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 $a,b,c,$ 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 $a + b + c + d = 4 + 12 + 2 + 32 = 050$ .

Solution2

Note: a+b+c+d=a+a+8+(a+4)/4+4a+16 =(25a+100)/4 = 25(a+4)/4 . Therefore, we know that a+4/4 has to be an integer. since a>0, the smaller integer we can have is 2, so a=4. Therefore, the smallest value is 50.

~Please convert to latex

Mock AIME 2 2006-2007 (Problems, Source)
Preceded by First Question	Followed by Problem 2
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Mock AIME 2 2006-2007 Problems/Problem 1

Contents

Problem

Solution

Solution2

See Also