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

m (Undo revision 185645 by Ryanjwang (talk))
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==Solution 2==
 
==Solution 2==
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Suppose there exist positive integral <math>a</math> and <math>b</math> such that <math>4f(a) = f(b)</math>.
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Suppose there exist positive integers <math>a</math> and <math>b</math> such that <math>4f(a) = f(b)</math>.
  
 
Thus, <math>4a^2 + 4a = b^2 + b</math>, or <math>(2a+1)^2 = b^2 + b + 1</math>. Then in order for the original equation to be true, <math>b^{2} + b + 1</math> would have to be a perfect square. Completing the square of <math>b^{2} + b + 1</math> results in <math>(b+1/2)^{2} + 3/4</math>. Thus, <math>b^{2} + b + 1</math>  is not a perfect square, and thus there is no <math>b</math> that satisfies <math>4f(a) = f(b)</math>.
 
Thus, <math>4a^2 + 4a = b^2 + b</math>, or <math>(2a+1)^2 = b^2 + b + 1</math>. Then in order for the original equation to be true, <math>b^{2} + b + 1</math> would have to be a perfect square. Completing the square of <math>b^{2} + b + 1</math> results in <math>(b+1/2)^{2} + 3/4</math>. Thus, <math>b^{2} + b + 1</math>  is not a perfect square, and thus there is no <math>b</math> that satisfies <math>4f(a) = f(b)</math>.
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==Solution 3==
 +
The given equation is <math>4f(a) = f(b)</math> or, <math>4a^2 + 4a = b^2 + b</math> or, <math>4a(a+1) = b(b+1)</math>.
 +
 +
For positive integers <math>a</math> and <math>b</math>,  <math>4a(a+1)</math> is an even number because it is divisible by <math>2</math> whereas <math>b(b+1)</math> is an odd number (the product of two consecutive numbers is an odd number). Therefore, <math>a</math> and <math>b</math> both cannot be positive integers.
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-ad1b314
  
 
==Alternate Solutions?==
 
==Alternate Solutions?==

Revision as of 08:47, 28 September 2023

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.

Solution 2

Suppose there exist positive integers $a$ and $b$ such that $4f(a) = f(b)$.

Thus, $4a^2 + 4a = b^2 + b$, or $(2a+1)^2 = b^2 + b + 1$. Then in order for the original equation to be true, $b^{2} + b + 1$ would have to be a perfect square. Completing the square of $b^{2} + b + 1$ results in $(b+1/2)^{2} + 3/4$. Thus, $b^{2} + b + 1$ is not a perfect square, and thus there is no $b$ that satisfies $4f(a) = f(b)$.

Solution 3

The given equation is $4f(a) = f(b)$ or, $4a^2 + 4a = b^2 + b$ or, $4a(a+1) = b(b+1)$.

For positive integers $a$ and $b$, $4a(a+1)$ is an even number because it is divisible by $2$ whereas $b(b+1)$ is an odd number (the product of two consecutive numbers is an odd number). Therefore, $a$ and $b$ both cannot be positive integers.

-ad1b314

Alternate Solutions?

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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