MathDB

13

Part of 2005 AIME Problems

Problems(2)

Particle with Interesting Movements

Source:

11/13/2005
A particle moves in the Cartesian Plane according to the following rules: 1. From any lattice point (a,b) (a,b), the particle may only move to (a \plus{} 1,b), (a,b \plus{} 1), or (a \plus{} 1,b \plus{} 1). 2. There are no right angle turns in the particle's path. How many different paths can the particle take from (0,0) (0,0) to (5,5) (5,5)?
analytic geometry
Polynomial with Integer Coefficients

Source:

11/13/2005
Let P(x)P(x) be a polynomial with integer coefficients that satisfies P(17)=10P(17)=10 and P(24)=17P(24)=17. Given that P(n)=n+3P(n)=n+3 has two distinct integer solutions n1n_1 and n2n_2, find the product n1⋅n2n_1\cdot n_2.
algebrapolynomialquadraticsVietafunctionAMCAIME