tmua.me Medium Set A Paper 2 · Question 20

A student begins a proof that there are no positive integers $a$ and $b$ satisfying

a^2+b^2=3ab.

Assume that a solution exists, and choose one for which $a+b$ is as small as possible. By interchanging $a$ and $b$ if necessary, take $a\leq b$ . In fact, $a<b$ , since $a=b$ would imply $a=0$ .

The student will complete the proof by finding a number $c$ such that

$c$ is a positive integer;
$c<a$ ;
$c^2+a^2=3ca$ .

The pair $c,a$ would then be a solution with a smaller sum.

Which expression for $c$ completes the proof?

3a-b

3b-a

b-a

2a-b

\dfrac{a+b}{2}

a+b

examday

tmua.me Medium Set A Paper 2 · Q20