5

Part of 1999 IMO Shortlist

Problems(2)

IMO ShortList 1999, geometry problem 5

Source: IMO ShortList 1999, geometry problem 5

ABC

\Omega

its incircle and

\Omega_{a}, \Omega_{b}, \Omega_{c}

three circles orthogonal to

\Omega

passing through

(B,C),(A,C)

(A,B)

respectively. The circles

\Omega_{a}

\Omega_{b}

C'

; in the same way we obtain the points

B'

A'

. Prove that the radius of the circumcircle of

A'B'C'

is half the radius of

\Omega

geometryIMO Shortlist

IMO ShortList 1999, number theory problem 5

Source: IMO ShortList 1999, number theory problem 5

n,k

be positive integers such that n is not divisible by 3 and

k \geq n

. Prove that there exists a positive integer

m

which is divisible by

n

and the sum of its digits in decimal representation is

k

number theorydecimal representationsum of digitsIMO Shortlist