USAMO 2002 — Days 1 and 2

Let $S$ be a set with $2002$ elements, and let $N$ be an integer with $0\le N\le 2^{2002}$. Prove that every subset of $S$ can be colored black or white so that:
(a) the union of any two white subsets is white;
(b) the union of any two black subsets is black;
(c) exactly $N$ subsets are white.

Problem proposer credited in the source: Gabriel Carroll.

Let $ABC$ be a triangle such that
$$\left(\cot\frac A2\right)^2+\left(2\cot\frac B2\right)^2+\left(3\cot\frac C2\right)^2=\left(\frac{6s}{7r}\right)^2,$$
where $s$ is its semiperimeter and $r$ its inradius. Prove that $ABC$ is similar to a triangle whose side lengths are positive integers with greatest common divisor $1$, and determine those integers.

Prove that every monic polynomial (a polynomial with leading coefficient $1$) of degree $n$ with real coefficients is the average of two monic polynomials of degree $n$, each having $n$ real roots.

Determine all functions $f:\mathbb R\to\mathbb R$ such that
$$f(x^2-y^2)=xf(x)-yf(y)$$
for every pair of real numbers $x,y$.

Let $a,b$ be integers greater than $2$. Prove that there are a positive integer $k$ and a finite sequence of positive integers $n_1,\ldots,n_k$ such that $n_1=a$, $n_k=b$, and $n_in_{i+1}$ is divisible by $n_i+n_{i+1}$ for every $1\le i\lt k$.

Problem proposer credited in the source: Gabriel Carroll.

An $n\times n$ sheet of stamps permits removal of blocks of three adjacent stamps in one row or one column. Tearing is allowed only along perforations separating adjacent stamps, and each block must come out in one piece. Let $b(n)$ be the smallest number of blocks that can be removed so that no further block can be removed. Prove that there are real constants $c,d$ such that
$$\frac17n^2-cn\le b(n)\le\frac15n^2+dn$$
for every positive integer $n$.