Sample Questions

1. (1992 Algebra) In a sequence of positive numbers, each term except the first two is the sum of all its predecessors. The eleventh term of the sequence is 1000 and the first term is 1. What is the second term?
2. (1996 Algebra) What is the largest six-digit number whose digits are all different such that the sum of the squares of the digits in the even-numbered positions equals the sum of the squares of the digits in the odd-numbered positions?
3. (1996 Geometry) A point is chosen from the interior of a right triangle. What is the probability that it is closer to a vertex of the triangle than it is to a midpoint of one of the triangle's sides?
4. (1992 Geometry) Let A, B, C, D, and E denote the vertices of a regular pentagon in the plane. If a line is drawn through each pair of these points, into how many regions is the plane divided?
5. (1996 Calculus) Several ants want to get from the origin to the point (1,1). The nth ant starts at the origin, moving a distance 1/n in the positive x direction and then 1/n in the positive y direction, and repeats this procedure until reaching (1,1). Denoting P(n) the length of the path traveled by the nth ant, calculate the limit of P(n) as n goes to infinity.
6. (1996 Pair) A palindrome is a positive integer such that when its digits are reversed, the result equals the original number. How many five-digit palindromes exist in base 7?
7. (1996 Team) Determine the remainder when (0! + 1! + ... + 64!)² is divided by 5.
8. (1990 Advanced Topics) How many sets of the form {p, p + 2, p + 4} exist with all three integers prime?
9. (1992 Advanced Topics) Suppose a₁, a₂, .., a₁₁ are distinct positive integers less than 21. What is the largest integer d such that we can say with certainty that two of the a_i differ by exactly d?
10. (1991 Calculus) Let f(x) = 2^x + 4^x + 8^x + 16^x + 32^x + 64^x + 128^x. Evaluate f'(0)/ln(16).