MathDB

10

Part of 2015 AIME Problems

Problems(2)

Absolute Value of Third-Degree Polynomial is 12

Source: 2015 AIME I Problem 10

3/20/2015
Let f(x)f(x) be a third-degree polynomial with real coefficients satisfying ∣f(1)∣=∣f(2)∣=∣f(3)∣=∣f(5)∣=∣f(6)∣=∣f(7)∣=12.|f(1)|=|f(2)|=|f(3)|=|f(5)|=|f(6)|=|f(7)|=12. Find ∣f(0)∣|f(0)|.
absolute valuealgebrapolynomialAMCAIMEAIME I
Recursion

Source: 2015 AIME 2 Problem 10

3/26/2015
Call a permutation a1,a2,…,ana_1,a_2,\ldots,a_n quasi-increasing if ak≤ak+1+2a_k\le a_{k+1}+2 for each 1≤k≤n−11\le k\le n-1. For example, 5432154321 and 1425314253 are quasi-increasing permutations of the integers 1,2,3,4,51,2,3,4,5, but 4512345123 is not. Find the number of quasi-increasing permutations of the integers 1,2,…,71,2,\ldots,7.
probability