MathDB

8

Part of 2007 AIME Problems

Problems(2)

Cubic Polynomial

Source: AIME I 2007 #8

3/15/2007
The polynomial P(x)P(x) is cubic. What is the largest value of kk for which the polynomials Q1(x)=x2+(k−29)x−kQ_{1}(x) = x^{2}+(k-29)x-k and Q2(x)=2x2+(2k−43)x+kQ_{2}(x) = 2x^{2}+(2k-43)x+k are both factors of P(x)P(x)?
algebrapolynomialquadraticsnumber theorygreatest common divisoralgorithmleast common multiple
Basic Rectangles

Source: AIME II 2007 #8

3/29/2007
A rectangular piece of of paper measures 4 units by 5 units. Several lines are drawn parallel to the edges of the paper. A rectangle determined by the intersections of some of these lines is called basic if (i) all four sides of the rectangle are segments of drawn line segments, and (ii) no segments of drawn lines lie inside the rectangle. Given that the total length of all lines drawn is exactly 2007 units, let NN be the maximum possible number of basic rectangles determined. Find the remainder when NN is divided by 1000.
geometryrectanglecalculusintegrationderivativequadraticsmodular arithmetic