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?
Let \(x,x',y,y'\) be integers such that \(x+\sqrt{d}y=x'+\sqrt{d}y'\), where \(d\) is a number that is not a square. Show that \(x=x'\) and \(y=y'\).
Show that if \(u_1\) and \(u_2\) are solutions to Pell’s equation, then \(u_1u_2\) is also a solution to Pell’s equation. What can you conclude about the number of solutions, if there are any?
Find all integer solutions to \(x^2+y^2-1=4xy\).
In a bag we have \(99\) red balls and \(99\) blue balls. We take balls from the bag, two balls at a time:
If the two balls are of the same colour, then we put in a red ball to the bag.
If the two balls are of different colour, we return a blue ball to the bag.
Regardless, after each step, one ball is lost from the bag, so eventually there will be only one ball. What is the colour of this last ball?
You have an \(8\times 8\) chessboard coloured in the usual way. You can pick any \(2\times 1\) or \(1\times 2\) piece and flip the white tiles to black tiles and vice-versa. Is it possible to finish with \(63\) white pieces and \(1\) black piece?
We start with the point \(S=(1,3)\) of the plane. We generate a sequence of points with coordinates \((x_n,y_n)\) with the following rule: \[x_0=1,y_0=3\qquad x_{n+1}=\frac{x_n+y_n}{2}\qquad y_{n+1}=\frac{2x_ny_n}{x_n+y_n}\] Is the point \((3,2)\) in the sequence?
Four black dots are drawn on a whiteboard. On the dots we write the numbers \(10\), \(20\), \(30\), and \(40\) (one number on each dot). We then repeat the following move any number of times: choose one dot, decrease its number by \(3\), and increase the number on each of the other three dots by \(1\). After some number of moves, is it possible for all four dots to show the number \(25\) simultaneously?
Every year the citizens of the planet “Lotsofteeth" enter a contest
to see who has the most teeth.
This year the judge notices:
Nobody has 0 teeth (everyone has at least 1).
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.