2017 USAJMO Problems/Problem 2
Consider the equation
(a) Prove that there are infinitely many pairs of positive integers satisfying the equation.
(b) Describe all pairs of positive integers satisfying the equation.
We have , which can be expressed as . At this point, we think of substitution. A substitution of form is slightly derailed by the leftover x and y terms, so instead, seeing the in front, we substitute . This leaves us with: Rearranging, we have . To satisfy this equation in integers, must obviously be a th power, and further inspection shows that it must also be odd. Also, since it is a square and all odd squares are 1 mod 8, every odd sixth power gives a solution. Since the problem asks for positive integers, the pair does not work. We go to the next highest odd th power, or . In this case, , so the LHS is , so . Using the original substitution yields as the first solution. We have shown part a by showing that there are an infinite number of positive integer solutions for , which can then be manipulated into solutions for .
To solve part (b), we look back at the original method of generating solutions. Define and to be the pair representing the th solution. Since the th odd number is , . It follows that . From our original substitution, .
Solution 2 (and motivation)
First, we shall prove a lemma:
PROOF: Expanding and simplifying the right side, we find that which proves our lemma.
Now, we have that Rearranging and getting rid of the denominator, we have that Factoring, we have Dividing both sides, we have Now, since the LHS is the 6th power of a rational number, and the RHS is an integer, the RHS must be a perfect 6th power. Define . By inspection, must be a positive odd integer satistisfying . We also have Now, we can solve for and in terms of : and . Now we have: and it is trivial to check that this parameterization works for all such (to keep and integral), which implies part (a).
MOTIVATION FOR LEMMA: I expanded the LHS, noticed the coefficients were , and immediately thought of binomial coefficients. Looking at Pascal's triangle, it was then easy to find and prove the lemma.
So named because it is a mix of solutions 1 and 2 but differs in other aspects. After fruitless searching, let , . Clearly . Then, Change the LHS of the original equation to and change the RHS to . Therefore . Let , and note that is an integer. Therefore . Because , is odd, and thus is greater than , since . Therefore, substituting for and , we get:
Simplified Evan Chen's Solution
Let , , such that . It's obvious , so
By plugging it in to the original equation, and simplifying, we get
Since d is an integer, we got:
1. Since , a and b can't both be even
2. If both a and b are odd. will be even. The left hand side (*) has at least 7 factors of 2. Evaluating the (*) RHS term by term, we get:
the (*) RHS has only 4 factors of 2. Contradiction.
3. Thus, has to be odd. We want to prove
We know: So We can assume if
Since is odd,
contradiction. So, .
If , obviously (*) will work, d will be an integer.
So (*) . The proof is complete.
Let , where is a positive integer. We will show that there is precisely one solution to the equation such that .
If , we have
The numerator is a multiple of , so is an integer multiple of . Thus, is also an integer, and we conclude that this pair satisfies the system of equations. Because this works for any positive integer , we conclude that there are infinitely many solutions to the equation.
We will now prove that these are the only possible solutions. Suppose the contrary, that there are solutions with a different ratio between and . As before, set , but this time in simplest terms. Then,
For this to be rational, must divide the expression in brackets and thus must divide . However, and are relatively prime, so must divide , and is therefore odd.
Next, set , where is a positive integer between and , inclusive. We have
The sum in the brackets must be divisible by , which is a power of greater than or equal to . Let be this sum. Then,
where represents and represents . Therefore, we must have
For , this is equal to , so we conclude that (and thus ). Then,
For this to be integral, the expression in brackets must be a multiple of . However, there are three terms that are odd in this expression, , , and . Thus, we have a contradiction, and we conclude that the only solutions are of the form stated above.
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