3

Part of 2011 Iran MO (2nd Round)

Problems(2)

BD'+CE'=D'E' if BD+CE=DE

Source: Iran second round 2011

l

intersects the extension of

AB

D

D

B

A

) and the extension of

AC

E

E

C

A

ABC

. Suppose that reflection of line

l

to perpendicular bisector of side

BC

intersects the mentioned extensions in

D'

E'

respectively. Prove that if

BD+CE=DE

BD'+CE'=D'E'

geometrygeometric transformationreflectioncircumcircleincenterperpendicular bisectorgeometry proposed

increasing sequence

Source: Iran second round 2011

Find all increasing sequences

a_1,a_2,a_3,...

of natural numbers such that for each

i,j\in \mathbb N

, number of the divisors of

i+j

a_i+a_j

is equal. (an increasing sequence is a sequence that if

i\le j

a_i\le a_j

inductionnumber theory proposednumber theory