10th AIME 1992

1. Find the sum of all positive rationals a/30 (in lowest terms) which are < 10.

2. How many positive integers > 9 have their digits strictly increasing from left to right?

3. At the start of a weekend a player has won the fraction 0.500 of the matches he has played. After playing another four matches, three of which he wins, he has won more than the fraction 0.503 of his matches. What is the largest number of matches he could have won before the weekend?

4. The binomial coefficients nCm can be arranged in rows (with the nth row nC0, nC1, ... nCn) to form Pascal's triangle. In which row are there three consecutive entries in the ratio 3 : 4 : 5?

5. Let S be the set of all rational numbers which can be written as 0.abcabcabcabc... (where the integers a, b, c are not necessarily distinct). If the members of S are all written in the form r/s in lowest terms, how many different numerators r are required?

6. How many pairs of consecutive integers in the sequence 1000, 1001, 1002, ... , 2000 can be added without a carry? (For example, 1004 and 1005, but not 1005 and 1006.)

7. ABCD is a tetrahedron. Area ABC = 120, area BCD = 80. BC = 10 and the faces ABC and BCD meet at an angle of 30^o. What is the volume of ABCD?

8. If A is the sequence a₁, a₂, a₃, ... , define ΔA to be the sequence a₂ - a₁, a₃ - a₂, a₄ - a₃, ... . If Δ(ΔA) has all terms 1 and a₁₉ = a₉₂ = 0, find a₁.

9. ABCD is a trapezoid with AB parallel to CD, AB = 92, BC = 50, CD = 19, DA = 70. P is a point on the side AB such that a circle center P touches AD and BC. Find AP.

10. A is the region of the complex plane {z : z/40 and 40/w have real and imaginary parts in (0, 1)}, where w is the complex conjugate of z (so if z = a + ib, then w = a - ib). (Unfortunately, there does not appear to be any way of writing z with a bar over it in HTML4). Find the area of A to the nearest integer.

11. L, L' are the lines through the origin that pass through the first quadrant (x, y > 0) and make angles π/70 and π/54 respectively with the x-axis. Given any line M, the line R(M) is obtained by reflecting M first in L and then in L'. Rⁿ(M) is obtained by applying R n times. If M is the line y = 19x/92, find the smallest n such that Rⁿ(M) = M.

12. The game of Chomp is played with a 5 x 7 board. Each player alternately takes a bite out of the board by removing a square any and any other squares above and/or to the left of it. How many possible subsets of the 5 x 7 board (including the original board and the empty set) can be obtained by a sequence of bites?

13. The triangle ABC has AB = 9 and BC/CA = 40/41. What is the largest possible area for ABC?

14. ABC is a triangle. The points A', B', C' are on sides BC, CA, AB and AA', BB', CC' meet at O. Also AO/A'O + BO/B'O + CO/C'O = 92. Find (AO/A'O)(BO/B'O)(CO/C'O).

15. How many integers n in {1, 2, 3, ... , 1992} are such that m! never ends in exactly n zeros?

Answers

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