8

Part of 1998 IMO Shortlist

Problems(2)

IMO ShortList 1998, number theory problem 8

Source: IMO ShortList 1998, number theory problem 8

a_{0},a_{1},a_{2},\ldots

be an increasing sequence of nonnegative integers such that every nonnegative integer can be expressed uniquely in the form

a_{i}+2a_{j}+4a_{k}

i,j

k

are not necessarily distinct. Determine

a_{1998}

number theoryInteger sequenceAdditive combinatoricsAdditive Number TheoryIMO Shortlist

A difficult problem [tangent circles in right triangles]

Source: IMO ShortList 1998, geometry problem 8; Yugoslav TST 1999

ABC

be a triangle such that

\angle A=90^{\circ }

\angle B<\angle C

. The tangent at

A

to the circumcircle

\omega

ABC

BC

D

E

be the reflection of

A

BC

X

be the foot of the perpendicular from

A

BE

Y

be the midpoint of the segment

AX

BY

intersect the circle

\omega

Z

. Prove that the line

BD

is tangent to the circumcircle of triangle

ADZ

. [hide="comment"] Edited by Orl.

geometrycircumcirclereflectioncomplex numbersIMO Shortlistgeometry solvedharmonic division