### Romanian Master of Mathematics 2008-20 (RMM) + SHL 2016-19 27p

geometry problems from Romanian Master of Mathematics (RMM) and Shortlists below
with aops links in the names

RMM Shortlists in aops: 2016, 2017, 2018, 2019
RMM geometry Shortlists in aops here

2008 - 2020

Let ABC be an equilateral triangle. P is a variable point internal to the triangle and its perpendicular distances to the sides are denoted  by a2, b2  and c2 for positive real numbers a, b and c. Find the locus of points P so that a, b and c can be the sides of a non-degenerate triangle.
UK
RMM  2009 p3
Given four points A1, A2, A3, A4 in the plane, no three collinear, such that A1A2·A3A4 = A1A3· A2A4 = A1A4·A2A3, denote by Oi the circumcenter of ∆Aj Ak Al, with {i , j ,k, l} = {1,2,3,4}. Assuming Ai ≠ Oi for all indices i , prove that the four lines AiOi are concurrent or parallel.

Nikolai Ivanov Beluhov, Bulgaria
RMM  2010 p3
Let A1A2A3A4 be a convex quadrilateral with no pair of parallel sides. For each i = 1, 2, 3, 4, define ωi to be the circle touching the quadrilateral  externally, and which is tangent to the lines Ai-1Ai , Ai Ai+1 and Ai+1 Ai+2 (indices are considered modulo 4, so A0 = A4, A5= A1 and A6 = A2). Let Ti be the point of tangency of ωi with the side Ai Ai+1. Prove that the lines A1A2, A3A4 and T2T4 are concurrent if and only if the lines A2A3, A4A1 and T1T3 are concurrent.
Pavel Kozhevnikov, Russia
RMM  2011 p3
A triangle ABC is inscribed in a circle ω. A variable line l chosen parallel to BC meets segments AB, AC at points D, E respectively, and meets ω at points K, L (where D lies between K and E). Circle γ1 is tangent to the segments KD and BD and also tangent to ω, while circle γ2 is tangent to the segments LE and CE and also tangent to ω. Determine the locus, as l varies, of the meeting point of the common inner tangents to γ1 and γ2.
Vasily Mokin and Fedor Ivlev, Russia
RMM  2012 p2
Given a non-isosceles triangle ABC, let D, E, and F denote the midpoints of the sides BC, CA, and AB respectively. The circle BCF  and the line BE meet again at P, and the circle ABE and the line AD meet again at Q. Finally, the lines DP and FQ meet at R. Prove that the centroid G of the triangle ABC lies on the circle PQR.
David Monk, United Kingdom
RMM  2012 p6
Let ABC be a triangle and let I and O denote its incentre and circumcentre respectively. Let  ωA be the circle through B and C which is tangent to the incircle of the triangle ABC, the circles ωB and ωC are defined similarly. The circles ωB and ωc meet at a point A΄ distinct from A, the points B΄ and C΄ are defined similarly. Prove that the lines AA΄, BB΄ and CC΄ are concurrent at a point on the line IO.

Fedor Ivlev, Russia
RMM  2013 p3
Let ABCD be a quadrilateral inscribed in a circle ω. The lines AB and CD meet at P, the lines AD and BC meet at Q, and the diagonals AC and BD meet at R. Let M be the midpoint of the segment PQ, and let K be the common point of the segment MR and the circle ω. Prove that the circumcircle of the triangle KPQ and ω are tangent to one another.
Medeubek Kungozhin, Russia
In 2014 it did not take place.

RMM  2015 p4
Let ABC be a triangle, and let D be the point where the incircle meets side BC. Let Jb and Jc be the incentres of the triangles ABD and ACD, respectively. Prove that the circumcentre of the triangle AJbJc lies on the angle bisector of <BAC.
Fedor Ivlev, Russia
RMM  2016 p1
Let ABC be a triangle and let D be a point on the segment BC, D ≠B and D ≠ C. The circle ABD meets the segment AC again at an interior point E. The circle ACD meets the segment AB again at an interior point F. Let A΄ be the reflection of A in the line BC. The lines A΄C and DE meet at P, and the lines A΄B and DF meet at Q. Prove that the lines AD, BP and CQ are concurrent (or all parallel).

RMM  2016 p5
A convex hexagon A1B1A2B2A3B3 is inscribed in a circle of radius R. The diagonals A1B2, A2B3, and A3B1 concur at X. For i = 1,2,3, let ωi be the circle tangent to the segments XAi and XBi, and to the arc AiBi of  not containing other vertices of the hexagon,  let ri be the radius of ωi.
(a) Prove that R ≥ r1 + r2 + r3.
(b) If R = r1 + r2 + r3 , prove that the six points where the circles ωi touch the diagonals A1B2, A2B3, A3B1 are concyclic.

RMM  2017 p6
Let ABCD be any convex quadrilateral and let P, Q, R, S be points on the segments AB, BC, CD, and DA, respectively. It is given  that the segments PR and QS dissect ABCD into four quadrilaterals, each of which has perpendicular diagonals. Show that the points P, Q, R, S are concyclic.

Nikolai Beluhov
RMM  2018 p1
Let $ABCD$ be a cyclic quadrilateral an let $P$ be a point on the side $AB.$ The diagonals $AC$ meets the segments $DP$ at $Q.$ The line through $P$ parallel to $CD$ mmets the extension of the side $CB$ beyond $B$ at $K.$ The line through $Q$ parallel to $BD$ meets the extension of the side $CB$ beyond $B$ at $L.$ Prove that the circumcircles of the triangles $BKP$ and $CLQ$ are tangent .

RMM  2018 p6
Fix a circle $\Gamma$, a line $\ell$ to tangent $\Gamma$, and another circle $\Omega$ disjoint from $\ell$ such that $\Gamma$ and $\Omega$ lie on opposite sides of $\ell$. The tangents to $\Gamma$ from a variable point $X$ on $\Omega$ meet $\ell$ at $Y$ and $Z$. Prove that, as $X$ varies over $\Omega$, the circumcircle of $XYZ$ is tangent to two fixed circles.

Let $ABCD$ be an isosceles trapezoid with $AB\parallel CD$. Let $E$ be the midpoint of $AC$. Denote by $\omega$ and $\Omega$ the circumcircles of the triangles $ABE$ and $CDE$, respectively. Let $P$ be the crossing point of the tangent to $\omega$ at $A$ with the tangent to $\Omega$ at $D$. Prove that $PE$ is tangent to $\Omega$.

Jakob Jurij Snoj, Slovenia
Prove that for every positive integer $n$ there exists a (not necessarily convex) polygon with no three collinear vertices, which admits exactly $n$ diffferent triangulations.

(A triangulation is a dissection of the polygon into triangles by interior diagonals which have no common interior points with each other nor with the sides of the polygon)

Let $ABC$ be a triangle with a right angle at $C$. Let $I$ be the incentre of triangle $ABC$, and let $D$ be the foot of the altitude from $C$ to $AB$. The incircle $\omega$ of triangle $ABC$ is tangent to sides $BC$, $CA$, and $AB$ at $A_1$, $B_1$, and $C_1$, respectively. Let $E$ and $F$ be the reflections of $C$ in lines $C_1A_1$ and $C_1B_1$, respectively. Let $K$ and $L$ be the reflections of $D$ in lines $C_1A_1$ and $C_1B_1$, respectively.
Prove that the circumcircles of triangles $A_1EI$, $B_1FI$, and $C_1KL$ have a common point.

2016 - 2019 SHL (shortlist)

RMM  2016 SHL G1 (also Romania TST1 p1 2016)
Two circles,  $\omega_1$ and $\omega_2$, centred at $O1$ and $O2$, respectively, meet at points $A$ and $B$. A line through $B$ meets $\omega_1$ again at $C$, and $\omega_2$ again at $D$. The tangents to  $\omega_1$ and $\omega_2$ at $C$ and $D$, respectively, meet at $E$, and the line $AE$ meets the circle $\omega$ through $A, O_1, O_2$ again at $F$. Prove that the length of the segment $EF$ is equal to the diameter of $\omega$.

RMM  2017 SHL G1 (also Romania TST1 p1 2017)
Let $ABCD$ be a trapezium, $AD\parallel BC$, and let $E,F$ be points on the sides$AB$ and $CD$, respectively. The circumcircle of $AEF$ meets $AD$ again at $A_1$, and the circumcircle of $CEF$ meets $BC$ again at $C_1$. Prove that $A_1C_1,BD,EF$ are concurrent.

Alexander Kuznetsov, Russia
Let $ABC$ be a triangle. Consider the circle $\omega_B$ internally tangent to the sides $BC$ and $BA$, and to the circumcircle of the triangle $ABC$, let $P$ be the point of contact of the two circles. Similarly, consider the circle $\omega_C$ internally tangent to the sides $CB$ and $CA$, and to the circumcircle of the triangle $ABC$, let $Q$ be the point of contact of the two circles. Show that the incentre of the triangle $ABC$ lies on the segment $PQ$ if and only if $AB + AC = 3BC$.

Luis Eduardo Garcia Hernandez, Mexico
RMM  2017 SHL G3 (also Romania TST3 p4 2017)
Let $ABCD$ be a convex quadrilateral and let $P$ and $Q$ be variable points inside this quadrilateral so that $\angle APB=\angle CPD=\angle AQB=\angle CQD$. Prove that the lines $PQ$ obtained in this way all pass through a fixed point , or they are all parallel.

Sergey Berlov, Russia
Let $ABC$ be a triangle and let $H$ be the orthogonal projection of $A$ on the line $BC$. Let $K$ be a point on the segment $AH$ such that $AH = 3 KH$. Let $O$ be the circumcenter of triangle $ABC$ and let $M$ and $N$ be the midpoints of sides $AC$ and $AB$ respectively. The lines $KO$ and $MN$ meet at a point $Z$ and the perpendicular at $Z$ to $OK$ meets lines $AB, AC$ at $X$ and $Y$ respectively. Show that $\angle XKY = \angle CKB$.

Italy
Let $\triangle ABC$ be a triangle, and let $S$ and $T$ be the midpoints of the sides $BC$ and $CA$, respectively. Suppose $M$ is the midpoint of the segment $ST$ and the circle $\omega$ through $A, M$ and $T$ meets the line $AB$ again at $N$. The tangents of $\omega$ at $M$ and $N$ meet at $P$. Prove that $P$ lies on $BC$ if and only if the triangle $ABC$ is isosceles with apex at $A$.
Reza Kumara, Indonesia
Let $BM$ be a median in an acute-angled triangle $ABC$. A point $K$ is chosen on the line through $C$ tangent to the circumcircle of $\vartriangle BMC$ so that $\angle KBC = 90^o$. The segments $AK$ and $BM$ meet at $J$. Prove that the circumcenter of $\vartriangle BJK$ lies on the line $AC$.

Aleksandr Kuznetsov, Russia
RMM  2019 SHL G2
Let $ABC$ be an acute-angled triangle. The line through $C$ perpendicular to $AC$ meets the external angle bisector of $\angle ABC$ at $D$. Let $H$ be the foot of the perpendicular from $D$ onto $BC$. The point $K$ is chosen on $AB$ so that $KH \parallel AC$. Let $M$ be the midpoint of $AK$. Prove that $MC = MB + BH$.

Giorgi Arabidze, Georgia
RMM  2019 SHL G3 (EMC 2019 P3)
Let $ABC$ be an acute-angled triangle with $AB \ne AC$, and let $I$ and $O$ be its incenter and circumcenter, respectively. Let the incircle touch $BC, CA$ and $AB$ at $D, E$ and $F$, respectively. Assume that the line through $I$ parallel to $EF$, the line through $D$ parallel to$AO$, and the altitude from $A$ are concurrent. Prove that the concurrency point is the orthocenter of the triangle $ABC$.

Petar Nizic-Nikolac, Croatia
Let $\Omega$ be the circumcircle of an acute-angled triangle $ABC$. Let $D$ be the midpoint of the minor arc $AB$ of $\Omega$. A circle $\omega$ centered at $D$ is tangent to $AB$ at $E$. The tangents to $\omega$ through $C$ meet the segment $AB$ at $K$ and $L$, where $K$ lies on the segment $AL$. A circle $\Omega_1$ is tangent to the segments $AL, CL$, and also to $\Omega$ at point $M$. Similarly, a circle $\Omega_2$ is tangent to the segments $BK, CK$, and also to $\Omega$ at point $N$. The lines $LM$ and $KN$ meet at $P$. Prove that $\angle KCE = \angle LCP$.

Poland
Let $\Omega$ be the circumcircle of an acute-angled triangle $ABC$. A point $D$ is chosen on the internal bisector of $\angle ACB$ so that the points $D$ and $C$ are separated by $AB$. A circle $\omega$ centered at $D$ is tangent to the segment $AB$ at $E$. The tangents to $\omega$ through $C$ meet the segment $AB$ at $K$ and $L$, where $K$ lies on the segment $AL$. A circle $\Omega_1$ is tangent to the segments $AL, CL$, and also to $\Omega$ at point $M$. Similarly, a circle $\Omega_2$ is tangent to the segments $BK, CK$, and also to $\Omega$ at point $N$. The lines $LM$ and $KN$ meet at $P$. Prove that $\angle KCE = \angle LCP$.

Poland
A quadrilateral $ABCD$ is circumscribed about a circle with center $I$. A point $P \ne I$ is chosen inside $ABCD$ so that the triangles $PAB, PBC, PCD,$ and $PDA$ have equal perimeters. A circle $\Gamma$ centered at $P$ meets the rays $PA, PB, PC$, and $PD$ at $A_1, B_1, C_1$, and $D_1$, respectively. Prove that the lines $PI, A_1C_1$, and $B_1D_1$ are concurrent.

Ankan Bhattacharya, USA

RMM 2019 original P4 (removed due to leak)
Let there be an equilateral triangle $ABC$ and a point $P$ in its plane such that $AP<BP<CP.$ Suppose that the lengths of segments $AP,BP$ and $CP$ uniquely determine the side of $ABC$. Prove that $P$ lies on the circumcircle of triangle $ABC.$

source: rmms.lbi.ro