Difference between revisions of "1977 Canadian MO Problems/Problem 1"

(Added another solution that uses simpler algebra.)
m (unfortunately, the equation does not imply that either term is zero, just that their difference is zero)
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== Problem ==
 
== Problem ==
 
 
If <math>f(x)=x^2+x,</math> prove that the equation <math>4f(a)=f(b)</math> has no solutions in positive integers <math>a</math> and <math>b.</math>
 
If <math>f(x)=x^2+x,</math> prove that the equation <math>4f(a)=f(b)</math> has no solutions in positive integers <math>a</math> and <math>b.</math>
 
  
 
== Solution ==
 
== Solution ==
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In order for both <math>a</math> and <math>b</math> to be integers, the [[discriminant]] must be a [[perfect square]].  However, since <math>b^2< b^2+b+1 <(b+1)^2,</math> the quantity <math>b^2+b+1</math> cannot be a perfect square when <math>b</math> is an integer. Hence, when <math>b</math> is a positive integer, <math>a</math> cannot be.
 
In order for both <math>a</math> and <math>b</math> to be integers, the [[discriminant]] must be a [[perfect square]].  However, since <math>b^2< b^2+b+1 <(b+1)^2,</math> the quantity <math>b^2+b+1</math> cannot be a perfect square when <math>b</math> is an integer. Hence, when <math>b</math> is a positive integer, <math>a</math> cannot be.
 
== Alternate Solution ==
 
Write out the expanded form of <math>4f(a)=f(b)</math>:
 
 
<math>4a^2+4a=b^2+b</math>
 
 
Now simply factor it to get: <math>4a(a+1)=b(b+1)</math>
 
 
Subtract <math>b(b+1)</math> from both sides: <math>4a(a+1)-b(b+1)=0</math>
 
 
For the left side to equal <math>0</math>, <math>4a</math> or <math>a+1</math> must be <math>0</math> AND <math>b</math> or <math>b+1</math> must be <math>0</math>.
 
 
Set each one equal to <math>0</math> to find the possible solutions:
 
 
<math>4a=0</math>
 
 
<math>a+1=0</math>
 
 
<math>b=0</math>
 
 
<math>b+1=0</math>
 
 
Thus, <math>a</math> must be <math>0</math> or <math>-1</math>. The same applies to <math>b</math>. None of these solutions are greater than <math>0</math>.
 
 
<math>\therefore</math> There are no positive solutions for <math>a</math> or <math>b</math>, Q.E.D.
 
  
 
{{alternate solutions}}
 
{{alternate solutions}}
  
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== See also ==
 
{{Old CanadaMO box|before=First question|num-a=2|year=1977}}
 
{{Old CanadaMO box|before=First question|num-a=2|year=1977}}
  
[[Category:Olympiad Algebra Problems]]
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[[Category:Intermediate Algebra Problems]]

Revision as of 10:26, 7 September 2008

Problem

If $f(x)=x^2+x,$ prove that the equation $4f(a)=f(b)$ has no solutions in positive integers $a$ and $b.$

Solution

Directly plugging $a$ and $b$ into the function, $4a^2+4a=b^2+b.$ We now have a quadratic in $a.$

Applying the quadratic formula, $a=\frac{-1\pm \sqrt{b^2+b+1}}{2}.$

In order for both $a$ and $b$ to be integers, the discriminant must be a perfect square. However, since $b^2< b^2+b+1 <(b+1)^2,$ the quantity $b^2+b+1$ cannot be a perfect square when $b$ is an integer. Hence, when $b$ is a positive integer, $a$ cannot be.

Alternate solutions are always welcome. If you have a different, elegant solution to this problem, please add it to this page.

See also

1977 Canadian MO (Problems)
Preceded by
First question
1 2 3 4 5 6 7 8 Followed by
Problem 2
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