Difference between revisions of "Mock USAMO by probability1.01 dropped problems"

(Problem 2)
(Problem 1)
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== Problem 1 ==
 
== Problem 1 ==
 
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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 $n>1$ be a fixed positive integer, and let $a_1,a_2,\ldots,a_n$ be distinct positive integers. We define $S_k=a_1^k+a_2^k+\cdots+a_n^k$. Prove that there are no distinct positive integers $p,q,r$ for which $S_p,S_q,S_r$ is a geometric sequence.

Solution

Problem 2

In triangle $ABC$, $AB \not= AC$, let the incircle touch $BC$, $CA$, and $AB$ at $D$, $E$, and $F$ respectively. Let $P$ be a point on $AD$ on the opposite side of $EF$ from $D$. If $EP$ and $AB$ meet at $M$, and $FP$ and $AC$ meet at $N$, prove that $MN$, $EF$, and $BC$ concur. Reason: The whole incircle business seemed rather artificial. Besides, it wasn’t that difficult.

Mock usamo.png

Solution

Problem 3

Solution

Problem 4

Solution

Problem 5

Solution

Problem 6

Solution

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