VMO 2002

Solve the following equation: √(4 - 3√(10 - 3x)) = x - 2.

Determine for which $n$ positive integer the equation: $a + b + c + d = n \sqrt {abcd}$ has positive integer solutions.

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?

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?

Find all positive integers n for which the equation a + b + c + d = n√(abcd) has a solution in positive integers.

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.