Difference between revisions of "Mock AIME 4 2006-2007 Problems/Problem 12"
(→Solution) |
|||
(2 intermediate revisions by 2 users not shown) | |||
Line 3: | Line 3: | ||
==Solution== | ==Solution== | ||
− | {{ | + | 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>. | ||
+ | Based on our observations, we now conjecture* that for every integer <math>x\ge 4</math>, <math>f(x)=2^{\lfloor\frac{x-4}{3}\rfloor}</math> So plugging in <math>x=2007</math>, we get <math>f(2007)=2^{1001} \rightarrow 2+1001=1003, 1+3=\boxed{004}</math> | ||
+ | *Insert proof of conjecture here | ||
+ | ---- | ||
+ | |||
+ | *[[Mock AIME 4 2006-2007 Problems/Problem 13| Next Problem]] | ||
+ | *[[Mock AIME 4 2006-2007 Problems/Problem 11| Previous Problem]] | ||
*[[Mock AIME 4 2006-2007 Problems]] | *[[Mock AIME 4 2006-2007 Problems]] |
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