MathDB

11

Part of 2007 AIME Problems

Problems(2)

Summation Pattern Problem

Source: AIME I 2007 #11

3/15/2007
For each positive integer pp, let b(p)b(p) denote the unique positive integer kk such that ∣k−p∣<12|k-\sqrt{p}|<\frac{1}{2}. For example, b(6)=2b(6) = 2 and b(23)=5b(23)=5. If S=∑p=12007b(p)S = \textstyle\sum_{p=1}^{2007}b(p), find the remainder when S is divided by 1000.
algebrapolynomialAMCaxusus
Rolling tube

Source: AIME II 2007 #11

3/29/2007
Two long cylindrical tubes of the same length but different diameters lie parallel to each other on a flat surface. The larger tube has radius 7272 and rolls along the surface toward the smaller tube, which has radius 2424. It rolls over the smaller tube and continues rolling along the flat surface until it comes to rest on the same point of its circumference as it started, having made one complete revolution. If the smaller tube never moves, and the rolling occurs with no slipping, the larger tube ends up a distance xx from where it starts. The distance xx can be expressed in the form aπ+bc,a\pi+b\sqrt{c}, where a,a, b,b, and cc are integers and cc is not divisible by the square of any prime. Find a+b+c.a+b+c.
geometrygeometric transformationrotationsymmetryAMCAIME