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Difference between revisions of "2017 USAJMO Problems/Problem 2"

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(b) Describe all pairs <math>(x,y)</math> of positive integers satisfying the equation.
 
(b) Describe all pairs <math>(x,y)</math> of positive integers satisfying the equation.
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==Solution 1==
  
 
==Solution 1==
 
==Solution 1==

Revision as of 21:07, 4 April 2021

Problem:

Consider the equation \[\left(3x^3 + xy^2 \right) \left(x^2y + 3y^3 \right) = (x-y)^7.\]

(a) Prove that there are infinitely many pairs $(x,y)$ of positive integers satisfying the equation.

(b) Describe all pairs $(x,y)$ of positive integers satisfying the equation.

Solution 1

Solution 1

We have $(3x^3+xy^2)(yx^2+3y^3)=(x-y)^7$, which can be expressed as $xy(3x^2+y^2)(x^2+3y^2)=(x-y)^7$. At this point, we

think of substitution. A substitution of form $a=x+y, b=x-y$ is slightly derailed by the leftover x and y terms, so

instead, seeing the xy in front, we substitute $x=a+b, y=a-b$. This leaves us with $(a^2-b^2)(4a^2+4ab+4b^2)(4a^2- 4ab+4b^2)=128b^7$, so $(a^2-b^2)(a^2+ab+b^2)(a^2-ab+b^2)=8b^7$. Expanding yields $a^6-b^6=8b^7$. Rearranging, we have

$b^6(8b+1)=a^6$. To satisfy this equation in integers, $8b+1$ must obviously be a $6th$ 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 $(a,b)=(0,0)$ does not work. We go to the next highest odd

$6th$ power, $3^6$ or $729$. In this case, $b=91$, so the LHS is $91^6\cdot3^6=273^6$, so $a=273$. Using the original

substitution yields $(x,y)=(364,182)$ as the first solution. We have shown part a by showing that there are an infinite

number of positive integer solutions for $(a,b)$, which can then be manipulated into solutions for $(x,y)$. To solve part

b, we look back at the original method of generating solutions. Define $a_n$ and $b_n$ to be the pair representing the nth

solution. Since the nth odd number is $2n+1$, $b_n=\frac{(2n+1)^6-1}{8}$. It follows that $a_n=(2n+1)b_n=\frac{(2n+1)^7- (2n+1)}{8}$. From our original substitution, $(x,y)=\left(\frac{(2n+1)^7+(2n+1)^6-2n-2}{8}, \frac{(2n+1)^7-(2n+1)^6-2n} {8}\right)$.

Solution 2 (and motivation)

First, we shall prove a lemma:

LEMMA:

\[\left(3x^3 + xy^2 \right) \left(x^2y + 3y^3 \right) = \frac{(x+y)^6-(x-y)^6}{4}\] PROOF: Expanding and simplifying the right side, we find that \[\frac{(x+y)^6-(x-y)^6}{4}=\frac{12x^5y+40x^3y^3+12xy^5}{4}\]\[=3x^5y+10x^3y^3+3xy^5\]\[=\left(3x^3 + xy^2 \right) \left(x^2y + 3y^3 \right)\] which proves our lemma.


Now, we have that \[\frac{(x+y)^6-(x-y)^6}{4}=(x-y)^7\] Rearranging and getting rid of the denominator, we have that \[(x+y)^6=4(x-y)^7+(x-y)^6\] Factoring, we have \[(x+y)^6=(x-y)^6(4(x-y)+1)\]Dividing both sides, we have \[\left(\frac{x+y}{x-y}\right)^6=4x-4y+1\] 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 $a=\frac{x+y}{x-y}$. By inspection, $a$ must be a positive odd integer satistisfying $a \geq 3$. We also have \[a^6=4x-4y+1\] Now, we can solve for $x$ and $y$ in terms of $a$: $x-y=\frac{a^6-1}{4}$ and $x+y=a(x-y)=\frac{a(a^6-1)}{4}$. Now we have: \[(x,y)=\left(\frac{(a+1)(a^6-1)}{8},\frac{(a-1)(a^6-1)}{8}\right)\] and it is trivial to check that this parameterization works for all such $a$ (to keep $x$ and $y$ integral), which implies part (a).


MOTIVATION FOR LEMMA: I expanded the LHS, noticed the coefficients were $(3,10,3)$, and immediately thought of binomial coefficients. Looking at Pascal's triangle, it was then easy to find and prove the lemma.

-sunfishho

Solution $(1.5, x)$ where $|x|>0$

So named because it is a mix of solutions 1 and 2 but differs in other aspects. After fruitless searching, let $x+y=a$, $x-y=b$. Clearly $a, b > 0$. Then $x=\frac{a+b}{2}, y=\frac{a-b}{2}, x^2+y^2=\frac{a^2+b^2}{2}, x^2=\frac{a^2+2ab+b^2}{4}, y^2=\frac{a^2-2ab+b^2}{4}, xy=\frac{a^2-b^2}{4}$.

Change the LHS to $xy(3x^2+y^2)(x^2+3y^2)=(a-b)(a+b)(a^2+ab+b^2)(a^2-ab+b^2)*\frac{1}{4}=(a^3-b^3)(a^3+b^3)*\frac{1}{4}=(a^6-b^6)*\frac{1}{4}$. Change the RHS to $b^7$. Therefore $(\frac{a}{b})^6=4b+1$. Let $n=\frac{a}{b}$, and note that $n$ is an integer. Therefore $a=n(\frac{n^6-1}{4}), b=\frac{n^6-1}{4}$. Because $n^6 \equiv 1 (\mod{4})$, $n$ is odd and $>1$ because $b>0$. Therefore, substituting for $x$ and $y$ we get: $x=\frac{(n^4+n^2+1)(n-1)(n+1)^2}{8}, y=\frac{(n+1)(n-1)^2(n^4+n^2+1)}{8}$.

Simplified Evan Chen's Solution

Let $x = da$, $y = db$, such that $gcd(a,b) = 1$. It's obvious $x > y$, so $a > b$

By plugging it in to the original equation, and simplifying, we get

\[d = \frac{ab(a^2 + 3b^2)(3a^2 + b^2)}{(a-b)^7}\]

Since d is an integer, we got:

\[(a-b)^7 \mid ab(a^2 + 3b^2)(3a^2 + b^2) \: \: \: \: \: (*)\]

1. Since $gcd(a,b) = 1$, a and b can't both be even

2. If both a and b are odd. $a-b$ will be even. The left hand side (*) has at least 7 factors of 2. Evaluating the (*) RHS term by term, we get: \[a \equiv 1 \pmod{2}\] \[b \equiv 1 \pmod{2}\] \[a^2 + 3b^2 \equiv 4 \pmod{8}\] \[3a^2 + b^2 \equiv 4 \pmod{8}\]

the (*) RHS has only 4 factors of 2. Contradiction.

3. Thus, $a-b$ has to be odd. We want to prove $a-b=1$

We know: \[a \equiv b \pmod{a-b}\] So \[ab(a^2 + 3b^2)(3a^2 + b^2) \equiv 16a^6 \pmod{a-b}\] We can assume if $a-b \neq 1$

Since $a-b$ is odd, $gcd(16, a-b) = 1$

Since $gcd(a, a-b) = 1$, $gcd(a^6, a-b) = 1$

\[ab(a^2 + 3b^2)(3a^2 + b^2) \equiv 16a^6 \neq 0 \pmod{a-b}\] contradiction. So, $a-b = 1$.

If $a-b=1$, obviously (*) will work, d will be an integer.

So (*) $\Leftrightarrow$ $a-b=1$. The proof is complete.

-Alexlikemath


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See also

2017 USAJMO (Problems • Resources)
Preceded by
Problem 1 		Followed by
Problem 3
1 • 2 • 3 • 4 • 5 • 6
All USAJMO Problems and Solutions
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