MathDB

9

Part of 2003 AIME Problems

Problems(2)

Balanced integers

Source:

5/11/2006
An integer between 1000 and 9999, inclusive, is called balanced if the sum of its two leftmost digits equals the sum of its two rightmost digits. How many balanced integers are there?
AMCAIME
Two Polynomials

Source:

12/26/2006
Consider the polynomials P(x)=x6−x5−x3−x2−xP(x)=x^{6}-x^{5}-x^{3}-x^{2}-x and Q(x)=x4−x3−x2−1.Q(x)=x^{4}-x^{3}-x^{2}-1. Given that z1,z2,z3,z_{1},z_{2},z_{3}, and z4z_{4} are the roots of Q(x)=0,Q(x)=0, find P(z1)+P(z2)+P(z3)+P(z4).P(z_{1})+P(z_{2})+P(z_{3})+P(z_{4}).
algebrapolynomial