13th AIME 1995

1. Starting with a unit square, a sequence of square is generated. Each square in the sequence has half the side-length of its predecessor and two of its sides bisected by its predecessor's sides as shown. Find the total area enclosed by the first five squares in the sequence.

2. Find the product of the positive roots of √1995 x^{log₁₉₉₅x} = x².

3. A object moves in a sequence of unit steps. Each step is N, S, E or W with equal probability. It starts at the origin. Find the probability that it reaches (2, 2) in less than 7 steps.

4. Three circles radius 3, 6, 9 touch as shown. Find the length of the chord of the large circle that touches the other two.

5. Find b if x⁴ + ax³ + bx² + cx + d has 4 non-real roots, two with sum 3 + 4i and the other two with product 13 + i.

6. How many positive divisors of n² are less than n but do not divide n, if n = 2³¹3¹⁹?

7. Find (1 - sin t)(1 - cos t) if (1 + sin t)(1 + cos t) = 5/4.

8. How many ordered pairs of positive integers x, y have y < x ≤ 100 and x/y and (x+1)/(y+1) integers?

9. ABC is isosceles as shown with the altitude AM = 11. AD = 10 and ∠BDC = 3 ∠BAC. Find the perimeter of ABC.

10. What is the largest positive integer that cannot be written as 42a + b, where a and b are positive integers and b is composite?

11. A rectangular block a x 1995 x c, with a ≤ 1995 ≤ c is cut into two non-empty parts by a plane parallel to one of the faces, so that one of the parts is similar to the original. How many possibilities are there for (a, c)?

12. OABCD is a pyramid, with ABCD a square, OA = OB = OC = OD, and ∠AOB = 45^o. Find cos θ, where θ is the angle between two adjacent triangular faces.

13. Find ∑₁¹⁹⁹⁵ 1/f(k), where f(k) is the closest integer to k^¼.

14. O is the center of the circle. AC = BD = 78, OA = 42, OX = 18. Find the area of the shaded area.

15. A fair coin is tossed repeatedly. Find the probability of obtaining five consecutive heads before two consecutive tails.

Answers

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