作者LimSinE (r=e^theta)
看板IMO_Taiwan
標題Re: APMO-2010
時間Mon Apr 5 23:06:00 2010
Problem 2.
For a positive integer k, call an integer a pure k-th power if it can
be represented as m^k for some integer m. Show that for every positive
integer n there exist n distinct positive integers such that their sum
is a pure 2009-th power, and their product is a pure 2010-th power.
Set a1,...,an distinct.
N = a1^2010 + a2^2010 + ... + an^2010
Define Xi = N^(2010*2008) ai^2010
Then
X1+X2+...+Xn = N^(2010*2008) * N = (N^2009)^2009, a pure 2009-th power
X1X2...Xn = (N^2008n a1a2...an) ^ 2010, a pure 2010-th power...
好像太簡單了...
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r=e^theta
即使有改變，我始終如一。
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※ 編輯: LimSinE 來自: 219.68.26.196 (04/06 23:03)