_1r_2q=rq_1q_2$.

2010 Germany Team Selection Test, 2

Tags: geometric transformation , rotation , geometry

Determine all $n \in \mathbb{Z}^+$ such that a regular hexagon (i.e. all sides equal length, all interior angles same size) can be partitioned in finitely many $n-$gons such that they can be composed into $n$ congruent regular hexagons in a non-overlapping way upon certain rotations and translations.

1941 Eotvos Mathematical Competition, 3

Tags: equilateral , geometry , hexagon

The hexagon $ABCDEF$ is inscribed in a circle. The sides $AB$, $CD$ and $EF$ are all equal in length to the radius. Prove that the midpoints of the other three sides determine an equilateral triangle.

2010 Contests, 1

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

Let $ABC$ be a triangle in which $BC<AC$. Let $M$ be the mid-point of $AB$, $AP$ be the altitude from $A$ on $BC$, and $BQ$ be the altitude from $B$ on to $AC$. Suppose that $QP$ produced meets $AB$ (extended) at $T$. If $H$ is the orthocenter of $ABC$, prove that $TH$ is perpendicular to $CM$.

May Olympiad L2 - geometry, 2004.3

Tags: angle , geometry

We have a pool table $8$ meters long and $2$ meters wide with a single ball in the center. We throw the ball in a straight line and, after traveling $29$ meters, it stops at a corner of the table. How many times did the ball hit the edges of the table? Note: When the ball rebounds on the edge of the table, the two angles that form its trajectory with the edge of the table are the same.

2015 China Team Selection Test, 1

Tags: angle bisector , geometry , perpendicular bisector

$\triangle{ABC}$ is isosceles with $AB = AC >BC$. Let $D$ be a point in its interior such that $DA = DB+DC$. Suppose that the perpendicular bisector of $AB$ meets the external angle bisector of $\angle{ADB}$ at $P$, and let $Q$ be the intersection of the perpendicular bisector of $AC$ and the external angle bisector of $\angle{ADC}$. Prove that $B,C,P,Q$ are concyclic.