Difference between revisions of "Mock AIME 2 2006-2007 Problems/Problem 1"
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== Problem == | == Problem == | ||
| − | A positive integer is called a dragon if it can be | + | A [[positive integer]] is called a ''dragon'' if it can be written as the sum of four positive integers <math>\displaystyle a,b,c,</math> and <math>\displaystyle d</math> such that <math>\displaystyle a+4=b-4=4c=d/4.</math> Find the smallest dragon. |
==Solution== | ==Solution== | ||
| − | { | + | |
| + | From <math>4c = \frac{d}4</math> we have that 16 [[divisor | divides]] <math>d</math>. From <math>a + 4 = \frac d4</math> we have <math>d \geq 20</math>. Minimizing <math>d</math> minimizes <math>a, b</math> and <math>c</math> and consequently minimizes our dragon. The smallest possible choice is <math>d = 32</math>, from which <math>a = 4, b = 12</math> and <math>c = 2</math> so our desired number is <math>a + b + c + d = 4 + 12 + 2 + 32 = 050</math>. | ||
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Revision as of 12:24, 15 September 2006
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 
.
