1. A discrete graph is complete if there is an edge connecting any pair of vertices. How many edges does a complete graph with 10 vertices have?
2. Let R be a ring with a multiplicative identity. If U is an additive subgroup of R such that ur in U for all u in U and for all r in R, then U is said to be a right ideal of R. If R has exactly two right ideals, which of the following must be true?
I. R is commutative.
II. R is a division ring (that is, all elements except the additive identity have multiplicative inverses).
III. R is infinite.
3. For which value of n are there exactly two abelian groups of order n up to isomorphism?
4. Consider a segment of length 10. Points A and B are chosen randomly such that A and B divide the segment into three smaller segments. What is the probability that the three smaller segments could form the sides of a triangle?
5. Let k be the number of real solutions of the equation e^x + x - 2 = 0 in the interval [0, 1], and let n be the number of real solutions that are not in [0, 1]. Which of the following is true?
