WMC 2018 Senior Problem 10

Level 2

Let a, b and c be real numbers that satisfy the following equations:

$a+\frac { 1 }{ bc } =\frac { 1 }{ 3 } ,\quad b+\frac { 1 }{ ac } =\frac { 1 }{ 6 } ,\quad c+\frac { 1 }{ ab } =\frac { 1 }{ 2 }$

Find the value of $\frac{c+b}{c+a}$ .

\frac { 7 }{ 12 }

\frac { 5 }{ 6 }

\frac { 2 }{ 9 }

\frac { 4 }{ 5 }

\frac { 1 }{ 2 }

1 solution

X X
Oct 14, 2018

Let $1+\dfrac1{abc}=t$

Then, $at=\dfrac13,bt=\dfrac16,ct=\dfrac12$

$\dfrac{c+b}{c+a}=\dfrac{ct+bt}{ct+at}=\dfrac45$

abc+1=bc/3 abc+1=ac/6 abc+1=ab/2 So bc/3=ac/6, hence a=2b ac/6=ab/2, hence c=3b So (c+b)/(c+a)=(3b+b)/(3b+2b)=4/5

David Webb - 2 years, 5 months ago