Index of Blogs and Courses
Problem 6 IMO 1988: Let a and b be positive integers
Show that (a2+ b2)/(ab +1) is the square of an integer.
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Problem 6 in the 1988 International Mathematical Olympiad paper has almost reached a legendry status. The problem is considered extremely difficult to solve - most solutions require a high level of mathematical sophistication or are long and tedious.
Using school level maths, I obtain a general term that may be used to provide a complete list of solutions.
First we notice that the expression (a2+ b2)/(ab +1) is symmetric in a and b. The problem also has some trivial solutions - for example, a = b = 1 and
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Problem 6 in the 1988 International Mathematical Olympiad paper has almost reached a legendry status. The problem is considered extremely difficult to solve - most solutions require a high level of mathematical sophistication or are long and tedious.
Using school level maths, I obtain a general term that may be used to provide a complete list of solutions.
First we notice that the expression (a2+ b2)/(ab +1) is symmetric in a and b. The problem also has some trivial solutions - for example, a = b = 1 and
a = 0, b = 1 (b = 0 and a = 1 is also a solution).
We can also find a special case solution following the simple process described in the slide
Derivation of the general solution: In the following, I shall derive a general expression for a and b from which the rest of solutions may be obtained. This is discussed in the following slides:
We can also find a special case solution following the simple process described in the slide
Derivation of the general solution: In the following, I shall derive a general expression for a and b from which the rest of solutions may be obtained. This is discussed in the following slides:
As a physicist, I need to check if eq.10 may be simplified further for some special conditions and how that might effect the answer in eq.11. The obvious case to consider is when x and m are large - this is discussed int he next slide:
It was satisfying for me to be able to provide a general solution for all possible combinations of a and b. Besides the solutions (a = 0, 1 or 2) and b = a3 (eq. 3), we have found a new class of solutions such that a must be the cube of a real positive number and b is defined by eq.10 (solutions of the problem in the form asked in IMO 1988).
