Olympiad problems

2002 AIME Problems, 4

Consider the sequence defined by $a_k=\frac 1{k^2+k}$ for $k\ge 1.$ Given that $a_m+a_{m+1}+\cdots+a_{n-1}=1/29,$ for positive integers $m$ and $n$ with $m<n$, find $m+n.$

2015 AMC 10, 23

Tags: quadratic , function , polynomial , vieta , sfft , algebra

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

2015 AMC 12/AHSME, 18

Tags: quadratic , function , polynomial , vieta , sfft , algebra

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

2010 Purple Comet Problems, 24

Tags: sfft , special factorizations

Find the number of ordered pairs of integers $(m, n)$ that satisfy $20m-10n = mn$.