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Find the positive solution to \(\frac 1{x^2-10x-29}+\frac1{x^2-10x-45}-\frac 2{x^2-10x-69}=0\)

 

Let n be the smallest positive integer that is a multiple of 75 and has exactly 75 positive integral divisors, including 1and itself. Find n/75

 Sep 21, 2018

Best Answer 

 #1
avatar+988 
+3

1. 

 

Let \(a = x^2 - 10x - 29\)


\(\frac{1}{a} + \frac{1}{a - 16} - \frac{2}{a - 40} = 0.\\  (a - 16)(a - 40) + a(a - 40) - 2(a)(a - 16) = 0 \\ -64a + 40 \times 16 = 0, \Rightarrow a = 10. \\\)

 

\(10 = x^2 - 10x - 29 \Longleftrightarrow 0 = (x - 13)(x + 3)\) The positive root is \(\boxed{13}.\)

 

2. 

 

https://artofproblemsolving.com/wiki/index.php?title=1990_AIME_Problems/Problem_5

 

I hope this helped,

 

Gavin

.
 Sep 21, 2018
edited by GYanggg  Sep 21, 2018
 #1
avatar+988 
+3
Best Answer

1. 

 

Let \(a = x^2 - 10x - 29\)


\(\frac{1}{a} + \frac{1}{a - 16} - \frac{2}{a - 40} = 0.\\  (a - 16)(a - 40) + a(a - 40) - 2(a)(a - 16) = 0 \\ -64a + 40 \times 16 = 0, \Rightarrow a = 10. \\\)

 

\(10 = x^2 - 10x - 29 \Longleftrightarrow 0 = (x - 13)(x + 3)\) The positive root is \(\boxed{13}.\)

 

2. 

 

https://artofproblemsolving.com/wiki/index.php?title=1990_AIME_Problems/Problem_5

 

I hope this helped,

 

Gavin

GYanggg Sep 21, 2018
edited by GYanggg  Sep 21, 2018

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