Mock USAMO by probability1.01 dropped problems
Problem 1
Let be a fixed positive integer, and let
be distinct positive integers. We define
. Prove that there are no distinct positive integers
for which
is a geometric sequence.
Problem 2
In triangle ,
, let the incircle touch
,
, and
at
,
, and
respectively. Let
be a point on
on the opposite
side of
from
. If
and
meet at
, and
and
meet
at
, prove that
,
, and
concur.
Reason: The whole incircle business seemed rather artificial. Besides, it wasn’t that difficult.