MathDB

RMO 2015

Source:

12/10/2015

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.

RMO 2015geometryincentercircumcircleratio

Question 5

Source:

12/6/2015

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.

RMO 2015

equilateral triangle from Indian RMO

Source: CRMO 2015 Region 3 (West Bengal) p5

9/30/2018

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.

geometrycircumcircleincenterEquilateral Triangle

RMO 2015 Karnataka geometry II, intersecting circles

Source: CRMO 2015 region 4 (Karnataka) p5

9/30/2018

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

.

geometrycirclesequal segments

Incentres in right triangle

Source: RMO 2015 Mumbai Region

9/3/2018

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.

geometry

5

Problems(5)