40th Vietnam 2002


A1. Solve the following equation: √(4 - 3√(10 - 3x)) = x - 2.
A2. ABC is an isosceles triangle with AB = AC. O is a variable point on the line BC such that the circle center O radius OA does not have the lines AB or AC as tangents. The lines AB, AC meet the circle again at M, N respectively. Find the locus of the orthocenter of the triangle AMN.
A3. m < 2001 and n < 2002 are fixed positive integers. A set of distinct real numbers are arranged in an array with 2001 rows and 2002 columns. A number in the array is bad if it is smaller than at least m numbers in the same column and at least n numbers in the same row. What is the smallest possible number of bad numbers in the array?
B1. If all the roots of the polynomial x³ + a x² + bx + c are real, show that 12ab + 27c ≤ 6a³ + 10(a² - 2b)^3/2. When does equality hold?
B2. Find all positive integers n for which the equation a + b + c + d = n√(abcd) has a solution in positive integers.
B3. n is a positive integer. Show that the equation 1/(x - 1) + 1/(2²x - 1) + ... + 1/(n²x - 1) = 1/2 has a unique solution x_n > 1. Show that as n tends to infinity, x_n tends to 4.

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