MathDB

23

Part of 2015 AMC 10

Problems(2)

Roots of Quadratic are Integers

Source: 2015 AMC 12A #18/10A #23

2/4/2015
The zeroes of the function f(x)=x2−ax+2af(x)=x^2-ax+2a are integers. What is the sum of all possible values of aa?
<spanclass=′latex−bold′>(A)</span>7<spanclass=′latex−bold′>(B)</span>8<spanclass=′latex−bold′>(C)</span>16<spanclass=′latex−bold′>(D)</span>17<spanclass=′latex−bold′>(E)</span>18<span class='latex-bold'>(A) </span>7\qquad<span class='latex-bold'>(B) </span>8\qquad<span class='latex-bold'>(C) </span>16\qquad<span class='latex-bold'>(D) </span>17\qquad<span class='latex-bold'>(E) </span>18
quadraticsfunctionalgebrapolynomialVietaSFFTAMC
Factorials ending in zero

Source: 2015 AMC 10 B Problem 23

2/26/2015
Let nn be a positive integer greater than 4 such that the decimal representation of n!n! ends in kk zeros and the decimal representation of (2n)!(2n)! ends in 3k3k zeros. Let ss denote the sum of the four least possible values of nn. What is the sum of the digits of ss?
<spanclass=′latex−bold′>(A)</span>7<spanclass=′latex−bold′>(B)</span>8<spanclass=′latex−bold′>(C)</span>9<spanclass=′latex−bold′>(D)</span>10<spanclass=′latex−bold′>(E)</span>11 <span class='latex-bold'>(A) </span>7\qquad<span class='latex-bold'>(B) </span>8\qquad<span class='latex-bold'>(C) </span>9\qquad<span class='latex-bold'>(D) </span>10\qquad<span class='latex-bold'>(E) </span>11
factorialmodular arithmeticfloor functionLaTeXAMC