MathDB

1

Part of 2012 IMC

Problems(2)

IMC 2012 Day 1, Problem 1

Source:

7/28/2012
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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Albert Einstein and Homer Simpson

Source: IMC 2012, Day 2, Problem 1

7/29/2012
Consider a polynomial f(x)=x2012+a2011x2011+⋯+a1x+a0.f(x)=x^{2012}+a_{2011}x^{2011}+\dots+a_1x+a_0. Albert Einstein and Homer Simpson are playing the following game. In turn, they choose one of the coefficients a0,a1,…,a2011a_0,a_1,\dots,a_{2011} and assign a real value to it. Albert has the first move. Once a value is assigned to a coefficient, it cannot be changed any more. The game ends after all the coefficients have been assigned values. Homer's goal is to make f(x)f(x) divisible by a fixed polynomial m(x)m(x) and Albert's goal is to prevent this. (a) Which of the players has a winning strategy if m(x)=x−2012m(x)=x-2012? (b) Which of the players has a winning strategy if m(x)=x2+1m(x)=x^2+1?
Proposed by Fedor Duzhin, Nanyang Technological University.
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