USA AIME 1986 1 What is the sum of the solutions to the equation 4 ? x = 12 7? 4 ? x ? 2 Evaluate the product ( ? 5 + ? 6 + ? 7)(? ? 5 + ? 6 + ? 7)( ? 5? ? 6 + ? 7)( ? 5 + ? 6? ? 7). 3 If tanx + tan y = 25 and cotx + cot y = 30, what is tan(x + y)? 4 Determine 3x4 + 2x5 if x1, x2, x3, x4, and x5 satisfy the system of equations below. 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 5 What is that largest positive integer n for which n3 + 100 is divisible by n + 10? 6 The pages of a book are numbered 1 through n. When the page numbers of the book were added, one of the page numbers was mistakenly added twice, resulting in an incorrect sum of