Some of the following are immediate; others may take more time — and require ingenuity and a deeper understanding. "Elementary" is not the same as "Simple." I hope you will even find it fun to think about some of these. You may find these interesting to discuss with friends.

- Prove that the product of two odd integers is also odd.
- If k > 0 is an integer, is k(k+1)(k+2) always divisible by 6?
- List these numbers from smallest to largest:
2
^{121}, 9^{55}, 7^{88}, u_{0}:= number of seconds since the birth of our universe. - If 3
^{1025}is divided by 5, what is the remainder? - If a, b > 0, show that (ab)
^{1/2}≤ (a + b)/2. - Is the square root of 7 a rational number?
- Can cos nx be written in the form
a
_{0}+ a_{1}cos x + a_{2}cos^{2}x + ...+ a_{n}cos^{n}x ? - Can the function sin x be written as a polynomial in x? How about
2
^{ x}? - Find a formula for
S
_{n}:= 1^{2}+ 2^{2}+... + n^{2}. - Does 0.99999... = 1.0000...?
- If c > 0 is a real number, prove there is an integer N so that Nc > 1.
- (a) Among all triangles inscribed in a fixed circle, show that
equilateral triangles have the largest area.

(b) Among all triangles in the plane with fixed area A_{0}, show that equilateral triangles have the smallest perimeter. [Equivalently, if you fix the perimeter, then equilateral triangles have the largest area.]

- Since Math 103 is a prerequisite, you should feel comfortable with all the material from it. Here are two old Final Exams: Fall 2007, Fall 2008.
- Describe the real numbers x that satisfy |x − 2| < 3.
- Sketch the points (x,y) in the plane where |x − y| > 1.
- a). How many real roots does
x
^{4}+ x^{2}− 2x + 2 = 0 have?

b). Find all points (x,y) in the plane that satisfy x^{2}- 2xy + 5y^{2}= 0. - Show that √(7 + 2√6) − √(7 − 2√6) = 2.
- Solve log
_{9}(5 − 3x) = -1/2 for x. - Let A = (-6,3), B = (2,7), and C be the vertices of a triangle.
Say the altitudes through the vertices A and B intersect at Q = (2,-1).
Find the coordinates of C.

[ The*altitude*through a vertex of a triangle is a straight line through the vertex that is perpendicular to the opposite side — or an extension of the opposite side. Although not needed here, the three altitudes always intersect in a single point, sometimes called the orthocenter of the triangle.] - Let y = f(x) describe a smooth curve in the plane (-∞ < x <∞) that does not pass through the origin. Say the point P = (a,b) on the curve is closest to the origin. Show that the straight line from the origin to P is perpendicular to the curve.
- Let f(x) be a continuous function that satisfies
∫
_{0}^{x}f(t) dt = c - cos(x^{2}). Find the function f(t) and the constant c.