# 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.*