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 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.
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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==
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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>.
 
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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==See Also==
 
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{{Mock AIME box|year=2006-2007|n=2|before=First Question|num-a=2}}
*[[Mock AIME 2 2006-2007/Problem 2 | Next Problem]]
 
 
 
*[[Mock AIME 2 2006-2007]]
 
  
 
[[Category:Introductory Number Theory Problems]]
 
[[Category:Introductory Number Theory Problems]]
 
[[Category:Introductory Algebra Problems]]
 
[[Category:Introductory Algebra Problems]]

Latest revision as of 10:49, 4 April 2012

Problem

A positive integer is called a dragon if it can be written as the sum of four positive integers $a,b,c,$ and $d$ such that $a+4=b-4=4c=d/4.$ Find the smallest dragon.

Solution

From $4c = \frac{d}4$ we have that 16 divides $d$. From $a + 4 = \frac d4$ we have $d \geq 20$. Minimizing $d$ minimizes $a, b$ and $c$ and consequently minimizes our dragon. The smallest possible choice is $d = 32$, from which $a = 4, b = 12$ and $c = 2$ so our desired number is $a + b + c + d = 4 + 12 + 2 + 32 = 050$.

See Also

Mock AIME 2 2006-2007 (Problems, Source)
Preceded by
First Question
Followed by
Problem 2
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