geometry problems from Romanian Master of Mathematics (RMM)

with aops links in the names

2008 - 2019

and 2018 SHL (shortlist)

and 2018 SHL (shortlist)

RMM 2008 p1

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 a

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 a

^{2}, b^{2}and c^{2}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.
by UK

RMM 2009 p3

Given four points A

Given four points A

_{1}, A_{2}, A_{3}, A_{4}in the plane, no three collinear, such that A_{1}A_{2}·A_{3}A_{4}= A_{1}A_{3}· A_{2}A_{4}= A_{1}A_{4}·A_{2}A_{3}, denote by O_{i}the circumcenter of ∆A_{j}A_{k}A_{l}, with {i , j ,k, l} = {1,2,3,4}. Assuming A_{i}≠ O_{i}for all indices i , prove that the four lines A_{i}O_{i}are concurrent or parallel.
by Nikolai Ivanov Beluhov,
Bulgaria

RMM 2010 p3

Let A

Let A

_{1}A_{2}A_{3}A_{4}be a convex quadrilateral with no pair of parallel sides. For each i = 1, 2, 3, 4, define ω_{i}to be the circle touching the quadrilateral externally, and which is tangent to the lines A_{i}_{-1}A_{i}, A_{i}A_{i}_{+1 }and A_{i}_{+1 }A_{i}_{+2 }(indices are considered modulo 4, so A_{0}= A_{4}, A_{5}= A_{1}and A_{6}= A_{2}). Let Ti be the point of tangency of ω_{i}with the side A_{i}A_{i}_{+1}. Prove that the lines A_{1}A_{2}, A_{3}A_{4}and T_{2}T_{4}are concurrent if and only if the lines A_{2}A_{3}, A_{4}A_{1}and T_{1}T_{3}are concurrent.
by Pavel Kozhevnikov,
Russia

RMM 2011 p3

A triangle ABC is inscribed in a circle ω. A variable line

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 γ_{1}is tangent to the segments KD and BD and also tangent to ω, while circle γ_{2}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 γ_{1}and γ_{2}.
by 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.

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.

by David Monk, United
Kingdom

RMM 2012 p6

Let ABC be a triangle and let I and O denote its incentre and circumcentre respectively. Let ω

Let ABC be a triangle and let I and O denote its incentre and circumcentre respectively. Let ω

_{A}be the circle through B and C which is tangent to the incircle of the triangle ABC, the circles ω_{B}and ω_{C}are defined similarly. The circles ω_{B}and ω_{c }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.by 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.

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.

by 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 J

RMM 2015 p4

Let ABC be a triangle, and let D be the point where the incircle meets side BC. Let J

_{b}and J_{c }be the incentres of the triangles ABD and ACD, respectively. Prove that the circumcentre of the triangle AJ_{b}J_{c}lies on the angle bisector of <BAC.
by 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).

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 A

A convex hexagon A

_{1}B_{1}A_{2}B_{2}A_{3}B_{3}is inscribed in a circle of radius R. The diagonals A_{1}B_{2}, A_{2}B_{3}, and A_{3}B_{1}concur at X. For i = 1,2,3, let ω_{i}be the circle tangent to the segments XA_{i}and XB_{i}, and to the arc A_{i}B_{i}of not containing other vertices of the hexagon, let r_{i}be the radius of ω_{i}.
(a) Prove that R ≥ r

_{1}+ r_{2}+ r_{3}.
(b) If R = r

_{1}+ r_{2}+ r_{3}, prove that the six points where the circles ω_{i}touch the diagonals A_{1}B_{2}, A_{2}B_{3}, A_{3}B_{1}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.

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.

by 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 .

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 $\angle XKY = \angle CKB$.

RMM 2018 SHL G2

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 $\Omega$.

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 $ABCD$ 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 $ABC$ 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 $\angle XKY = \angle CKB$.

by 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 $A$.

by Reza Kumara, Indonesia

by 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)

source: rmms.lbi.ro

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