A nice Indian RMO problem from 2025

The RMO (short for regional maths olympiad) is the test required to be passed, to be eligible for participating in the national level maths olympiad. The top 50 or so from the national olympiad then go through 2-3 months of training and testing and top 6 are then selected to represent India in the international maths olympiad.

The following nice problem appeared in the RMO 2025. Seems like a difficult one for the levels expected in the RMO.

Show that there are no positive rationals $x,y$ such that

$$ x + y + \frac{1}{x} + \frac{1}{y} = 2025$$

Scroll down for a solution.

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Solution:

Let $x = \frac{a}{b}$ and $y = \frac{c}{d}$ where $\gcd(a,b) = \gcd(c,d) = 1$

Rewriting as 

$$ x + \frac{1}{x} + y + \frac{1}{y} = 2025 $$

we get

$$ (a^2 + b^2) cd + (c^2 + d^2) ab = 2025 abcd$$

This implies that $(c^2 + d^2)ab$ is divisible by $cd$.

Now $c(c^2 + d^2) - d (cd) = c^3$ and $d(c^2 + d^2) - c (cd) = d^3$

Thus gcd of $c^2 + d^2$ and $cd$ divides both $c^3$ and $d^3$, and thus must be $1$.

Therefore $cd$ divides $ab$.

In a similar fashion, $ab$ divides $cd$.

Thus we must have that $ab = cd$

So we can now rewrite as

$$a^2 + b^2 + c^2 + d^2 = 2025 ab = 0 \mod 3$$

Now exactly zero or two of $a,b,c,d$ must be divisible by $3$, which gives $$a^2 + b^2 + c^2 + d^2 \neq 0 \mod 3$$

Thus there cannot be such rational $x,y$.

A very nice problem!