Olympiad problems

2015 AMC 12/AHSME, 18

Tags: quadratics , function , algebra , polynomial , Vieta , SFFT , AMC

The zeroes of the function $f(x)=x^2-ax+2a$ are integers. What is the sum of all possible values of $a$? $\textbf{(A) }7\qquad\textbf{(B) }8\qquad\textbf{(C) }16\qquad\textbf{(D) }17\qquad\textbf{(E) }18$

1996 AIME Problems, 8

Tags: SFFT

The harmonic mean of two positive numbers is the reciprocal of the arithmetic mean of their reciprocals. For how many ordered pairs of positive integers $(x,y)$ with $x<y$ is the harmonic mean of $x$ and $y$ equal to $6^{20}.$

2014 AIME Problems, 14

Tags: quadratics , algebra , polynomial , symmetry , SFFT , linear algebra , Contrived

Let $m$ be the largest real solution to the equation \[\frac{3}{x-3}+\frac{5}{x-5}+\frac{17}{x-17}+\frac{19}{x-19}= x^2-11x-4.\] There are positive integers $a,b,c$ such that $m = a + \sqrt{b+\sqrt{c}}$. Find $a+b+c$.

2006 Stanford Mathematics Tournament, 6

Tags: SFFT

Let $a,b,c$ be real numbers satisfying: \[ab-a=b+119\] \[bc-b=c+59\] \[ca-c=a+71\] Determine all possible values of $a+b+c$.