9th AIME 1991

1. m, n are positive integers such that mn + m + n = 71, m²n + mn² = 880, find m² + n².

2. The rectangle ABCD has AB = 4, BC = 3. The side AB is divided into 168 equal parts by points P₁, P₂, ... , P₁₆₇ (in that order with P₁ next to A), and the side BC is divided into 168 equal parts by points Q₁₆₇, Q₁₆₆, ... , Q₁ (in that order with Q₁ next to C). The parallel segments P₁Q₁, P₂Q₂, ... , P₁₆₇Q₁₆₇ are drawn. Similarly, 167 segments are drawn between AD and DC, and finally the diagonal AC is drawn. Find the sum of the lengths of the 335 parallel segments.

3. Expand (1 + 0.2)¹⁰⁰⁰ by the binomial theorem to get a⁰ + a¹ + ... + a¹⁰⁰⁰, where aⁱ = 1000Ci (0.2)ⁱ. Which is the largest term?

4. How many real roots are there to (1/5) log₂x = sin(5πx) ?

5. How many fractions m/n, written in lowest terms, satisfy 0 < m/n < 1 and mn = 20! ?

6. The real number x satisfies [x + 0.19] + [x + 0.20] + [x + 0.21] + ... + [x + 0.91] = 546. Find [100x].

7. Consider the equation x = √19 + 91/(√19 + 91/(√19 + 91/(√19 + 91/(√19 + 91/x)))). Let k be the sum of the absolute values of the roots. Find k².

8. For how many reals b does x² + bx + 6b have only integer roots?

9. If sec x + tan x = 22/7, find cosec x + cot x.

10. The letter string AAABBB is sent electronically. Each letter has 1/3 chance (independently) of being received as the other letter. Find the probability that using the ordinary text order the first three letters come rank strictly before the second three. (For example, ABA ranks before BAA, but after AAB.)

11. 12 equal disks are arranged without overlapping, so that each disk covers part of a circle radius 1 and between them they cover every point of the circle. Each disk touches two others. (Note that the disks are not required to cover every point inside the circle.) Find the total area of the disks.

12. ABCD is a rectangle. P, Q, R, S lie on the sides AB, BC, CD, DA respectively so that PQ = QR = RS = SP. PB = 15, BQ = 20, PR = 30, QS = 40. Find the perimeter of ABCD.

13. m red socks and n blue socks are in a drawer, where m + n ≤ 1991. If two socks are taken out at random, the chance that they have the same color is 1/2. What is the largest possible value of m?

14. A hexagon is inscribed in a circle. Five sides have length 81 and the other side has length 31. Find the sum of the three diagonals from a vertex on the short side.

15. Let S_n be the minimum value of ∑ √((2k-1)² + a_k²) for positive reals a₁, a₂, ... , a_n with sum 17. Find the values of n for which S_n is integral.

Answers

To avoid possible copyright problems, I have changed the wording, but not the substance, of the problems.