+654 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
+981 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
+981 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
