14th AIME 1996

1. The square below is magic. It has a number in each cell. The sums of each row and column and of the two main diagonals are all equal. Find x.

2. For how many positive integers n < 1000 is [log₂n] positive and even?

3. Find the smallest positive integer n for which (xy - 3x - 7y - 21)ⁿ has at least 1996 terms.

4. A wooden unit cube rests on a horizontal surface. A point light source a distance x above an upper vertex casts a shadow of the cube on the surface. The area of the shadow (excluding the part under the cube) is 48. Find x.

5. The roots of x³ + 3x² + 4x - 11 = 0 are a, b, c. The equation with roots a+b, b+c, c+a is x³ + rx² + sx + t = 0. Find t.

6. In a tournament with 5 teams each team plays every other team once. Each game ends in a win for one of the two teams. Each team has ½ chance of winning each game. Find the probability that no team wins all its games or loses all its games.

7. 2 cells of a 7 x 7 board are painted black and the rest white. How many different boards can be produced (boards which can be rotated into each other do not count as different).

8. The harmonic mean of a, b > 0 is 2ab/(a + b). How many ordered pairs m, n of positive integer with m < n have harmonic mean 6²⁰?

9. There is a line of lockers numbered 1 to 1024, initially all closed. A man walks down the line, opens 1, then alternately skips and opens each closed locker (so he opens 1, 3, 5, ... , 1023). At the end of the line he walks back, opens the first closed locker, then alternately skips and opens each closed locker (so he opens 1024, skips 1022 and so on). He continues to walk up and down the line until all the lockers are open. Which locker is opened last?

10. Find the smallest positive integer n such that tan 19n^o = (cos 96^o + sin 96^o)/(cos 96^o - sin 96^o).

11. Let the product of the roots of z⁶ + z⁴ + z³ + z² + 1 = 0 with positive imaginary part be r(cos θ^o + i sin θ^o). Find θ.

12. Find the average value of |a₁ - a₂| + |a₃ - a₄| + |a₅ - a₆| + |a₇ - a₈| + |a₉ - a₁₀| for all permutations a₁, a₂, ... , a₁₀ of 1, 2, ... , 10.

13. AB = √30, BC = √15, CA = √6. M is the midpoint of BC. ∠ADB = 90^o. Find area ADB/area ABC.

14. A 150 x 324 x 375 block is made up of unit cubes. Find the number of cubes whose interior is cut by a long diagonal of the block.

15. ABCD is a parallelogram. ∠BAC = ∠CBD = 2 ∠DBA. Find ∠ACB/∠AOB, where O is the intersection of the diagonals.

Answers

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