MathDB

10

Part of 2015 Harvard-MIT Mathematics Tournament

Problems(2)

2015 Algebra #10: Contrived System of Equations

Source:

3/28/2015
Find all ordered 4-tuples of integers (a,b,c,d)(a,b,c,d) (not necessarily distinct) satisfying the following system of equations: \begin{align*}a^2-b^2-c^2-d^2&=c-b-2\\2ab&=a-d-32\\2ac&=28-a-d\\2ad&=b+c+31.\end{align*}
algebrasystem of equations
2015 Geometry #10

Source:

12/23/2016
Let G\mathcal{G} be the set of all points (x,y)(x,y) in the Cartesian plane such that 0≤y≤80\le y\le 8 and (x−3)2+31=(y−4)2+8y(8−y).(x-3)^2+31=(y-4)^2+8\sqrt{y(8-y)}. There exists a unique line ℓ\ell of negative slope tangent to G\mathcal{G} and passing through the point (0,4)(0,4). Suppose ℓ\ell is tangent to G\mathcal{G} at a unique point PP. Find the coordinates (α,β)(\alpha, \beta) of PP.