### India Regional (RMO) 1990 - 2018 105p

geometry problems from Indian Mathematical Olympiads (RMO)
with aops links in the names

In 2011 it was renamed from RMO to CRMO,
as a first round named as PreRMO was added.

Indian RMO 2000-10, CRMO 2011-18 with solutions

1990 - 2018

1990 India RMO p3
A square sheet of paper $ABCD$ is so folded that $B$ falls on the mid point of $M$ of $CD$. Prove that the crease will divide $BC$ in the ratio $5 : 3$.

1990 India RMO p5
$P$ is any point inside a triangle $ABC$. The perimeter of the triangle $AB + BC + Ca = 2s$. Prove that $s < AP +BP +CP < 2s$.

1990 India RMO p8
If the circumcenter and centroid of a triangle coincide, prove that it must be equilateral.

1991 India RMO p1
Let $P$ be an interior point of a triangle $ABC$ and $AP,BP,CP$ meet the sides $BC,CA,AB$ in $D,E,F$ respectively. Show that $\frac{AP}{PD} = \frac{AF}{FB} + \frac{AE}{EC}.$

1991 India RMO p5
Take any point $P_1$ on the side $BC$ of a triangle $ABC$ and draw the following chain of lines: $P_1P_2$ parallel to $AC$; $P_2P_3$ parallel to $BC$; $P_3P_4$ parallel to $AB$ ; $P_4P_5$ parallel to $CA$; and $P_5P_6$ parallel to $BC$, Here, $P_2,P_5$ lie on $AB$; $P_3,P_6$ lie on $CA$ and $P_4$ on $BC$> Show that $P_6P_1$ is parallel to $AB$.

1992 India RMO p4
$ABCD$ is a cyclic quadrilateral with $AC \perp BD$; $AC$ meets $BD$ at $E$. Prove that $EA^2 + EB^2 + EC^2 + ED^2 = 4 R^2$ where $R$ is the radius of the circumscribing circle.

1992 India RMO p5
$ABCD$ is a quadrilateral and $P,Q$ are the midpoints of $CD, AB, AP, DQ$ meet at $X$ and $BP, CQ$ meet at $Y$. Prove that $A[ADX]+A[BCY] = A[PXOY]$.

1992 India RMO p8
The cyclic octagon $ABCDEFGH$ has sides $a,a,a,a,b,b,b,b$ respectively. Find the radius of the circle that circumscribes $ABCDEFGH.$

1993 India RMO p1
Let $ABC$ be an acute angled triangle and $CD$ be the altitude through $C$. If $AB = 8$ and $CD = 6$, find the distance between the midpoints of $AD$ and $BC$.

1993 India RMO p4
Let $ABCD$ be a rectangle with $AB = a$ and $BC = b$. Suppose $r_1$ is the radius of the circle passing through $A$ and $B$ touching $CD$; and similarly $r_2$ is the radius of the circle passing through $B$ and $C$ and touching $AD$. Show that $r_1 + r_2 \geq \frac{5}{8} ( a + b) .$

1994 India RMO p2
In a triangle $ABC$, the incircle touches the sides $BC, CA, AB$ at $D, E, F$ respectively. If the radius if the incircle is $4$ units and if $BD, CE , AF$ are consecutive integers, find the sides of the triangle $ABC$.

1994 India RMO p6
Let $AC$ and $BD$ be two chords of a circle with center $O$ such that they intersect at right angles inside the circle at the point $M$. Suppose $K$ and $L$ are midpoints of the chords $AB$ and $CD$ respectively. Prove that $OKML$ is a parallelogram.

1995 India RMO p1
In triangle $ABC$, $K$ and $L$ are points on the side $BC$ ($K$ being closer to $B$ than $L$) such that $BC \cdot KL = BK \cdot CL$ and $AL$ bisects $\angle KAC$. Show that $AL \perp AB.$

1996 India RMO p1
The sides of a triangle are three consecutive integers and its inradius is $4$. Find the circumradius.

1997 India RMO p1
Let $P$ be an interior point of a triangle $ABC$ and let $BP$ and $CP$ meet $AC$ and $AB$ in $E$ and $F$ respectively. IF $S_{BPF} = 4$,$S_{BPC} = 8$ and $S_{CPE} = 13$, find $S_{AFPE}.$

1997 India RMO p4
In a quadrilateral $ABCD$, it is given that $AB$ is parallel to $CD$ and the diagonals $AC$ and $BD$ are perpendicular to each other. Show that (a) $AD \cdot BC \geq AB \cdot CD$ (b) $AD + BC \geq AB + CD.$

1998 India RMO p1
Let $ABCD$ be a convex quadrilateral in which $\angle BAC = 50^{\circ}, \angle CAD = 60^{\circ}$and $\angle BDC = 25^{\circ}$. If $E$ is the point of intersection of $AC$ and $BD$, find $\angle AEB$.

1998 India RMO p4
Let $ABC$ be a triangle with $AB = AC$ and $\angle BAC = 30^{\circ}$, Let $A'$ be the reflection of $A$ in the line $BC$; $B'$ be the reflection of $B$ in the line $CA$; $C'$ be the reflection of $C$ in line $AB$, Show that $A'B'C'$ is an equilateral triangle.

1999 India RMO p1
Prove that the inradius of a right angled triangle with integer sides is an integer.

1999 India RMO p3
Let $ABCD$ be a square and $M,N$ points on sides $AB, BC$ respectively such that $\angle MDN = 45^{\circ}$. If $R$ is the midpoint of $MN$ show that $RP =RQ$ where $P,Q$ are points of intersection of $AC$ with the lines $MD, ND$.

2000 India RMO p1
Let $AC$ be a line segment in the plane and $B$ a points between $A$ and $C$. Construct isosceles triangles $PAB$ and $QAC$ on one side of the segment $AC$ such that $\angle APB = \angle BQC = 120^{\circ}$ and an isosceles triangle $RAC$ on the other side of $AC$ such that $\angle ARC = 120^{\circ}.$ Show that $PQR$ is an equilateral triangle.

The internal bisector of angle $A$ in a triangle $ABC$ with $AC > AB$ meets the circumcircle $\Gamma$ of the triangle in $D$. Join$D$ to the center $O$ of the circle $\Gamma$ and suppose that $DO$ meets $AC$ in $E$, possibly when extended. Given that $BE$ is perpendicular to $AD$, show that $AO$ is parallel to $BD$.

2001 India RMO p1
Let $BE$ and $CF$ be the altitudes of an acute triangle $ABC$ with $E$ on $AC$ and $F$ on $AB$. Let $O$ be the point of intersection of $BE$ and $CF$. Take any line $KL$ through $O$ with $K$ on $AB$ and $L$ on $AC$. Suppose $M$ and $N$ are located on $BE$ and $CF$ respectively. such that $KM$ is perpendicular to $BE$ and $LN$ is perpendicular to $CF$. Prove that $FM$ is parallel to $EN$.

2001 India RMO p5
In a triangle $ABC$, $D$ is a point on $BC$ such that $AD$ is the internal bisector of $\angle A$. Suppose $\angle B = 2 \angle C$ and $CD =AB$. prove that $\angle A = 72^{\circ}$.

2002 India RMO p1
In an acute triangle $ABC$ points $D,E,F$ are located on the sides $BC,CA, AB$ such that $\frac{CD}{CE} = \frac{CA}{CB} , \frac{AE}{AF} = \frac{AB}{AC} , \frac{BF}{FD} = \frac{BC}{BA}$. Prove that $AD,BE,CF$ are altitudes of triangle $ABC$.

2002 India RMO p5
The circumference of a circle is divided into eight arcs by a convex quadrilateral $ABCD$ with four arcs lying inside the quadrilateral and the remaining four lying outside it. The lengths of the arcs lying inside the quadrilateral are denoted by $p,q,r,s$ in counter-clockwise direction. Suppose $p+r = q+s$. Prove that $ABCD$ is cyclic.

2003 India RMO p1
Let $ABC$ be a triangle in which $AB =AC$ and $\angle CAB = 90^{\circ}$. Suppose that $M$ and $N$ are points on the hypotenuse $BC$ such that $BM^2 + CN^2 = MN^2$. Prove that $\angle MAN = 45^{\circ}$.

2003 India RMO p5
Suppose $P$ is an interior point of a triangle $ABC$ such that the ratios $\frac{d(A,BC)}{d(P,BC)} , \frac{d(B,CA)}{d(P,CA)} , \frac{d(C,AB)}{d(P,AB)}$ are all equal. Find the common value of these ratios. $d(X,YZ)$ represents the perpendicular distance from $X$ to the line $YZ$.

2004 India RMO p1
Consider in the plane a circle $\Gamma$ with centre $O$ and a line $l$ not intersecting the circle. Prove that there is a unique point $Q$ on the perpendicular drawn from $O$ to line $l$, such that for any point $P$ on the line $l, PQ$ represents the length of the tangent from $P$ to the given circle.

2004 India RMO p5
Let $ABCD$ be a quadrilateral, $X$ and $Y$ be the midpoints of $AC$ and $BD$ respectively and lines through $X$ and $Y$ respectively parallel to $BD, AC$ meet in $O$. Let $P,Q,R,S$ be the midpoints of $AB, BC, CD, DA$ respectively. Prove that
a) $APOS$ and $APXS$ have the same area
b) $APOS, BQOP, CROQ, DSOR$ have the same area.

2005 India RMO p1
Let $ABCD$ be a convex quadrilateral, $P,Q, R,S$ are the midpoints of $AB, BC, CD, DA$ respectively such that triangles $AQR, CSP$ are equilateral. Prove that $ABCD$ is a rhombus. Find its angles.

2005 India RMO p5
In a triangle ABC, D is midpoint of BC . If $\angle ADB = 45 ^{\circ}$ and $\angle ACD = 30^{\circ}$, determine  $\angle BAD.$

2006 India RMO p1
Let $ABC$ be an acute-angled triangle and let $D,E,F$ be the feet of perpendiculars from $A,B,C$ respectively to $BC,CA,AB .$ Let the perpendiculars from  $F$ to $CB,CA,AD,BE$ meet them in $P,Q,M,N$ respectively. Prove that the points $P,Q,M,N$ are collinear.

2006 India RMO p5
Let $ABCD$ be a quadrilateral in which $AB$ is parallel to $CD$ and perpendicular to $AD, AB = 3CD;$ and the area of the quadrilateral is $4$. if a circle can be drawn touching all the four sides of the quadrilateral, find its radius.

2007 India RMO p1
Let $ABC$ be an acute-angled triangle; $AD$ be the bisector of $\angle BAC$ with $D$ on $BC$; and $BE$ be the altitude from $B$ on $AC$.
Show that $\angle CED > 45^\circ .$

2007 India RMO p5
A trapezium $ABCD$, in which $AB$ is parallel to $CD$, is inscribed in a circle with centre $O$. Suppose the diagonals $AC$ and $BD$ of the trapezium intersect at $M$, and $OM = 2$.
(a) If $\angle AMB$ is $60^\circ ,$ find, with proof, the difference between the lengths of the parallel sides.
(b) If $\angle AMD$ is $60^\circ ,$ find, with proof, the difference between the lengths of the parallel sides.

2008 India RMO p1
Let $ABC$ be an acute angled triangle; let $D,F$ be the midpoints of $BC,AB$ respectively. Let the perpendicular from $F$ to $AC$ and the perpendicular from $B$ ti $BC$ meet in $N$: Prove that $ND$ is the circumradius of $ABC$.

2009 India RMO p1
Let $ABC$ be a triangle in which $AB = AC$ and let $I$ be its in-centre. Suppose $BC = AB + AI$. Find $\angle{BAC}$

2009 India RMO p5
A convex polygon is such that the distance between any two vertices does not exceed $1$.
(i) Prove that the distance between any two points on the boundary of the polygon does not exceed $1$.
(ii) If $X$ and $Y$ are two distinct points inside the polygon, prove that there exists a point $Z$ on the boundary of the polygon such that $XZ + YZ\le1$.

2010 India RMO p1
Let $ABCDEF$ be a convex hexagon in which diagonals $AD, BE, CF$ are concurrent at  $O$. Suppose $[OAF]$ is geometric mean of $[OAB]$ and $[OEF]$ and $[OBC]$ is geometric mean of $[OAB]$ and $[OCD]$. Prove that $[OED]$ is the geometric mean of $[OCD]$ and $[OEF]$.
(Here $[XYZ]$ denotes are of $\triangle XYZ$)

2010 India RMO p5
Let $ABC$ be a triangle in which $\angle A = 60^\circ$. Let $BE$ and $CF$ be the bisectors of $\angle B$ and $\angle C$ with $E$ on $AC$ and $F$ on $AB$. Let $M$ be the reflection of $A$ in line $EF$. Prove that $M$ lies on $BC$.

2011 India RMO p1
Let $ABC$ be a triangle. Let $D, E, F$ be points respectively on the segments $BC, CA, AB$ such that $AD, BE, CF$ concur at the point $K$. Suppose $\frac{BD}{DC} = \frac {BF}{FA}$ and $\angle ADB = \angle AFC$. Prove that $\angle ABE = \angle CAD$.

2011 India RMO p5
Let $ABC$ be a triangle and let $BB_1,CC_1$ be respectively the bisectors of $\angle{B},\angle{C}$ with $B_1$ on $AC$ and $C_1$ on $AB$, Let $E,F$ be the feet of perpendiculars drawn from $A$ onto $BB_1,CC_1$ respectively. Suppose $D$ is the point at which the incircle of $ABC$ touches $AB$. Prove that $AD=EF$

Let $ABC$ be an acute angled scalene triangle with circumcentre $O$ and orthocentre $H.$ If $M$ is the midpoint of $BC,$ then show that $AO$ and $HM$ intersect on the circumcircle of $ABC.$

Let $ABCD$ be a convex quadrilateral. Let $E,F,G,H$ be the midpoints of $AB,BC,CD,DA$ respectively. If $AC,BD,EG,FH$ concur at a point $O,$ prove that $ABCD$ is a parallelogram.

2012 India RMO Region 1 p1
Let $ABC$ be a triangle and $D$ be a point on the segment $BC$ such that $DC = 2BD$. Let $E$ be the mid-point of $AC$. Let $AD$ and $BE$ intersect in $P$. Determine the ratios $BP:PE$ and $AP:PD$.

2012 India RMO Region 1 p5
Let $ABC$ be a triangle. Let $BE$ and $CF$ be internal angle bisectors of $\angle B$ and $\angle C$ respectively with $E$ on $AC$ and $F$ on $AB$. Suppose $X$ is a point on the segment $CF$
such that $AX$ perpendicular $CF$; and $Y$ is a point on the segment $BE$ such that $AY$ perpendicular $BE$. Prove that $XY = (b + c-a)/2$ where $BC = a, CA = b$and $AB = c$.

2012 India RMO Region 2 p1
Let $ABCD$ be a unit square. Draw a quadrant of a circle with $A$ as centre and $B,D$ as end points of the arc. Similarly, draw a quadrant of a circle with $B$ as centre and $A,C$ as end points of the arc. Inscribe a circle $\Gamma$ touching the arc $AC$ internally, the arc $BD$ internally and also touching the side $AB$. Find the radius of the circle $\Gamma$?

2012 India RMO Region 2 p5
Let $ABC$ be a triangle. Let $D, E$ be a points on the segment $BC$ such that $BD =DE = EC$. Let $F$ be the mid-point of $AC$. Let $BF$ intersect $AD$ in $P$ and $AE$ in $Q$ respectively. Determine $BP:PQ$.

2012 India RMO Region 3 p1
Let $ABCD$ be a unit square. Draw a quadrant of the a circle with $A$ as centre and $B,D$ as end points of the arc. Similarly, draw a quadrant of a circle with $B$ as centre and $A,C$ as end points of the arc. Inscribe a circle $\Gamma$ touching arcs $AC$ and $BD$ both externally and also touching the side $CD$. Find the radius of $\Gamma$.

2012 India RMO Region 3 p5
Let $ABC$ be a triangle. Let $D,E$ be points on the segment $BC$ such that $BD=DE=EC$. Let $F$ be the mid-point of $AC$. Let $BF$ intersect $AD$ in $P$ and $AE$ in $Q$ respectively. Determine the ratio of the area of the triangle $APQ$ to that of the quadrilateral $PDEQ$.

2012 India RMO Region 4 p1
Let $ABCD$ be a unit square. Draw a quadrant of the a circle with $A$ as centre and $B,D$ as end points of the arc. Similarly, draw a quadrant of a circle with $B$ as centre and $A,C$ as end points of the arc. Inscribe a circle $\Gamma$ touching the arc $AC$ externally, the arc $BD$ externally and also touching the side $AD$. Find the radius of $\Gamma$.

2012 India RMO Region 4 p5
Let $ABC$ be a triangle. Let $E$ be a point on the segment $BC$ such that $BE = 2EC$. Let $F$ be the mid-point of $AC$. Let $BF$ intersect $AE$ in $Q$. Determine $BQ:QF$.

2012 India RMO Region 5 p4 Mumbai
$H$ is the orthocentre of an acute–angled triangle $ABC$. A point $E$ is taken on the line segment $CH$ such that $ABE$ is a right–angled triangle. Prove that the area of the triangle $ABE$ is the geometric mean of the areas of triangles $ABC$ and $ABH$.

2012 India RMO Region 5 p7 Mumbai
On the extension of chord $AB$ of a circle centroid at $O$ a point $X$ is taken and tangents $XC$ and $XD$ to the circle are drawn from it with $C$ and $D$ lying on the circle, let $E$ be the midpoint of the line segment $CD$. If $\angle OEB = 140^o$ then determine with proof the magnitude of $\angle AOB$.

Let $ABCD$ be a convex quadrilateral such that $\angle ADC=\angle BCD>90^{\circ}$. Let $E$ be the point of intersection of $AC$ and the line through $B$ parallel to $AD;$ let $F$ be the point of intersection of $BD$ and the line through $A$ parallel to $BC.$ Prove that $EF\parallel CD.$

Let $AL$ and $BK$ be the angle bisectors in a non-isosceles triangle $ABC,$ where $L$ lies on $BC$ and $K$ lies on $AC.$ The perpendicular bisector of $BK$ intersects the line $AL$ at $M$. Point $N$ lies on the line $BK$ such that $LN$ is parallel to $MK.$ Prove that $LN=NA.$

2013 India RMO Region 1 p1
Let $ABC$ be an acute-angled triangle. The circle $\Gamma$ with $BC$ as diameter intersects $AB$ and $AC$ again at $P$ and $Q$, respectively. Determine $\angle BAC$ given that the orthocenter of triangle $APQ$ lies on $\Gamma$.

2013 India RMO Region 1 p5
Let $ABC$ be a triangle with $\angle A=90^{\circ}$ and $AB=AC$. Let $D$ and $E$ be points on the segment $BC$ such that $BD:DE:EC = 3:5:4$. Prove that $\angle DAE= 45^{\circ}$

2013 India RMO Region 2 p2
In a triangle $ABC$, $AD$ is the altitude from $A$, and $H$ is the orthocentre. Let $K$ be the centre of the circle passing through $D$ and tangent to $BH$ at $H$. Prove that the line $DK$ bisects $AC$.

2013 India RMO Region 2 p4
Let $ABC$ be a triangle with $\angle A=90^{\circ}$ and $AB=AC$. Let $D$ and $E$ be points on the segment $BC$ such that $BD:DE:EC = 1:2:\sqrt{3}$. Prove that $\angle DAE= 45^{\circ}$

2013 India RMO Region 3 p3
In an acute-angled triangle $ABC$ with $AB < AC$, the circle $\omega$ touches $AB$ at $B$ and passes through $C$ intersecting $AC$ again at $D$. Prove that the orthocentre of triangle $ABD$ lies on $\omega$ if and only if it lies on the perpendicular bisector of $BC$.

2013 India RMO Region 3 p5
Let $ABC$ be a triangle which it not right-angled. De fine a sequence of triangles $A_iB_iC_i$, with $i \ge 0$, as follows: $A_0B_0C_0$ is the triangle $ABC$ and, for $i \ge 0$,  $A_{i+1},B_{i+1},C_{i+1}$ are the reflections of the orthocentre of triangle $A_iB_iC_i$ in the sides $B_iC_i$,$C_iA_i$,$A_iB_i$, respectively. Assume that $\angle A_m = \angle A_n$ for some distinct natural numbers $m,n$. Prove that $\angle A = 60^{\circ}$

2013 India RMO Region 4 p1
Let $\omega$ be a circle with centre $O$. Let $\gamma$ be another circle passing through $O$ and intersecting $\omega$ at points $A$ and $B$. $A$ diameter $CD$ of $\omega$ intersects $\gamma$ at a point $P$ different from $O$. Prove that $\angle APC= \angle BPD$

2013 India RMO Region 4 p5
In a triangle $ABC$, let $H$ denote its orthocentre. Let $P$ be the reflection of $A$ with respect to $BC$. The circumcircle of triangle $ABP$ intersects the line $BH$ again at $Q$, and the circumcircle of triangle $ACP$ intersects the line $CH$ again at $R$. Prove that $H$ is the incentre of triangle $PQR$.

2013 India RMO Region 5 p1 Mumbai
Let $ABC$ be an isosceles triangle with $AB=AC$ and let $\Gamma$ denote its circumcircle. A point $D$ is on arc $AB$ of $\Gamma$ not containing $C$. A point $E$ is on arc $AC$ of $\Gamma$ not containing $B$. If $AD=CE$ prove that $BE$ is parallel to $AD$.

In a triangle $ABC$, points $D$ and $E$ are on segments $BC$ and $AC$ such that $BD=3DC$ and $AE=4EC$. Point $P$ is on line $ED$ such that $D$ is the midpoint of segment $EP$. Lines $AP$ and $BC$ intersect at point $S$. Find the ratio $BS/SD$.

In acute $\triangle ABC,$ let $D$ be the foot of perpendicular from $A$ on $BC$. Consider points $K, L, M$ on segment $AD$ such that $AK= KL= LM= MD$. Suppose the sum of the areas of the shaded region equals the sum of the areas of the unshaded regions in the following picture. Prove that $BD= DC$.
2014 India RMO Region 1 p5
Let $ABC$ be an acute angled triangle with $H$ as its orthocentre. For any point $P$ on the circumcircle of triangle $ABC$, let $Q$ be the point of intersection of the line $BH$ with line $AP$. Show that there is a unique point $X$ on the circumcircle of triangle $ABC$ such that for every $P$ other than $B,C$, the circumcircle of $HPQ$ passes through $X$.

2014 India RMO Region 2 p1
In an acute-angled triangle $ABC, \angle ABC$ is the largest angle. The perpendicular bisectors of $BC$ and $BA$ intersect AC at $X$ and $Y$ respectively. Prove that circumcentre of triangle $ABC$ is incentre of triangle $BXY$ .

2014 India RMO Regions 2 p5
Let $ABC$ be a triangle with $AB > AC$. Let $P$ be a point on the line $AB$ beyond $A$ such that $AP +P C = AB$. Let $M$ be the mid-point of $BC$ and let $Q$ be the point on the side $AB$ such that $CQ \perp AM$. Prove that $BQ = 2AP.$

2014 India RMO Region 3 p1
Let $ABC$ be a triangle with $\angle ABC$ as the largest angle. Let $R$ be its circumcenter. Let the circumcircle of triangle $ARB$ cut $AC$ again at $X$. Prove that $RX$ is perpendicular to $BC$.

2014 India RMO Region 3 p5
Let $ABC$ be a triangle and let $X$ be on $BC$ such that $AX=AB$. let $AX$ meet circumcircle $\omega$ of triangle $ABC$ again at $D$. prove that circumcentre of triangle $BDX$ lies on $\omega$.

2014 India RMO Region 4 p1
Let $ABCD$ be a isosceles trapezium having an incircle with $AB$ parallel to $CD$.
Let $CE$ be the perpendicular from $C$ on $AB$.
prove that  $CE^2 = AB. CD$

2014 India RMO Region 4 p5
Let $ABC$ be a triangle and $I$ be its incentre. let the incircle of $ABC$ touch $BC$ at $D$. Let incircle of triangle $ABD$ touch $AB$ at $E$ and incircle of triangle $ACD$ touch $BC$ at $F$.
Prove that $B,E,I,F$ are concyclic

2014 India RMO Region 5 p3  Mumbai
Let $ABC$ be an acute-angled triangle in which $\angle ABC$ is the largest angle. Let $O$ be its circumcentre. The perpendicular bisectors of $BC$ and $AB$ meet $AC$ at $X$ and $Y$ respectively. The internal angle bisectors of $\angle AXB$ and $\angle BYC$ meet $AB$ and $BC$ at $D$ and $E$ respectively. Prove that $BO$ is perpendicular to  $AC$ if $DE$ is parallel to $AC$.
Let $D,E,F$ be the points of contact of the incircle of an acute-angled triangle $ABC$ with $BC,CA,AB$ respectively. Let $I_1,I_2,I_3$ be the incentres of the triangles $AFE, BDF, CED$, respectively. Prove that the lines $I_1D, I_2E, I_3F$ are concurrent.

2015 India RMO Region 1 p1
In a cyclic quadrilateral $ABCD$, let the diagonals $AC$ and $BD$ intersect at $X$. Let the circumcircles of triangles $AXD$ and $BXC$ intersect again at $Y$ . If $X$ is the incentre of triangle $ABY$ , show that $\angle CAD = 90^o$.

2015 India RMO Region 1 p5
Let $ABC$ be a right triangle with $\angle B = 90^{\circ}$.Let $E$ and $F$ be respectively the midpoints of $AB$ and $AC$.Suppose the incentre $I$ of $ABC$ lies on the circumcircle of triangle $AEF$, find the ratio $BC:AB$.

2015 India RMO Region 2 p1
Let $ABC$ be a triangle. Let $B'$ and $C'$ denote the reflection of $B$ and $C$ in the internal angle bisector of angle $A$. Show that the triangles $ABC$ and $AB'C'$ have the same incenter.

2015 India RMO Region 2 p5
Two circles $X$ and $Y$ in the plane intersect at two distinct points $A$ and $B$ such that the centre of $Y$ lies on $X$. Let points $C$ and $D$ be on $X$ and $Y$ respectively, so that $C, B$ and $D$ are collinear. Let point $E$ on $Y$ be such that $DE$ is parallel to $AC$. Show that $AE = AB$.

2015 India RMO Region 3 p1
Two circles $\Gamma$ and $\Sigma$, with centers $O$ and $P$, respectively, are such that $P$ lies on $\Gamma$. Let $A$ be a point on $\Sigma$, and let $M$ be the midpoint of $AP$. Let $B$ be another point on $\Sigma$, such that $AB // OM$. Then prove that the midpoint of $AB$ lies on $\Gamma$.

2015 India RMO Region 3 p5
Let $ABC$ be a triangle with circumcircle $\Gamma$ and incenter $I.$ Let the internal angle bisectors of $\angle A,\angle B,\angle C$ meet $\Gamma$ in $A',B',C'$ respectively. Let $B'C'$ intersect $AA'$ at $P,$ and $AC$ in $Q.$ Let $BB'$ intersect $AC$ in $R.$ Suppose the quadrilateral $PIRQ$ is a kite; that is, $IP=IR$ and $QP=QR.$ Prove that $ABC$ is an equilateral triangle.

2015 India RMO Region 4 p1
Let $ABC$ be a triangle. Let $B'$ denote the reflection of $b$ in the internal angle bisector $l$ of $\angle A$.Show that the circumcentre of the triangle $CB'I$ lies on the line $l$ where $I$ is the incentre of $ABC$.

2015 India RMO Region 4 p5
Two circles $\Gamma$ and $\Sigma$ intersect at two distinct points $A$ and $B$. A line through $B$ intersects $\Gamma$ and $\Sigma$ again at $C$ and $D$, respectively. Suppose that $CA=CD$. Show that the centre of $\Sigma$ lies on $\Gamma$.

2015 India RMO Region 5 p1 Mumbai
Let $ABCD$ be a convex quadrilateral with $AB=a$, $BC=b$, $CD=c$ and $DA=d$. Suppose
$a^2+b^2+c^2+d^2=ab+bc+cd+da,$ and the area of $ABCD$ is $60$ sq. units. If the length of one of the diagonals is $30$ units, determine the length of the other diagonal.

2015 India RMO Region 5 p5 Mumbai
Let $ABC$ be a right-angled triangle with $\angle B = 90^\circ$ and let $BD$ be the altitude from $B$ on to $AC$. Draw $DE \perp AB$ and $DF \perp BC$. Let $P, Q, R$ and $S$ be respectively the incentres of triangle $DF C, DBF, DEB$ and $DAE$. Suppose $S, R, Q$ are collinear. Prove that $P, Q, R, D$ lie on a circle.

The length of each side of a convex quadrilateral $ABCD$ is a positive integer. If the sum of the lengths of any three sides is divisible by the length of the remaining side then prove that some two sides of the quadrilateral have the same length.

2016 India RMO Regions 1 p1 Mumbai
Let $ABC$ be a right-angled triangle with $\angle B=90^{\circ}$. Let $I$ be the incenter of $ABC$. Draw a line perpendicular to $AI$ at $I$. Let it intersect the line $CB$ at $D$. Prove that $CI$ is perpendicular to $AD$ and prove that $ID=\sqrt{b(b-a)}$ where $BC=a$ and $CA=b$.

2016 India RMO Regions 1 p5 Mumbai
Let $ABC$ be a triangle with centroid $G$. Let the circumcircle of triangle $AGB$ intersect the line $BC$ in $X$ different from $B$; and the circucircle of triangle $AGC$ intersect the line $BC$ in $Y$ different from $C$. Prove that $G$ is the centroid of triangle $AXY$.

2016 India RMO Regions 2 p1 Telangana
Let $ABC$ be a right angled triangle with $\angle B=90^{\circ}$. Let $I$ be the incentre of triangle $ABC$. Suppose $AI$ is extended to meet $BC$ at $F$ . The perpendicular on $AI$ at $I$ is extended to meet $AC$ at $E$ . Prove that $IE = IF$.

2016 India RMO Regions 2 p5 Telangana
Let $ABC$ be a right angled triangle with $\angle B=90^{\circ}$. Let $AD$ be the bisector of angle $A$ with $D$ on $BC$ . Let the circumcircle of triangle $ACD$ intersect $AB$ again at $E$; and let the circumcircle of triangle $ABD$ intersect $AC$ again at $F$ . Let $K$ be the reflection of $E$ in the line $BC$ . Prove that $FK = BC$.

2016 India RMO Regions 3 p1  Karnataka and WB
Let $ABC$ be a triangle and $D$ be the mid-point of $BC$. Suppose the angle bisector of $\angle ADC$ is tangent to the circumcircle of triangle $ABD$ at $D$. Prove that  $\angle A=90^{\circ}$.

2016 India RMO Regions 3 p5  Karnataka and WB
Let $ABC$ be a right-angled triangle with $\angle B=90^{\circ}$. Let $I$ be the incentre if $ABC$. Extend $AI$ and $CI$; let them intersect $BC$ in $D$ and $AB$ in $E$ respectively. Draw a line perpendicular to $AI$ at $I$ to meet $AC$ in $J$, draw a line perpendicular to $CI$ at $I$ to meet $AC$ at $K$. Suppose $DJ=EK$. Prove that  $BA=BC$.

2016 India RMO Regions 4 p1 Odisha
Let $ABC$ be an isosceles triangle with $AB=AC.$ Let $\Gamma$ be its circumcircle and let $O$ be the centre of $\Gamma$ . let $CO$ meet $\Gamma$ in $D .$  Draw a line parallel to $AC$ thrugh $D.$ Let it intersect $AB$ at  $E.$  Suppose $AE : EB=2:1$ .Prove that $ABC$ is an equilateral triangle.

2016 India RMO Regions 4 p5 Odisha
Let $ABC$ be a triangle , $AD$ an altitude and  $AE$ a median . Assume $B,D,E,C$ lie in that order on the line $BC$ . Suppose the incentre of  triangle  $ABE$ lies on $AD$ and he incentre of  triangle  $ADC$ lies on $AE$ . Find ,with proof ,the angles of  triangle  $ABC$ .[/quote]

2016 India RMO Region Delhi p1
Given are two circles $\omega_1,\omega_2$ which intersect at points $X,Y$. Let $P$ be an arbitrary point on $\omega_1$. Suppose that the lines $PX,PY$ meet $\omega_2$ again at points $A,B$ respectively. Prove that the circumcircles of all triangles $PAB$ have the same radius.

2016 India RMO Region Delhi p3
Two circles $C_1$ and $C_2$ intersect each other at points $A$ and $B$. Their external common tangent (closer to $B$) touches $C_1$ at $P$ and $C_2$ at $Q$. Let $C$ be the reflection of $B$ in line $PQ$. Prove that $\angle CAP=\angle BAQ$.
Let $\triangle ABC$ be scalene, with $BC$ as the largest side. Let $D$ be the foot of the perpendicular from $A$ on side $BC$. Let points $K,L$ be chosen on the lines $AB$ and $AC$ respectively, such that $D$ is the midpoint of segment $KL$. Prove that the points $B,K,C,L$ are concyclic if and only if $\angle BAC=90^{\circ}$.

2017 India RMO Regions Odisha p1
Let $AOB$ be a given angle less than $180^{\circ}$ and let $P$ be an interior point of the angular region determined by $\angle AOB$. Show, with proof, how to construct, using only ruler and compass, a line segment $CD$ passing through $P$ such that $C$ lies on the way $OA$ and $D$ lies on the ray $OB$, and $CP:PD=1:2$.

Let $\Omega$ be a circle with a chord $AB$ which is not a diameter. $\Gamma_{1}$ be a circle on one side of $AB$ such that it is tangent to $AB$ at $C$ and internally tangent to $\Omega$ at $D$. Likewise, let $\Gamma_{2}$ be a circle on the other side of $AB$ such that it is tangent to $AB$ at $E$ and internally tangent to $\Omega$ at $F$. Suppose the line $DC$ intersects $\Omega$ at $X \neq D$ and the line $FE$ intersects $\Omega$ at $Y \neq F$. Prove that $XY$ is a diameter of $\Omega$ .

Let $ABC$ be a triangle with integer sides in which $AB<AC$. Let the tangent to the circumcircle of triangle $ABC$ at $A$ intersect the line $BC$ at $D$. Suppose $AD$ is also an integer. Prove that $\gcd(AB,AC)>1$.

Let $ABC$ be an acute-angled triangle with $AB<AC$. Let $I$ be the incentre of triangle $ABC$, and let $D,E,F$ be the points where the incircle touches the sides $BC,CA,AB,$ respectively. Let $BI,CI$ meet the line $EF$ at $Y,X$ respectively. Further assume that both $X$ and $Y$ are outside the triangle $ABC$. Prove that
i) $B,C,Y,X$ are concyclic.
ii) $I$ is also the incentre of triangle $DYX$.

Let $ABC$ be an acute angled triangle and let $D$ be an interior point of the segment $BC$. Let the circumcircle of $ACD$ intersect $AB$ at $E$ ($E$ between $A$ and $B$) and let circumcircle of $ABD$ intersect $AC$ at $F$ ($F$ between $A$ and $C$). Let $O$ be the circumcenter of $AEF$. Prove that $OD$ bisects $\angle EDF$.

In a cyclic quadrilateral $ABCD$ with circumcenter $O$, the diagonals $AC$ and $BD$ intersect at $X$. Let the circumcircles of triangles $AXD$ and $BXC$ intersect at $Y$. Let the circumcircles of triangles $AXB$ and $CXD$ intersect at $Z$. If $O$ lies inside $ABCD$ and if the points $O,X,Y,Z$ are all distinct, prove that $O,X,Y,Z$ lie on a circle.

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