MathDB

1

Part of 2015 Harvard-MIT Mathematics Tournament

Problems(2)

2015 Algebra #1: Unique Polynomial

Source:

3/28/2015
Let QQ be a polynomial Q(x)=a0+a1x+⋯+anxn,Q(x)=a_0+a_1x+\cdots+a_nx^n, where a0,…,ana_0,\ldots,a_n are nonnegative integers. Given that Q(1)=4Q(1)=4 and Q(5)=152Q(5)=152, find Q(6)Q(6).
algebrapolynomial
2015 Geometry #1

Source:

12/23/2016
Let RR be the rectangle in the Cartesian plane with vertices at (0,0)(0,0), (2,0)(2,0), (2,1)(2,1), and (0,1)(0,1). RR can be divided into two unit squares, as shown. [asy]size(120); defaultpen(linewidth(0.7)); draw(origin--(2,0)--(2,1)--(0,1)--cycle^^(1,0)--(1,1));[/asy] Pro selects a point PP at random in the interior of RR. Find the probability that the line through PP with slope 12\frac{1}{2} will pass through both unit squares.