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

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We have <math>(3x^3+xy^2)(yx^2+3y^3)=(x-y)^7</math>, which can be expressed as <math>xy(3x^2+y^2)(x^2+3y^2)=(x-y)^7</math>. At this point, we think of substitution. A substitution of form <math>a=x+y, b=x-y</math> is slightly derailed by the leftover x and y terms, so instead, seeing the xy in front, we substitute <math>x=a+b, y=a-b</math>. This leaves us with <math>(a^2-b^2)(4a^2+4ab+4b^2)(4a^2-4ab+4b^2)=128b^7</math>, so <math>(a^2-b^2)(a^2+ab+b^2)(a^2-ab+b^2)=8b^7</math>. Expanding yields <math>a^6-b^6=8b^7</math>. Rearranging, we have <math>b^6(8b+1)=a^6</math>. To satisfy this equation in integers, <math>8b+1</math> must obviously be a <math>6th</math> 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 <math>(a,b)=(0,0)</math> does not work. We go to the next highest odd <math>6th</math> power, <math>3^6</math> or <math>729</math>. In this case, <math>b=91</math>, so the LHS is <math>91^6*3^6=273^6</math>, so <math>a=273</math>. Using the original substitution yields <math>(x,y)=(364,182)</math> as the first solution. We have shown part a by showing that there are an infinite number of positive integer solutions for <math>(a,b)</math>, which can then be manipulated into solutions for <math>(x,y)</math>. To solve part b, we look back at the original method of generating solutions. Define <math>a_n</math> and <math>b_n</math> to be the pair representing the nth solution. Since the nth odd number is <math>2n+1</math>, <math>b_n=\frac{(2n+1)^6-1}{8}</math>. It follows that <math>a_n=(2n+1)b_n=\frac{(2n+1)^7-(2n+1)}{8}</math>. From our original substitution, <math>(x,y)=(\frac{(2n+1)^7+(2n+1)^6-2n-2}{8}, \frac{(2n+1)^7-(2n+1)^6-2n}{8})</math>.
 
We have <math>(3x^3+xy^2)(yx^2+3y^3)=(x-y)^7</math>, which can be expressed as <math>xy(3x^2+y^2)(x^2+3y^2)=(x-y)^7</math>. At this point, we think of substitution. A substitution of form <math>a=x+y, b=x-y</math> is slightly derailed by the leftover x and y terms, so instead, seeing the xy in front, we substitute <math>x=a+b, y=a-b</math>. This leaves us with <math>(a^2-b^2)(4a^2+4ab+4b^2)(4a^2-4ab+4b^2)=128b^7</math>, so <math>(a^2-b^2)(a^2+ab+b^2)(a^2-ab+b^2)=8b^7</math>. Expanding yields <math>a^6-b^6=8b^7</math>. Rearranging, we have <math>b^6(8b+1)=a^6</math>. To satisfy this equation in integers, <math>8b+1</math> must obviously be a <math>6th</math> 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 <math>(a,b)=(0,0)</math> does not work. We go to the next highest odd <math>6th</math> power, <math>3^6</math> or <math>729</math>. In this case, <math>b=91</math>, so the LHS is <math>91^6*3^6=273^6</math>, so <math>a=273</math>. Using the original substitution yields <math>(x,y)=(364,182)</math> as the first solution. We have shown part a by showing that there are an infinite number of positive integer solutions for <math>(a,b)</math>, which can then be manipulated into solutions for <math>(x,y)</math>. To solve part b, we look back at the original method of generating solutions. Define <math>a_n</math> and <math>b_n</math> to be the pair representing the nth solution. Since the nth odd number is <math>2n+1</math>, <math>b_n=\frac{(2n+1)^6-1}{8}</math>. It follows that <math>a_n=(2n+1)b_n=\frac{(2n+1)^7-(2n+1)}{8}</math>. From our original substitution, <math>(x,y)=(\frac{(2n+1)^7+(2n+1)^6-2n-2}{8}, \frac{(2n+1)^7-(2n+1)^6-2n}{8})</math>.
 
{plshalp}
 
{plshalp}
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 +
==Solution 2 (and motivation)==
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First, we shall prove a lemma:
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 +
LEMMA:
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 +
<cmath>\left(3x^3 + xy^2 \right) \left(x^2y + 3y^3 \right) = \frac{(x+y)^6-(x-y)^6}{4}</cmath>
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PROOF: Expanding and simplifying the right side, we find that <cmath>\frac{(x+y)^6-(x-y)^6}{4}=\frac{12x^5y+40x^3y^3+12xy^5}{4}</cmath><cmath>=3x^5y+10x^3y^3+3xy^5</cmath><cmath>=\left(3x^3 + xy^2 \right) \left(x^2y + 3y^3 \right)</cmath>
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which proves our lemma.
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Now, we have that <cmath>\frac{(x+y)^6-(x-y)^6}{4}=(x-y)^7</cmath>
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Rearranging and getting rid of the denominator, we have that <cmath>(x+y)^6=4(x-y)^7+(x-y)^6</cmath>
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Factoring, we have <cmath>(x+y)^6=(x-y)^6(4(x-y)+1)</cmath>Dividing both sides, we have <cmath>(\frac{x+y}{x-y})^6=4x-4y+1</cmath>
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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 <math>a=\frac{x+y}{x-y}</math>. By inspection, <math>a</math> must be a positive odd integer satistisfying <math>a \geq 3</math>. We also have <cmath>a^6=4x-4y+1</cmath> Now, we can solve for <math>x</math> and <math>y</math> in terms of <math>a</math>:
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<math>x-y=\frac{a^6-1}{4}</math> and <math>x+y=a(x-y)=\frac{a(a^6-1)}{4}</math>.
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Now we have: <cmath>(x,y)=(\frac{(a+1)(a^6-1)}{8},\frac{(a-1)(a^6-1)}{8})</cmath>
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and it is trivial to check that this parameterization works for all such <math>a</math> (to keep <math>x</math> and <math>y</math> integral), which implies part (a).
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MOTIVATION FOR LEMMA:
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I expanded the LHS, noticed the coefficients were <math>(3,10,3)</math>, and immediately thought of binomial coefficients. Looking at Pascal's triangle, it was then easy to find and prove the lemma.
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-sunfishho
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{{MAA Notice}}
 
{{MAA Notice}}
  
 
==See also==
 
==See also==
 
{{USAJMO newbox|year=2017|num-b=1|num-a=3}}
 
{{USAJMO newbox|year=2017|num-b=1|num-a=3}}

Revision as of 00:41, 21 June 2017

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 (and motivation)

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 $91^6*3^6=273^6$, 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, $(x,y)=(\frac{(2n+1)^7+(2n+1)^6-2n-2}{8}, \frac{(2n+1)^7-(2n+1)^6-2n}{8})$. {plshalp}

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 \[(\frac{x+y}{x-y})^6=4x-4y+1\] 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: \[(x,y)=(\frac{(a+1)(a^6-1)}{8},\frac{(a-1)(a^6-1)}{8})\] 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

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. AMC logo.png

See also

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