Olympiad problems

2009 Moldova Team Selection Test, 1

[color=darkblue]Points $ X$, $ Y$ and $ Z$ are situated on the sides $ (BC)$, $ (CA)$ and $ (AB)$ of the triangles $ ABC$, such that triangles $ XYZ$ and $ ABC$ are similiar. Prove that circumcircle of $ AYZ$ passes through a fixed point.[/color]

2019 Belarus Team Selection Test, 5.2

Tags: moving points , geometry

Let $AA_1$ be the bisector of a triangle $ABC$. Points $D$ and $F$ are chosen on the line $BC$ such that $A_1$ is the midpoint of the segment $DF$. A line $l$, different from $BC$, passes through $A_1$ and intersects the lines $AB$ and $AC$ at points $B_1$ and $C_1$, respectively. Find the locus of the points of intersection of the lines $B_1D$ and $C_1F$ for all possible positions of $l$. [i](M. Karpuk)[/i]

2019 PUMaC Geometry A, 7

Tags: geometry

Let $ABCD$ be a trapezoid such that $AB||CD$ and let $P=AC\cap BD,AB=21,CD=7,AD=13,[ABCD]=168.$ Let the line parallel to $AB$ through $P$ intersect the circumcircle of $BCP$ in $X.$ Circumcircles of $BCP$ and $APD$ intersect at $P,Y.$ Let $XY\cap BC=Z.$ If $\angle ADC$ is obtuse, then $BZ=\frac{a}{b},$ where $a,b$ are coprime positive integers. Compute $a+b.$

Cono Sur Shortlist - geometry, 2020.G3.3

Tags: circumcircle , butterfly theorem , lines meeting at circmucircle , geometry

Let $ABC$ be an acute triangle such that $AC<BC$ and $\omega$ its circumcircle. $M$ is the midpoint of $BC$. Points $F$ and $E$ are chosen in $AB$ and $BC$, respectively, such that $AC=CF$ and $EB=EF$. The line $AM$ intersects $\omega$ in $D\neq A$. The line $DE$ intersects the line $FM$ in $G$. Prove that $G$ lies on $\omega$.

2018 Peru Iberoamerican Team Selection Test, P2

Tags: geometry , angle chasing , equal segments , angle

Let $ABC$ be a triangle with $AB = AC$ and let $D$ be the foot of the height drawn from $A$ to $BC$. Let $P$ be a point inside the triangle $ADC$ such that $\angle APB> 90^o$ and $\angle PAD + \angle PBD = \angle PCD$. The $CP$ and $AD$ lines are cut at $Q$ and the $BP$ and $AD$ lines cut into $R$. Let $T$ be a point in segment $AB$ such that $\angle TRB = \angle DQC$ and let S be a point in the extension of the segment $AP$ (on the $P$ side) such that $\angle PSR = 2 \angle PAR$. Prove that $RS = RT$.

1996 Irish Math Olympiad, 5

Tags: geometry

Show how to dissect a square into at most five pieces in such a way that the pieces can be reassembled to form three squares of (pairwise) distinct areas.

2016 Silk Road, 2

Tags: geometry , equal angles , circumcircle

Around the acute-angled triangle $ABC$ ($AC>CB$) a circle is circumscribed, and the point $N$ is midpoint of the arc $ACB$ of this circle. Let the points $A_1$ and $B_1$ be the feet of perpendiculars on the straight line $NC$, drawn from points $A$ and $B$ respectively (segment $NC$ lies inside the segment $A_1B_1$). Altitude $A_1A_2$ of triangle $A_1AC$ and altitude $B_1B_2$ of triangle $B_1BC$ intersect at a point $K$ . Prove that $\angle A_1KN=\angle B_1KM$, where $M$ is midpoint of the segment $A_2B_2$ .

1982 IMO Shortlist, 6

Tags: geometry , combinatorial geometry , combinatorics , length , polygon

Let $S$ be a square with sides length $100$. Let $L$ be a path within $S$ which does not meet itself and which is composed of line segments $A_0A_1,A_1A_2,A_2A_3,\ldots,A_{n-1}A_n$ with $A_0=A_n$. Suppose that for every point $P$ on the boundary of $S$ there is a point of $L$ at a distance from $P$ no greater than $\frac {1} {2}$. Prove that there are two points $X$ and $Y$ of $L$ such that the distance between $X$ and $Y$ is not greater than $1$ and the length of the part of $L$ which lies between $X$ and $Y$ is not smaller than $198$.

2023 Harvard-MIT Mathematics Tournament, 10

Tags: hmmt , geometry

Triangle $ABC$ has incenter $I$. Let $D$ be the foot of the perpendicular from $A$ to side $BC$. Let $X$ be a point such that segment $AX$ is a diameter of the circumcircle of triangle $ABC$. Given that $ID = 2$, $IA = 3$, and $IX = 4$, compute the inradius of triangle $ABC$.

1999 Harvard-MIT Mathematics Tournament, 2

Tags: geometry

A semicircle is inscribed in a semicircle of radius $2$ as shown. Find the radius of the smaller semicircle. [img]https://cdn.artofproblemsolving.com/attachments/c/1/c60cd40eaecfe417aca46ce4fd386fe22af85b.png[/img]

2014 Iran Geometry Olympiad (senior), 1:

Tags: geometry

ABC is a triangle with A=90 and C=30.Let M be the midpoint of BC. Let W be a circle passing through A tangent in M to BC. Let P be the circumcircle of ABC. W is intersecting AC in N and P in M. prove that MN is perpendicular to BC.

2006 IberoAmerican, 2

Tags: cyclic quadrilateral , projective geometry , geometry

[color=darkred]The sides $AD$ and $CD$ of a tangent quadrilateral $ABCD$ touch the incircle $\varphi$ at $P$ and $Q,$ respectively. If $M$ is the midpoint of the chord $XY$ determined by $\varphi$ on the diagonal $BD,$ prove that $\angle AMP = \angle CMQ.$[/color]

2019 JHMT, 3

Tags: geometry

Square $ABCD$ has side length of $2$. Quarter-circle arcs $BD$ (centered at $C$) and $AC$ (centered at $D$) divide $ABCD$ into four sections. The area of the smallest of the four sections that are formed can be expressed as $a - \frac{b\pi }{c} - \sqrt{d}$. Find abcd, where $a, b, c$ and $d$ are integers, $ \sqrt{d}$ is a written in simplestradical form, and $\frac{b}{c}$ is written in simplest form.

2025 CMIMC Geometry, 5

Tags: geometry

Let $\triangle{ABC}$ be an equilateral triangle. Let $E_{AB}$ be the ellipse with foci $A, B$ passing through $C,$ and in the parallel manner define $E_{BC}, E_{AC}.$ Let $\triangle{GHI}$ be a (nondegenerate) triangle with vertices where two ellipses intersect such that the edges of $\triangle{GHI}$ do not intersect those of $\triangle{ABC}.$ Compute the ratio of the largest sides of $\triangle{GHI}$ and $\triangle{ABC}.$

2019 HMNT, 8

Tags: geometry

In $\vartriangle ABC$, the external angle bisector of $\angle BAC$ intersects line $BC$ at $D$. $E$ is a point on ray $\overrightarrow{AC}$ such that $\angle BDE = 2\angle ADB$. If $AB = 10$, $AC = 12$, and $CE = 33$, compute $\frac{DB}{DE}$ .

2014 NIMO Problems, 8

Tags: probability , geometry , geometric transformation , reflection , perpendicular bisector

Aaron takes a square sheet of paper, with one corner labeled $A$. Point $P$ is chosen at random inside of the square and Aaron folds the paper so that points $A$ and $P$ coincide. He cuts the sheet along the crease and discards the piece containing $A$. Let $p$ be the probability that the remaining piece is a pentagon. Find the integer nearest to $100p$. [i]Proposed by Aaron Lin[/i]

2019-2020 Winter SDPC, 8

Tags: geometry

Let $ABC$ be a triangle with circumcircle $\Gamma$. If the internal angle bisector of $\angle A$ meets $BC$ and $\Gamma$ at $D$ and $E$ respectively. Let $O_1$ be the center of the circle through $A$ and $D$ tangent to $BC$, let the external angle bisector of $\angle A$ meet $\Gamma$ at $F$, and let $FO_1$ meet $\Gamma$ at some point $P \neq F$. Show that the circumcircle of $DEP$ is tangent to $BC$.

2012 BMT Spring, 6

Tags: cyclic quadrilateral , geometry

Let $ \text{ABCD} $ be a cyclic quadrilateral, with $ \text{AB} = 7 $, $ \text{BC} = 11 $, $ \text{CD} = 13 $, and $ \text{DA} = 17 $. Let the incircle of $ \text{ABD} $ hit $ \text{BD} $ at $ \text{R} $ and the incircle of $ \text{CBD} $ hit $ \text{BD} $ at $ \text{S} $. What is $ \text{RS} $?

2004 Germany Team Selection Test, 2

Tags: fixed point , geometry

Three distinct points $A$, $B$, and $C$ are fixed on a line in this order. Let $\Gamma$ be a circle passing through $A$ and $C$ whose center does not lie on the line $AC$. Denote by $P$ the intersection of the tangents to $\Gamma$ at $A$ and $C$. Suppose $\Gamma$ meets the segment $PB$ at $Q$. Prove that the intersection of the bisector of $\angle AQC$ and the line $AC$ does not depend on the choice of $\Gamma$.

1977 All Soviet Union Mathematical Olympiad, 235

Tags: combinatorial geometry , geometry

Given a closed broken line without self-intersections in a plane. Not a triple of its vertices belongs to one straight line. Let us call "special" a couple of line's segments if the one's extension intersects another. Prove that there is even number of special pairs.

2025 Caucasus Mathematical Olympiad, 7

Tags: geometry

From a point $O$ lying outside the circle $\omega$, two tangents are drawn touching $\omega$ at points $M$ and $N$. A point $K$ is chosen on the segment $MN$. Let points $P$ and $Q$ be the midpoints of segments $KM$ and $OM$ respectively. The circumcircle of triangle $MPQ$ intersects $\omega$ again at point $L$ ($L \neq M$). Prove that the line $LN$ passes through the centroid of triangle $KMO$.

2009 Hong Kong TST, 3

Tags: circumcircle , geometry

Let $ ABCDE$ be an arbitrary convex pentagon. Suppose that $ BD\cap CE \equal{} A'$, $ CE\cap DA \equal{} B'$, $ DA\cap EB \equal{} C'$, $ EB\cap AC \equal{} D'$ and $ AC\cap BD \equal{} E'$. Suppose also that $ eABD'\cap eAC'E \equal{} A''$, $ eBCE'\cap eBD'A \equal{} B''$, $ eCDA'\cap eCE'B \equal{} C''$, $ eDEB'\cap eDA'C \equal{} D''$, $ eEAC'\cap eEB'D \equal{} E''$. Prove that $ AA'', BB'', CC'', DD'', EE''$ are concurrent. (Here $ l_1\cap l_2 \equal{} P$ means that $ P$ is the intersection of lines $ l_1$ and $ l_2$. Also $ eA_1A_2A_3\cap eB_1B_2B_3 \equal{} Q$ means that $ Q$ is the intersection of the circumcircles of $ \Delta A_1A_2A_3$ and $ \Delta B_1B_2B_3$.)

1969 Spain Mathematical Olympiad, 2

Tags: complex number , locus , geometry

Find the locus of the affix $M$, of the complex number $z$, so that it is aligned with the affixes of $i$ and $iz$ .

2001 Estonia Team Selection Test, 6

Tags: incircle , circumcircle , geometry

Let $C_1$ and $C_2$ be the incircle and the circumcircle of the triangle $ABC$, respectively. Prove that, for any point $A'$ on $C_2$, there exist points $B'$ and $C'$ such that $C_1$ and $C_2$ are the incircle and the circumcircle of triangle $A'B'C'$, respectively.

Kyiv City MO Seniors 2003+ geometry, 2015.10.5

Tags: circles , equal segments , geometry

Circles ${{w} _ {1}}$ and ${{w} _ {2}}$ with centers at points ${{O} _ {1}}$ and ${{ O} _ {2}}$ intersect at points $A$ and $B$, respectively. Around the triangle ${{O} _ {1}} {{O} _ {2}} B$ circumscribe a circle $w$ centered at the point $O$, which intersects the circles ${{w } _ {1}}$ and ${{w} _ {2}}$ for the second time at points $K$ and $L$, respectively. The line $OA$ intersects the circles ${{w} _ {1}}$ and ${{w} _ {2}}$ at the points $M$ and $N$, respectively. The lines $MK$ and $NL$ intersect at the point $P$. Prove that the point $P$ lies on the circle $w$ and $PM = PN$. (Vadym Mitrofanov)