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IMC 2012 Day 1, Problem 1

Source:

July 28, 2012
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Problem Statement

For every positive integer nn, let p(n)p(n) denote the number of ways to express nn as a sum of positive integers. For instance, p(4)=5p(4)=5 because
4=3+1=2+2=2+1+1=1+1+1.4=3+1=2+2=2+1+1=1+1+1.
Also define p(0)=1p(0)=1.
Prove that p(n)−p(n−1)p(n)-p(n-1) is the number of ways to express nn as a sum of integers each of which is strictly greater than 1.
Proposed by Fedor Duzhin, Nanyang Technological University.
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