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polite numbers, sum of consecutive numbers, power of two

Source: Dutch NMO 2010 p2

September 6, 2019
number theorypower of 2consecutive

Problem Statement

A number is called polite if it can be written as m+(m+1)+...+n m + (m+1)+...+ n, for certain positive integers m<n m <n . For example: 1818 is polite, since 18=5+6+718 =5 + 6 + 7. A number is called a power of two if it can be written as 2ℓ2^{\ell} for some integer ℓ≥0\ell \ge 0. (a) Show that no number is both polite and a power of two. (b) Show that every positive integer is polite or a power of two.
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