

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 x^{3} + a x^{2} + bx + c are real, show that 12ab + 27c ≤ 6a^{3} + 10(a^{2}  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/(2^{2}x  1) + ... + 1/(n^{2}x  1) = 1/2 has a unique solution x_{n} > 1. Show that as n tends to infinity, x_{n} tends to 4.

