+0

Help

0
256
1
+641

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

#1
+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
+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

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