Olympiad problems

2021 Princeton University Math Competition, A2

Tags: geometry

Let $ABCD$ be a cyclic quadrilateral with circumcircle $\Gamma$, and let $E$ be the midpoint of the diagonal $BD$. Let $I_1, I_2, I_3, I_4$ be the centers of the circles inscribed into triangles $\vartriangle ABE$, $\vartriangle ADE$, $\vartriangle BCE$, $\vartriangle CDE$, in that order. Prove that the circles $AI_1I_2$ and $CI_3I_4$ intersect $\Gamma$ at diametrically opposite points. Remark: For a circle $C$ and points $X, Y \in C$, we say that $X$ and $Y$ are diametrically opposite if $XY$ is a diameter of $C$.

2013 BMT Spring, P2

Tags: geometry

From a point $A$ construct tangents to a circle centered at point $O$, intersecting the circle at $P$ and $Q$ respectively. Let $M$ be the midpoint of $PQ$. If $K$ and $L$ are points on circle $O$ such that $K, L$, and $A$ are collinear, prove $\angle MKO = \angle MLO$.

2016 Czech-Polish-Slovak Match, 3

Tags: circumcircle , centroid , geometry

Let $ABC$ be an acute-angled triangle with $AB < AC$. Tangent to its circumcircle $\Omega$ at $A$ intersects the line $BC$ at $D$. Let $G$ be the centroid of $\triangle ABC$ and let $AG$ meet $\Omega$ again at $H \neq A$. Suppose the line $DG$ intersects the lines $AB$ and $AC$ at $E$ and $F$, respectively. Prove that $\angle EHG = \angle GHF$.(Slovakia)

2013 AMC 12/AHSME, 23

Tags: geometry , geometric transformation , rotation , number theory , greatest common divisor

$ ABCD$ is a square of side length $ \sqrt{3} + 1 $. Point $ P $ is on $ \overline{AC} $ such that $ AP = \sqrt{2} $. The square region bounded by $ ABCD $ is rotated $ 90^{\circ} $ counterclockwise with center $ P $, sweeping out a region whose area is $ \frac{1}{c} (a \pi + b) $, where $a $, $b$, and $ c $ are positive integers and $ \text{gcd}(a,b,c) = 1 $. What is $ a + b + c $? $\textbf{(A)} \ 15 \qquad \textbf{(B)} \ 17 \qquad \textbf{(C)} \ 19 \qquad \textbf{(D)} \ 21 \qquad \textbf{(E)} \ 23 $

2012 Oral Moscow Geometry Olympiad, 2

Tags: congruent , polygon , interior , geometry

Two equal polygons $F$ and $F'$ are given on the plane. It is known that the vertices of the polygon $F$ belong to $F'$ (may lie inside it or on the border). Is it true that all the vertices of these polygons coincide?

2011 Benelux, 2

Tags: geometry , incenter , circumcircle , perpendicular bisector , cyclic quadrilateral , angle bisector

Let $ABC$ be a triangle with incentre $I$. The angle bisectors $AI$, $BI$ and $CI$ meet $[BC]$, $[CA]$ and $[AB]$ at $D$, $E$ and $F$, respectively. The perpendicular bisector of $[AD]$ intersects the lines $BI$ and $CI$ at $M$ and $N$, respectively. Show that $A$, $I$, $M$ and $N$ lie on a circle.

2020 Polish Junior MO Second Round, 2.

Tags: geometry , parallelogram , perpendicular bisector

Let $ABCD$ be the parallelogram, such that angle at vertex $A$ is acute. Perpendicular bisector of the segment $AB$ intersects the segment $CD$ in the point $X$. Let $E$ be the intersection point of the diagonals of the parallelogram $ABCD$. Prove that $XE = \frac{1}{2}AD$.

2007 Belarusian National Olympiad, 2

Tags: geometry

Circles $S_1$ and $S_2$ with centers $O_1$ and $O_2$, respectively, pass through the centers of each other. Let $A$ be one of their intersection points. Two points $M_1$ and $M_2$ begin to move simultaneously starting from $A$. Point $M_1$ moves along $S_1$ and point $M_2$ moves along $S_2$. Both points move in clockwise direction and have the same linear velocity $v$. (a) Prove that all triangles $AM_1M_2$ are equilateral. (b) Determine the trajectory of the movement of the center of the triangle $AM_1M_2$ and find its linear velocity.

1961 Putnam, B3

Tags: geometry , circumcircle

Consider four points in the plane, no three of which are collinear, and such that the circle through three of them does not pass through the fourth. Prove that one of the four points can be selected having the property that it lies inside the circle determined by the other three.

2018 JHMT, 9

Tags: geometry

In a trapezoid $ABCD$, $AD \parallel BC$ and $\angle A = 60^o$. Let $E$ be a point on $AB$, and let $O_1$ and $O_2$ be circumcenters of $\vartriangle AED$ and $\vartriangle BEC$, respectively. Let $\frac{\overline{O_1O_2}}{\overline{DC}}$ be $x$. $x^2$ is in the form $\frac{p}{q}$ where $p, q$ are relatively prime positive integers. Find $p + q$.

May Olympiad L2 - geometry, 2019.3

Tags: geometry , equal angles

On the sides $AB, BC$ and $CA$ of a triangle $ABC$ are located the points $P, Q$ and $R$ respectively, such that $BQ = 2QC, CR = 2RA$ and $\angle PRQ = 90^o$. Show that $\angle APR =\angle RPQ$.

Ukraine Correspondence MO - geometry, 2007.7

Tags: midpoint , geometry , isosceles

Let $ABC$ be an isosceles triangle ($AB = AC$), $D$ be the midpoint of $BC$, and $M$ be the midpoint of $AD$. On the segment $BM$ take a point $N$ such that $\angle BND = 90^o$. Find the angle $ANC$.

2006 Moldova Team Selection Test, 2

Tags: analytic geometry , trigonometry , geometry

Consider a right-angled triangle $ABC$ with the hypothenuse $AB=1$. The bisector of $\angle{ACB}$ cuts the medians $BE$ and $AF$ at $P$ and $M$, respectively. If ${AF}\cap{BE}=\{P\}$, determine the maximum value of the area of $\triangle{MNP}$.

2019 Stanford Mathematics Tournament, 7

Tags: geometry

Let $G$ be the centroid of triangle $ABC$ with $AB = 9$, $BC = 10,$ and $AC = 17$. Denote $D$ as the midpoint of $BC$. A line through $G$ parallel to $BC$ intersects $AB$ at $M$ and $AC$ at $N$. If $BG$ intersects $CM$ at $E$ and $CG$ intersects $BN$ at $F$, compute the area of triangle $DEF$.

Kyiv City MO 1984-93 - geometry, 1993.8.3

Tags: geometry , angle bisector , equal segments

In the triangle $ABC$, $\angle .ACB = 60^o$, and the bisectors $AA_1$ and $BB_1$ intersect at the point $M$. Prove that $MB_1 = MA_1$.

2019 IOM, 3

Tags: circumcircle , incenter , geometry

In a non-equilateral triangle $ABC$ point $I$ is the incenter and point $O$ is the circumcenter. A line $s$ through $I$ is perpendicular to $IO$. Line $\ell$ symmetric to like $BC$ with respect to $s$ meets the segments $AB$ and $AC$ at points $K$ and $L$, respectively ($K$ and $L$ are different from $A$). Prove that the circumcenter of triangle $AKL$ lies on the line $IO$. [i]Dušan Djukić[/i]

2015 BMT Spring, 7

Tags: geometry

Define $A = (1, 0, 0)$, $B = (0, 1, 0)$, and $P$ as the set of all points $(x, y, z)$ such that $x+y+z = 0$. Let $P$ be the point on $P$ such that $d = AP + P B$ is minimized. Find $d^2$.

2023 China National Olympiad, 2

Tags: geometry

Let $\triangle ABC$ be an equilateral triangle of side length 1. Let $D,E,F$ be points on $BC,AC,AB$ respectively, such that $\frac{DE}{20} = \frac{EF}{22} = \frac{FD}{38}$. Let $X,Y,Z$ be on lines $BC,CA,AB$ respectively, such that $XY\perp DE, YZ\perp EF, ZX\perp FD$. Find all possible values of $\frac{1}{[DEF]} + \frac{1}{[XYZ]}$.

1985 IMO Longlists, 31

Tags: ellipse , concurrency , conic , geometry

Let $E_1, E_2$, and $E_3$ be three mutually intersecting ellipses, all in the same plane. Their foci are respectively $F_2, F_3; F_3, F_1$; and $F_1, F_2$. The three foci are not on a straight line. Prove that the common chords of each pair of ellipses are concurrent.

2018 PUMaC Geometry B, 8

Tags: geometry , parallelogram , angle bisector

Let $ABCD$ be a parallelogram such that $AB = 35$ and $BC = 28$. Suppose that $BD \perp BC$. Let $\ell_1$ be the reflection of $AC$ across the angle bisector of $\angle BAD$, and let $\ell_2$ be the line through $B$ perpendicular to $CD$. $\ell_1$ and $\ell_2$ intersect at a point $P$. If $PD$ can be expressed in simplest form as $\frac{m}{n}$, find $m + n$.

2010 Chile National Olympiad, 5

Tags: geometry , geometric inequality , distance

Consider a line $ \ell $ in the plane and let $ B_1, B_2, B_3 $ be different points in $ \ell$. Let $ A $ be a point that is not in $ \ell$. Show that there is $ P, Q $ in $ {B_1, B_2, B_3} $ with $ P \ne Q $ so that the distance from $ A $ to $ \ell$ is greater than the distance from $ P $ to the line that passes through $ A $ and $ Q $.

1992 Tournament Of Towns, (349) 1

Tags: geometry , 3d geometry , combinatorics , combinatorial geometry

We are given a cube with edges of length $n$ cm. At our disposal is a long piece of insulating tape of width $1$ cm. It is required to stick this tape to the cube. The tape may freely cross an edge of the cube on to a different face but it must always be parallel to an edge of the cube. It may not overhang the edge of a face or cross over a vertex. How many pieces of the tape are necessary in order to completely cover the cube? (You may assume that $n$ is an integer.) (A Spivak)

1998 Bulgaria National Olympiad, 3

Tags: trigonometry , complex number , geometry

On the sides of a non-obtuse triangle $ABC$ a square, a regular $n$-gon and a regular $m$-gon ($m$,$n > 5$) are constructed externally, so that their centers are vertices of a regular triangle. Prove that $m = n = 6$ and find the angles of $\triangle ABC$.

2018 Polish MO Finals, 5

Tags: radical axis , projective geometry , geometry

An acute triangle $ABC$ in which $AB<AC$ is given. Points $E$ and $F$ are feet of its heights from $B$ and $C$, respectively. The line tangent in point $A$ to the circle escribed on $ABC$ crosses $BC$ at $P$. The line parallel to $BC$ that goes through point $A$ crosses $EF$ at $Q$. Prove $PQ$ is perpendicular to the median from $A$ of triangle $ABC$.

2014 Contests, 3

Tags: circumcircle , geometry

Let $ABC$ be a triangle and let $P$ be a point on $BC$. Points $M$ and $N$ lie on $AB$ and $AC$, respectively such that $MN$ is not parallel to $BC$ and $AMP N$ is a parallelogram. Line $MN$ meets the circumcircle of $ABC$ at $R$ and $S$. Prove that the circumcircle of triangle $RP S$ is tangent to $BC$.