Problems

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?

Today we will learn a really useful strategy for solving a certain kind of problems. This strategy is called the invariance principle, and after working through this sheet you’ll be able to recognise easily when we need to use an invariant to solve a problem. This strategy is applicable to kinds of problems where some task is repeatedly performed, and we wish to see if it is possible to transform our “initial state" into some given “final state". The key is to ask yourself:

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?

The topic of this problem sheet will be polynomials. Before we dive into the examples, let’s recap a few key concepts.

A polynomial in \(x\) is an expression formed by adding or subtracting monomials, which are terms of the form \(a x^n\), where \(a\) is a number called a coefficient, and \(n\) is a whole number (non-negative integer). Here, \(x\) is a variable that may represent a number. The degree of a polynomial \(f\), written as \(\deg(f)\) is the highest power of \(x\) appearing in the polynomial. For example: \(\deg(x^3+x^2+x)=3\).

We can perform several familiar operations on polynomials, which you may have seen before:

• Addition and subtraction: We add or subtract polynomials by looking at each power of \(x\) and adding or subtracting the corresponding coefficients. For example, if \[f(x) = x^4 + 3x - 1 \quad \text{and} \quad g(x) = x^3 + 2x + 5,\] then \(f(x) - g(x) = x^4 - x^3 + x - 6\).

• Multiplication: We use the distributive property, which means that every term in the first polynomial is multiplied by every term in the second polynomial. For example, if \[f(x) = x^2 + x + 1 \quad \text{and} \quad g(x) = x - 1,\] then \(f(x) g(x) = (x^2 + x + 1)(x - 1) = x^3 + x^2 + x - x^2 - x - 1 = x^3 - 1.\)

Let’s now present the examples. They have some very important techniques, so read them carefully before attempting the problems.

In this example we will discuss division with remainder. For polynomials \(f(x)\) and \(g(x)\) with \(\deg(f)\geq \deg(g)\) there always exists polynomials \(q(x)\) and \(r(x)\) such that \[f(x)=q(x)g(x)+r(x)\] and \(\deg(r)<\deg(g)\) or \(r(x)=0\). This should look very much like usual division of numbers, and just like in that case, we call \(f(x)\) the dividend, \(g(x)\) the divisor, \(q(x)\) the quotient, and \(r(x)\) the remainder. If \(r(x)=0\), we say that \(g(x)\) divides \(f(x)\), and we may write \(g(x)\mid f(x)\). Let \(f(x)=x^7-1\) and \(g(x)=x^3+x+1\). Is \(f(x)\) divisible by \(g(x)\)?

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?

Consider a graph with four vertices and where each vertex is connected to every other one (this is called the complete graph of four vertices, sometimes written as \(K_4\)). We write the numbers \(10,20,30,\) and \(40\) on the vertices. We play the following game: choose any vertex, and subtract three from that vertex, and add one to each of the three other vertices, so an example could be:

After playing this game for some number of steps, can we make the graph have the number \(25\) on each vertex?

Every year the citizens of the planet “Lotsofteeth" enter a contest in which they find the person with the most teeth. The judge notices that no one this year is toothless and that there are more people than the number of teeth in any single person. Is it true that there are two people with exactly the same number of teeth and why?