4th AIME 1986

1. Find the sum of the solutions to x^1/4 = 12/(7 - x^1/4).

2. Find (√5 + √6 + √7)(√5 + √6 - √7)(√5 - √6 + √7)(-√5 + √6 + √7).

3. Find tan(x+y) where tan x + tan y = 25 and cot x + cot y = 30.

4. 2x₁ + x₂ + x₃ + x₄ + x₅ = 6
x₁ + 2x₂ + x₃ + x₄ + x₅ = 12
x₁ + x₂ + 2x₃ + x₄ + x₅ = 24
x₁ + x₂ + x₃ + 2x₄ + x₅ = 48
x₁ + x₂ + x₃ + x₄ + 2x₅ = 96

Find 3x₄ + 2x₅.

5. Find the largest integer n such that n + 10 divides n³ + 100.

6. For some n, we have (1 + 2 + ... + n) + k = 1986, where k is one of the numbers 1, 2, ... , n. Find k.

7. The sequence 1, 3, 4, 9, 10, 12, 13, 27, ... includes all numbers which are a sum of one or more distinct powers of 3. What is the 100th term?

8. Find the integral part of ∑ log₁₀k, where the sum is taken over all positive divisors of 1000000 except 1000000 itself.

9. A triangle has sides 425, 450, 510. Lines are drawn through an interior point parallel to the sides, the intersections of these lines with the interior of the triangle have the same length. What is it?

10. abc is a three digit number. If acb + bca + bac + cab + cba = 3194, find abc.

11. The polynomial 1 - x + x² - x³ + ... - x¹⁵ + x¹⁶ - x¹⁷ can be written as a polynomial in y = x + 1. Find the coefficient of y².

12. Let X be a subset of {1, 2, 3, ... , 15} such that no two subsets of X have the same sum. What is the largest possible sum for X?

13. A sequence has 15 terms, each H or T. There are 14 pairs of adjacent terms. 2 are HH, 3 are HT, 4 are TH, 5 are TT. How many sequences meet these criteria?

14. A rectangular box has 12 edges. A long diagonal intersects 6 of them. The shortest distance of the other 6 from the long diagonal are 2√5 (twice), 30/√13 (twice), 15/√10 (twice). Find the volume of the box.

15. The triangle ABC has medians AD, BE, CF. AD lies along the line y = x + 3, BE lies along the line y = 2x + 4, AB has length 60 and angle C = 90^o. Find the area of ABC.

Answers

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