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?
So far we have discussed polynomials in one variable, i.e: with only an \(x\) as our variable. We can however, include as many as we want. For example, we can talk of a polynomial such as \[P(x,y)=x^2-y^2,\] where both \(x\) and \(y\) are variables. This is an example of an antisymmetric polynomial, which means that \(P(x,y)=-P(y,x)\) (i.e: switching \(x\) for \(y\) gives the original polynomial with a minus sign). Conversely, a polynomial \(Q(x,y)\) such that \(Q(x,y)=Q(y,x)\) is called symmetric. Show that every antisymmetric polynomial \(P(x,y)\) can be factored as \[P(x,y)=(x-y)Q(x,y),\] where \(Q(x,y)\) is a symmetric polynomial.
Solve the equation \[\left(x^2-3x+3\right)^2-3\left(x^2-3x+3\right)+3=x\]
Find all integer solutions to \(x^2+y^2-1=4xy\).
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.
Cut a square into three parts and use these pieces to form a triangle whose angles are all acute (i.e: less than \(90^\circ\))