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Indian Team Selection Test 2010 ST3 P2

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May 22, 2010
number theoryprime numbersnumber theory unsolved

Problem Statement

Call a positive integer good if either N=1N=1 or NN can be written as product of even number of prime numbers, not necessarily distinct. Let P(x)=(x−a)(x−b),P(x)=(x-a)(x-b), where a,ba,b are positive integers.
(a) Show that there exist distinct positive integers a,ba,b such that P(1),P(2),⋯ ,P(2010)P(1),P(2),\cdots ,P(2010) are all good numbers. (b) Suppose a,ba,b are such that P(n)P(n) is a good number for all positive integers nn. Prove that a=ba=b.
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