# Difference between revisions of "Mock AIME 4 2006-2007 Problems/Problem 12"

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Let <math>f(x)</math> denote the number of partitions that have an even number of even parts of <math>x</math>. Testing a few small values for <math>x</math>, we see that <math>f(1)=0, f(2)=0, f(3)=0, f(4)=1, f(5)=1, f(6)=2, f(7)=2, | Let <math>f(x)</math> denote the number of partitions that have an even number of even parts of <math>x</math>. Testing a few small values for <math>x</math>, we see that <math>f(1)=0, f(2)=0, f(3)=0, f(4)=1, f(5)=1, f(6)=2, f(7)=2, | ||

f(8)=4, f(9)=4...</math>. | f(8)=4, f(9)=4...</math>. |

## Latest revision as of 20:34, 27 September 2019

## Problem

The number of partitions of 2007 that have an even number of even parts can be expressed as , where and are positive integers and is prime. Find the sum of the digits of .

## Solution

Let denote the number of partitions that have an even number of even parts of . Testing a few small values for , we see that . Based on our observations, we now conjecture* that for every integer , So plugging in , we get

- Insert proof of conjecture here