Problem for 7th Grade

[MM]

Find all triplets of positive integer numbers a, b and c for which [tex]\frac{a}{a+2}=\frac{b^{2}}{b+4}=\frac{c^{3}}{c+10}[/tex].
Bulgarian Winter Mathematical Competition 2004 7th grade.

[broniran]

Since a is a positive integer [tex]\frac{a}{a+2}<1[/tex]. This implies that [tex]\frac{c^3}{c+10}<1\Rightarrow c<3[/tex].It's sufficient to check for [tex]c=1,2[/tex].
1 case: [tex]c=1[/tex], then [tex]\frac{c^3}{c+10}=\frac{1}{11}[/tex]. There doesn't exist a positive integer a such that [tex]\frac{a}{a+2}=\frac{1}{11}[/tex]
2 case: [tex]c=2[/tex], then [tex]\frac{c^3}{c+10}=\frac{4}{9}[/tex] but again there doesn't exist a positive integer a with the needed properties.
This means that there isn't a triple [tex](a,b,c)[/tex] of positive integers such that [tex]\frac{a}{a+2}=\frac{b^2}{b+10}=\frac{c^3}{c+10}[/tex]