MathDB

14

Part of 2007 AIME Problems

Problems(2)

Recursive Sequence

Source: AIME I 2007 #14

3/15/2007
Let a sequence be defined as follows: a1=3a_{1}= 3, a2=3a_{2}= 3, and for n≥2n \ge 2, an+1an−1=an2+2007a_{n+1}a_{n-1}= a_{n}^{2}+2007. Find the largest integer less than or equal to a20072+a20062a2007a2006\frac{a_{2007}^{2}+a_{2006}^{2}}{a_{2007}a_{2006}}.
AMC
Polynomial

Source: AIME II 2007 #14

3/29/2007
Let f(x)f(x) be a polynomial with real coefficients such that f(0)=1,f(0) = 1, f(2)+f(3)=125,f(2)+f(3)=125, and for all xx, f(x)f(2x2)=f(2x3+x).f(x)f(2x^{2})=f(2x^{3}+x). Find f(5).f(5).
algebrapolynomialquadraticsfunctionprobabilityfunctional equationAMC