1. True or False? Every integer $n>2$ occurs in some Pythagorean Triple. (Justify your answer.)

2. Give a geometric proof that $\sqrt{3}$ is irrational. (Hint: It might be easier to show that $1+\sqrt{3}$ is irrational, then deduce that $\sqrt{3}$ is also irrational.)

3. Use the Euclidean algorithm to find an integral solution to $270x+168y = 6$.

Solution sketches.

1. What is the height of a regular tetrahedron of side length 1?

2. Exercise 2.2.2 from the text.

3. Let $P$ be a polyhedron. Suppose $F_1, \ldots, F_k$ are the faces of $P$ that meet at vertex $V$, and that $A_1, \ldots, A_k$ are the angles of these faces at $V$. Define the defect at vertex $V$ to be ($360$-(sum of the angles $A_i$)). (For example, in a cube there are three squares meeting at any vertex, so the defect at any vertex is ($360-$($90+90+90$)) = $90$ degrees.) The total defect of $P$ is the sum of the defects at all of the vertices of $P$. Exercise: find the total defect of each of the Platonic solids.

Solution sketches.

1. It is not possible to construct an angle of $\pi/13$ radians with straightedge and compass. Show that it is nevertheless possible to trisect an angle of $\pi/13$ with straightedge and compass. (That is, if you are given an angle of $\pi/13$, then from it you can construct an angle of $\pi/39$.)

2. Show that if a polygon is constructible, then its area is a constructible number. (Hint: start with triangles.)

3. Show that if a regular polygon of circumradius 1 has constructible area, then it is possible to construct a copy of the polygon. (The circumradius is the radius of the circumscribing circle.)

Solution sketches.

1. The Euler characteristic of a sphere is $2$.

• What is the Euler characterstic of object that is a disjoint union of two spheres?
• Suppose you move two disjoint spheres closer together until they are tangent. What happens to the Euler characteristic at this moment? Is your answer unexpected? (If so, how do you explain it?)

2. Find the regular continued fraction expansion of $\sqrt{p}$ for the values $p=7,11,13$.

3. Find an integer solution to $x^2-py^2=1$ for each $p=7,11,13$.

Solution sketches.

1. Use Brahmagupta's method to find a solution to $x^2-Dy^2=1$, where $D=n^2+1$.

2. The quadratic mean of a sequence $a_1,\ldots,a_n$ is $$ \sqrt{\frac{a_1^2+\cdots+a_n^2}{n}}. $$ Find an integer $n>1$ such that the quadratic mean of the first $n$ positive integers is again an integer. That is, find $n>1$ such that $$ \sqrt{\frac{1^2+2^2+\cdots+(n-1)^2+n^2}{n}} $$ is a positive integer. (Hint: Reduce this problem to Pell's equation using the formula $1^2+2^2+\cdots+n^2=n(n+1)(2n+1)/6$.)

3. Find the volume formula for a tetrahedron by completing:
(i) Exercise 4.3.5 from the text.
(ii) Exercise 4.3.6 from the text.

Solution sketches.

1. Exercise 6.5.2 from the text.

2. Find all of the roots of the sextic $x^6 - 15x^2 - 4 = 0$ using an adapted form of the Cardano formula. (That is, let $y=x^2$, solve $y^3 - 15y - 4 = 0$ with the Cardano formula, then find $x$.) Which of your roots is equal to the root $x=2$?

3. Sketch the parabola $y=x^2-x$ and the hyperbola $xy=1$ together, and locate all (real) points of intersection of these curves.

Solution sketches.

2. Set up a quartic to find the points of intersection of the parabolas $y=x^2-2$ and $x=y^2-2$, then find the points of intersection.

3. Solve the system: $$\begin{array}{rl} x+y+z&=4\\ x^2+y^2+z^2&=4\\ x^3+y^3+z^3&=4\\ \end{array} $$

Solution sketches.

1. Which of the five groups contain which others? (Draw a diagram.)

2. For each of the five groups, state whether the following properties are preserved by the group: length, angle, area, parallelism of lines, the property that a set of points forms a circle, the property that a set of points forms a triangle, the property that a set of points forms a square.

3. Answer this question for each of the five groups: how many types of curves are there which are defined by equations of the form $Ax+By+C=0$, if curves are considered to be of the same type when a group element can transform one into the other?

Solution sketches.

1. Suppose that $f(x)$ is a polynomial of degree greater than $1$. Explain how to find the points at infinity on the curve $y=f(x)$.

2. Is there a (complex) polynomial $F(x,y)$ such that the projective completion of the curve defined by $F(x,y)=0$ has no points at infinity?

3. Everybody knows that $2$ points are sufficient to determine a line. How many points are sufficient to determine an irreducible conic in the projective plane over $\mathbb C$?

Solution sketches.

1. Exercise 8.7.1 from the text.

2. Exercise 8.7.2 from the text.

3. Find all points of intersection of the curves $y=x^2$ and $y=x^3$, and compute the intersection multiplicity at each point of intersection.

Solution sketches.