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May 23rd, 2014, 09:42 PM   #1
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Math Competition Problem

Any clues on how to solve this? I'm stuck
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May 24th, 2014, 02:29 AM   #2
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Quote:
 Originally Posted by Orlando895 Any clues on how to solve this? I'm stuck
Here's how i did it.

$5^5 = 3125 \$ so for positive integer $\ a \$ , $\ 1 \ \le \ a \ \le \ 4$

We check a maximum of four values of $a$

$a = 4$

$$4^5 + b^2 + c^2 = 2012$$

$$b^2 + c^2 = 988$$

$$\sqrt{988} < 32$$

$$\sqrt{ \frac{988}{2}} > 22$$

So WLOG , we test $\ 23 \ \le \ b \ \le \ 31$

Nine values , none of them work.

$a = 3$

$$3^5 + b^2 + c^2 = 2012$$

$$b^2 + c^2 = 1769$$

$$\sqrt{1769} < 43$$

$$\sqrt{\frac{1769}{2}} > 29$$

So WLOG , we test $\ 30 \ \le \ b \ \le \ 42$

Thirteen values , $37^2 + 20^2 = 1769$ works AND $37 - 20 = 17$ is prime

We can stop , no need to test any other values of $a$ since the conditions are satisfied.

THEREFORE

$$a = 3$$

$$b = 37$$

$$c = 20$$

$$a + b + c = 60$$

Coice E.

Note: $3^5 + 40^2 + 13^2 = 2012$ but $40 - 13 = 27$ is NOT prime so choice D is disqualified.

Last edited by agentredlum; May 24th, 2014 at 02:52 AM. Reason: The first edit was incorrect. It could be corrected but leads to more complicated and more tedious calculations.

 May 24th, 2014, 06:34 AM #3 Member   Joined: Nov 2013 Posts: 47 Thanks: 4 Thanks! My approach was similar and I was able to deduce that a had to be less than 5, I was just trying to come up with a more direct approach rather than resorting to a sort of trial and error method. But hey, at least it gets us the answer. Thanks again! Thanks from agentredlum
 May 24th, 2014, 07:56 AM #4 Math Team   Joined: Apr 2010 Posts: 2,778 Thanks: 361 For a = 4 we can also work as follows: If a = 4 then 4^5 + b^2 + c^2 = 2012. b^2 + c^2 = 988 if $\displaystyle \text{sign}(c) \ne \text{sign}(b)$ then b^2 + c^2 is odd. Since 988 is even, that's not an option. If the signs are equal, then b - c is even, so, since it's prime, it's two. Now, we just need to check that there are no solutions to b^2 + c^2 = (c+2)^2 + c^2 = 988. Thanks from agentredlum

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