| 1. How many of 1, 2, 3, ... , 1000 can be written as the difference of the squares of two non-negative integers? |
| 2. The 9 horizontal and 9 vertical lines on an 8 x 8 chessboard form r rectangles including s squares. Find s/r in lowest terms. |
| 3. M is a 2-digit number ab, and N is a 3-digit number cde. We have 9·M·N = abcde. Find M, N. |
| 4. Circles radii 5, 5, 8, k are mutually externally tangent. Find k. |
| 5. The closest approximation to r = 0.abcd (where any of a, b, c, d may be zero) of the form 1/n or 2/n is 2/7. How many possible values are there for r? |
| 6. A1A2...An is a regular polygon. An equilateral triangle A1BA2 is constructed outside the polygon. What is the largest n for which BA1An can be consecutive vertices of a regular polygon? |
| 7. A car travels at 2/3 mile/min due east. A circular storm starts with its center 110 miles due north of the car and travels southeast at 1/√2 miles/min. The car enters the storm circle at time t1 mins and leaves it at t2. Find (t1 + t2)/2. |
| 8. How many 4 x 4 arrays of 1s and -1s are there with all rows and all columns having zero sum? |
| 9. The real number x has 2 < x2 < 3 and the fractional parts of 1/x and x2 are the same. Find x12 - 144/x. |
| 10. A card can be red, blue or green, have light, medium or dark shade, and show a circle, square or triangle. There are 27 cards, one for each possible combination. How many possible 3-card subsets are there such that for each of the three characteristics (color, shade, shape) the cards in the subset are all the same or all different? |
| 11. Find [100(cos 1o + cos 2o + ... + cos 44o)/(sin 1o + sin 2o + ... + sin 44o)]. |
| 12. a, b, c, d are non-zero reals and f(x) = (ax + b)/(cx + d). We have f(19) = 19, f(97) = 97 and f(f(x)) = x for all x (except -d/c). Find the unique y not in the range of f. |
| 13. Let S = {(x, y) : | ||x| - 2| - 1| + | ||y| - 2| - 1| = 1. If S is made out of wire, what is the total length of wire is required? |
| 14. v, w are roots of z1997 = 1 chosen at random. Find the probability that |v + w| >= √(2 + √3). |
| 15. Find the area of the largest equilateral triangle that can be inscribed in a rectangle with sides 10 and 11. |
To avoid possible copyright problems, I have changed the wording, but not the substance, of the problems.
AIME home
© John Scholes
jscholes@kalva.demon.co.uk
1 Aug 2003
Last updated/corrected 1 Aug 03