IMO 1999 SL N2

Origin: ARM | Category: Number Theory

Problem

Prove that every positive rational number can be repre- sented in the form a3 + b3 c3 + d3 , where a, b, c, d are positive integers.

Solution

We first prove that every rational number in the interval (1, 2) can be represented in the form a3+b3 a3+d3 . Taking b, d such that b ̸= d and a = b + d, we get a2 −ab + b2 = a2 −ad + d2 and a3 + b3 a3 + d3 = (a + b)(a2 −ab + b2) (a + d)(a2 −ad + d2) = a + b a + d . For a given rational number 1 < m/n < 2 we can select a = m + n and b = 2m −n such that along with d = a −b we have a+b a+d = m n . This completes the proof of the first statement. For m/n outside of the interval we can easily select a rational number p/q such that 3 n m < p q < 2n m . In other words 1 < p3m q3n < 2. We now proceed to obtain a, b and d for p3m q3n as before, and we finally have p3m q3n = a3 + b3 a3 + d3 ⇒m n = (aq)3 + (bq)3 (ap)3 + (dp)3 . Thus we have shown that all positive rational numbers can be expressed in the form a3+b3 c3+d3 .