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I was working on the Volume 2 book published by the Art of Problem Solving company when I came across a problem that I just didn't know how to start. The problem is as follows:

Q: Are there integers m and n such that $5m^2 - 6mn + 7n^2 = 1985?$ (IMO 1985)

Here is the solution (I made it a spoiler if anyone wants to solve the problem first):

A: Multiplying the equation by 5 and completing the square to get perfect squares, we obtain $(5m-3n)^2 + 26n^2 = 9925$. Taking the equation (mod 13) to eliminate the $26n^2$, we have $(5m-3n)^2 \equiv 6 \pmod {13}$. But the squares mod 13 are 0, 1, 4, 9, 3, 12, and 10; since 6 is not a square [I think it should say "quadratic residue" instead of "square"], there can be no such m and n.

I completely understand each step taken in the solution but I am still confused as to what I should have done before and during my attempt at solving the problem in order to be able to think of that method during a timed test. Is it just practice and having exposure to many types of problems, and what would you guys have done when solving/seeing the problem for the first time?

Soham Konar
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    This is probably not very insightful but for me, the left side screams to be made into a square - which is what I would try doing first. – Qi Zhu Dec 31 '19 at 19:35
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    It's a standard reduction to Pell form that goes back to Legendre, see this answer. Here it's not much more than completing the square. – Bill Dubuque Dec 31 '19 at 19:55
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    Also worth knowing is completing a product (see the "Linked" questions there for many worked examples). – Bill Dubuque Dec 31 '19 at 20:00
  • Yes, that makes sense. Thank you to both of you. – Soham Konar Dec 31 '19 at 20:50
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    In answer to your more general question, there is no guaranteed way of solving a difficult problem. You usually have to try many different approaches before succeeding. One of the basic ideas is to ask yourself "do I know any result like this?", "what does this remind me of?". – almagest Jan 01 '20 at 19:22
  • Yeah that's what I usually do when solving problems like this one. – Soham Konar Jan 01 '20 at 19:23
  • maybe one of the basic ways to solve this kind of questions is to solve a quadratic equation, but this is a long and frustrating way to solve this particular problem – PNT Feb 12 '21 at 11:54

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