Easy to use what exactly?

Algebra Level 5

For some silly integer m m , the polynomial x 3 2011 x + m x^3 - 2011x + m has the three integer roots a , b , a, b, and c c . Find a + b + c |a| + |b| + |c| .


The answer is 98.

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1 solution

Parth Lohomi
Feb 5, 2015

With Vieta's formula , we know that a + b + c = 0 a+b+c = 0 , and a b + b c + a c = 2011. ab+bc+ac = -2011.

a , b , c 0 a,b,c\neq 0 since any one being zero will make the the other 2 ± 2011 . a = ( b + c ) . 2 \pm \sqrt{2011}. a = -(b+c). WLOG, let a b c . |a| \ge |b| \ge |c|. Then if a > 0 a > 0 , then b , c < 0 b,c < 0 and if a < 0 , b , c > 0. a < 0, b,c > 0.

a b + b c + a c = 2011 = a ( b + c ) + b c = a 2 + b c ab+bc+ac = -2011 = a(b+c)+bc = -a^2+bc

a 2 = 2011 + b c a^2 = 2011 + bc We know that b , c b, c have the same sign. So a 45. |a| \ge 45. ( 4 4 2 < 2011 (44^2<2011 and 4 5 2 = 2025 ) 45^2 = 2025) Also, b c bc maximize when b = c b = c if we fixed * *a. Hence, 2011 = a 2 b c > 3 4 a 2 . 2011 = a^2 - bc > \dfrac{3}{4}a^2.

So a 2 < ( 4 ) 2011 3 = 2681 + 1 3 . a ^2 < \dfrac{(4)2011}{3} = 2681+\dfrac{1}{3}.

5 2 2 = 2704 52^2 = 2704 so a 51. |a| \le 51.

Now we have limited a to 45 a 51. 45\le |a| \le 51.

Let's us analyze a 2 = 2011 + b c . a^2 = 2011 + bc.

Here is a table:

a |a| \implies a 2 = 2011 + b c a^2 = 2011 + bc

45 45 \implies 14 14

46 46 \implies 14 + 91 = 105 14 + 91 =105

47 47 \implies 105 + 93 = 198 105 + 93 = 198

48 48 \implies 198 + 95 = 293 198 + 95 = 293

49 49 \implies 293 + 97 = 390 293 + 97 = 390

We can tell we don't need to bother with 45, 105 = ( 3 ) ( 5 ) ( 7 ) 105 = (3)(5)(7) , So 46 won't work. 198 / 47 > 4 , 198/47 > 4, 198 is not divisible by 5, 198/6 = 33, which is too small to get 47. 293/48 > 6, 293 is not divisible by 7 or 8 or 9, we can clearly tell that 10 is too much.

Hence, a = 49 , a 2 2011 = 390. b = 39 , c = 10. |a| = 49, a^2 -2011 = 390. b = 39, c = 10.

Answer: 98 \boxed{98}

My 10 0 t h 100^{th} solution

Parth Lohomi - 6 years, 4 months ago

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Congrats bro !!!

A Former Brilliant Member - 6 years, 4 months ago

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