19th AIME1 2001

1. Find the sum of all positive two-digit numbers that are divisible by both their digits.

2. Given a finite set A of reals let m(A) denote the mean of its elements. S is such that m(S∪{1}) = m(S) - 13 and m(S∪{2001}) = m(S) + 27. Find m(S).

3. Find the sum of the roots of the polynomial x²⁰⁰¹ + (½ - x)²⁰⁰¹.

4. The triangle ABC has ∠A = 60^o, ∠B = 45^o. The bisector of ∠A meets BC at T where AT = 24. Find area ABC.

5. An equilateral triangle is inscribed in the ellipse x² + 4y² = 4, with one vertex at (0,1) and the corresponding altitude along the y-axis. Find its side length.

6. A fair die is rolled four times. Find the probability that each number is no smaller than the preceding number.

7. A triangle has sides 20, 21, 22. The line through the incenter parallel to the shortest side meets the other two sides at X and Y. Find XY.

8. A number n is called a double if its base-7 digits form the base-10 number 2n. For example, 51 is 102 in base 7. What is the largest double?

9. ABC is a triangle with AB = 13, BC = 15, CA = 17. Points D, E, F on AB, BC, CA respectively are such that AD/AB = α, BE/BC = β, CF/CA = γ, where α + β + γ = 2/3, and α² + β² + γ² = 2/5. Find area DEF/area ABC.

10. S is the array of lattice points (x, y, z) with x = 0, 1 or 2, y = 0, 1, 2, or 3 and z = 0, 1, 2, 3 or 4. Two distinct points are chosen from S at random. Find the probability that their midpoint is in S.

11. 5N points form an array of 5 rows and N columns. The points are numbered left to right, top to bottom (so the first row is 1, 2, ... , N, the second row N+1, ... , 2N, and so on). Five points, P₁, P₂, ... , P₅ are chosen, P₁ in the first row, P₂ in the second row and so on. P_i has number x_i. The points are now renumbered top to bottom, left to right (so the first column is 1, 2, 3, 4, 5 the second column 6, 7, 8, 9, 10 and so on). P_i now has number y_i. We find that x₁ = y₂, x₂ = y₁, x₃ = y₄, x₄ = y₅, x₅ = y₃. Find the smallest possible value of N.

12. Find the inradius of the tetrahedron vertices (6,0,0), (0,4,0), (0,0,2) and (0,0,0).

13. The chord of an arc of ∠d (where d < 120^o) is 22. The chord of an arc of ∠2d is x+20, and the chord of an arc of ∠3d is x. Find x.

14. How many different 19-digit binary sequences do not contain the subsequences 11 or 000?

15. The labels 1, 2, ... , 8 are randomly placed on the faces of an octahedron (one per face). Find the probability that no two adjacent faces (sharing an edge) have adjacent numbers, where 1 and 8 are also considered adjacent.

Answers

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