MathDB

11

Part of 2016 AIME Problems

Problems(2)

Polynomial FE

Source: 2016 AIME I #11

3/4/2016
Let P(x)P(x) be a nonzero polynomial such that (x−1)P(x+1)=(x+2)P(x)(x-1)P(x+1)=(x+2)P(x) for every real xx, and (P(2))2=P(3)\left(P(2)\right)^2 = P(3). Then P(72)=mnP(\tfrac72)=\tfrac{m}{n}, where mm and nn are relatively prime positive integers. Find m+nm + n.
2016 AIME IAIMEpolynomial
A-nice Problem

Source: 2016 AIME II #11

3/17/2016
For positive integers NN and kk, define NN to be kk-nice if there exists a positive integer aa such that aka^k has exactly NN positive divisors. Find the number of positive integers less than 10001000 that are neither 77-nice nor 88-nice.
AMCAIMEAIME II