Problems

Suppose that \((x_1,y_1),(x_2,y_2)\) are solutions to Pell’s equation \(x^2-dy^2 = 1\). Show that \((x_1x_2+dy_1y_2,x_1y_2+x_2y_2)\) also satisfies the same equation.

Suppose that \(x+y\sqrt{d}>1\) gives a solution to Pell’s equation. Show that \(x\geq 2\) and \(y\geq 1\). Can the bounds be achieved?

Can we obtain the polynomial \(h(x)=x\) by adding, subtracting, or multiplying the polynomials \(f(x)=x^2+x\) and \(g(x)=x^2+2\)?

Every year the citizens of the planet “Lotsofteeth" enter a contest to see who has the most teeth.
This year the judge notices:

1. Nobody has 0 teeth (everyone has at least 1).

2. There are more people in the contest than the most teeth that any one person has. (For example, if the most teeth anyone has is 27, then there are more than 27 people participating in the contest.)

Must there be two people who have exactly the same number of teeth? Explain why.

A cube net is a 2D shape that can be folded into a cube. For example, in the following diagram we show a cube net and the steps that fold it into a cube:

Imagine that you want to cover an endless floor with this cube net, so there are no gaps or overlaps, how would you lay them out? This is called covering or tiling the plane.

Cut a square into three parts and use these pieces to form a triangle whose angles are all acute (i.e: less than \(90^\circ\))