MathDB

Divisibility problem with fractions

Source: IMOTC 2015 Practice Test 1 Problem 1

7/11/2015

Find all positive integers

a,b

such that

\frac{a^2+b}{b^2-a}

and

\frac{b^2+a}{a^2-b}

are also integers.

number theoryAPMO

Concyclic incentres and excentres

Source: Indian Team Selection Test 2015 Day 1 Problem 1

7/11/2015

Let

ABCD

be a convex quadrilateral and let the diagonals

AC

and

BD

intersect at

O

. Let

I_1, I_2, I_3, I_4

be respectively the incentres of triangles

AOB, BOC, COD, DOA

. Let

J_1, J_2, J_3, J_4

be respectively the excentres of triangles

AOB, BOC, COD, DOA

opposite

O

. Show that

I_1, I_2, I_3, I_4

lie on a circle if and only if

J_1, J_2, J_3, J_4

lie on a circle.

geometry

Midpoints of arcs reflected on sides

Source: IMOTC 2015 Practice Test 2 Problem 1

7/11/2015

Let

ABC

be a triangle in which

CA>BC>AB

. Let

H

be its orthocentre and

O

its circumcentre. Let

D

and

E

be respectively the midpoints of the arc

AB

not containing

C

and arc

AC

not containing

B

. Let

D'

and

E'

be respectively the reflections of

D

in

AB

and

E

in

AC

. Prove that

O, H, D', E'

lie on a circle if and only if

A, D', E'

are collinear.

geometrycircumcirclegeometric transformationreflection

Reflection of incentre on a side

Source: Indian Team Selection Test 2015 Day 3 Problem 1

7/11/2015

In a triangle

ABC

, a point

D

is on the segment

BC

, Let

X

and

Y

be the incentres of triangles

ACD

and

ABD

respectively. The lines

BY

and

CX

intersect the circumcircle of triangle

AXY

at

P\ne Y

and

Q\ne X

, respectively. Let

K

be the point of intersection of lines

PX

and

QY

. Suppose

K

is also the reflection of

I

in

BC

where

I

is the incentre of triangle

ABC

. Prove that

\angle BAC=\angle ADC=90^{\circ}

.

geometryreflectionincenter

1

Problems(4)