Difference between revisions of "Mock AIME 2 2006-2007 Problems/Problem 1"
m (moved Mock AIME 2 2006-2007/Problem 1 to Mock AIME 2 2006-2007 Problems/Problem 1: correcting title) |
m |
||
Line 1: | Line 1: | ||
== Problem == | == Problem == | ||
− | A [[positive integer]] is called a ''dragon'' if it can be written as the sum of four positive integers <math> | + | A [[positive integer]] is called a ''dragon'' if it can be written as the sum of four positive integers <math>a,b,c,</math> and <math>d</math> such that <math>a+4=b-4=4c=d/4.</math> Find the smallest dragon. |
==Solution== | ==Solution== | ||
Line 8: | Line 8: | ||
---- | ---- | ||
− | *[[Mock AIME 2 2006-2007/Problem 2 | Next Problem]] | + | *[[Mock AIME 2 2006-2007 Problems/Problem 2 | Next Problem]] |
*[[Mock AIME 2 2006-2007]] | *[[Mock AIME 2 2006-2007]] |
Revision as of 15:35, 3 April 2012
Problem
A positive integer is called a dragon if it can be written as the sum of four positive integers and
such that
Find the smallest dragon.
Solution
From we have that 16 divides
. From
we have
. Minimizing
minimizes
and
and consequently minimizes our dragon. The smallest possible choice is
, from which
and
so our desired number is
.