HMMT February

Algebra Difficulty 5.1 AIME, harder Find the answer

Two positive rational numbers $x$ and $y$ , when written in lowest terms, have the property that the sum of their numerators is 9 and the sum of their denominators is 10 . What is the largest possible value of $x+y$ ?

Your answer

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Solution

For fixed denominators $a<b$ (with sum 10), we maximize the sum of the fractions by giving the smaller denominator as large a numerator as possible: $8 / a+1 / b$ . Then, if $a \geq 2$ , this quantity is at most $8 / 2+1 / 1=5$ , which is clearly smaller than the sum we get by setting $a=1$ , namely $8 / 1+1 / 9=73 / 9$ . So this is the answer.

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