Mock AIME 2 2006-2007 Problems/Problem 1

Problem

A positive integer is called a dragon if it can be written as the sum of four positive integers $\displaystyle a,b,c,$ and $\displaystyle d$ such that $\displaystyle a+4=b-4=4c=d/4.$ 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 $a + b + c + d = 4 + 12 + 2 + 32 = 050$ .

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Mock AIME 2 2006-2007

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Mock AIME 2 2006-2007 Problems/Problem 1

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