VMO 2002 — Division A

Solve the equation
$$\sqrt{4-3\sqrt{10-3x}}=x-2.$$

Let $ABC$ be an isosceles triangle with $AB=AC$. A point $O$ varies on the line $BC$, subject to the condition that the circle with center $O$ and radius $OA$ has neither $AB$ nor $AC$ as a tangent. The lines $AB,AC$ meet the circle again at $M,N$, respectively. Determine the locus of the orthocenter of triangle $AMN$.

Let $m\lt 2001$ and $n\lt 2002$ be positive integers. Distinct real numbers are placed in a table with $2001$ rows and $2002$ columns. An entry is called bad if it is smaller than at least $m$ entries in its column and at least $n$ entries in its row. What is the largest possible number of bad entries?

If all the real roots of $x^3+ax^2+bx+c$ are real, prove that
$$12ab+27c\le6a^3+10\sqrt{(a^2-2b)^3}.$$

Determine all positive integers $n$ for which the equation
$$a+b+c+d=n\sqrt{abcd}$$
has a solution in positive integers $a,b,c,d$.

Let $n$ be a positive integer. Prove that the equation
$$\frac1{x-1}+\frac1{2^2x-1}+\cdots+\frac1{n^2x-1}=\frac12$$
has a unique solution $x_n\gt 1$. Prove that $x_n$ tends to $4$ as $n$ tends to infinity.