Difference between revisions of "Mock USAMO by probability1.01 dropped problems"
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== Problem 1 == | == Problem 1 == | ||
− | + | Let <math>n>1</math> be a fixed positive integer, and let <math>a_1,a_2,\ldots,a_n</math> be distinct positive integers. We define <math>S_k=a_1^k+a_2^k+\cdots+a_n^k</math>. Prove that there are no distinct positive integers <math>p,q,r</math> for which <math>S_p,S_q,S_r</math> is a geometric sequence. | |
[[Mock USAMO by probability1.01 dropped problems/Problem 1|Solution]] | [[Mock USAMO by probability1.01 dropped problems/Problem 1|Solution]] |
Revision as of 02:40, 16 May 2009
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.