20th AIME2 2002

1. n is an integer between 100 and 999 inclusive, and so is n' the integer formed by reversing its digits. How many possible values are there for |n-n'|?

2. P (7,12,10), Q (8,8,1) and R (11,3,9) are three vertices of a cube. What is its surface area?

3. a, b, c are positive integers forming an increasing geometric sequence, b-a is a square, and log₆a + log₆b + log₆c = 6. Find a + b + c.

4. Hexagons with side 1 are used to form a large hexagon. The diagram illustrates the case n = 3 with three unit hexagons on each side of the large hexagon. Find the area enclosed by the unit hexagons in the case n = 202.

5. Find the sum of all positive integers n = 2^a3^b (a, b ≥ 0) such that n⁶ does not divide 6ⁿ.

6. Find the integer closest to 1000 ∑₃¹⁰⁰⁰⁰ 1/(n²-4).

7. Find the smallest n such that ∑₁ⁿ k² is a multiple of 200. You may assume ∑₁ⁿ k² = n(n+1)(2n+1)/6.

8. Find the smallest positive integer n for which there are no integer solutions to [2002/x] = n.

9. Let S = {1, 2, ... , 10}. Find the number of unordered pairs A, B, where A and B are disjoint non-empty subsets of S.

10. Find the two smallest positive values of x for which sin(x^o) = sin(x rad).

11. Two different geometric progressions both have sum 1 and the same second term. One has third term 1/8. Find its second term.

12. An unfair coin is tossed 10 times. The probability of heads on each toss is 0.4. Let a_n be the number of heads in the first n tosses. Find the probability that a_n/n ≤ 0.4 for n = 1, 2, ... , 9 and a₁₀/10 = 0.4.

13. ABC is a triangle, D lies on the side BC and E lies on the side AC. AE = 3, EC = 1, CD = 2, DB = 5, AB = 8. AD and BE meet at P. The line parallel to AC through P meets AB at Q, and the line parallel to BC through P meets AB at R. Find area PQR/area ABC.

14. Triangle APM has ∠A = 90^o and perimeter 152. A circle center O (on AP) has radius 19 and touches AM at A and PM at T. Find OP.

15. Two circles touch the x-axis and the line y = mx (m > 0). They meet at (9,6) and another point and the product of their radii is 68. Find m.

Answers

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