Let $S$ be a set with 2002 elements, and let $N$ be an integer with $0 \leq N \leq 2^{2002}$. Prove that it is possible to color every subset of $S$ either black or white so that the following conditions hold: (a) the union of any two white subsets is white; (b) the union of any two black subsets is black; (c) there are exactly $N$ white subsets.

Let $ABC$ be a triangle such that \[ \left( \cot \dfrac{A}{2} \right)^2 + \left( 2\cot \dfrac{B}{2} \right)^2 + \left( 3\cot \dfrac{C}{2} \right)^2 = \left( \dfrac{6s}{7r} \right)^2, \] where $s$ and $r$ denote its semiperimeter and its inradius, respectively. Prove that triangle $ABC$ is similar to a triangle $T$ whose side lengths are all positive integers with no common divisors and determine these integers.

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

Click for solution Let $ P$ be the given polynomial, and we're about to prove that there exist $ Q(x),R(x)\in\mathbb{R}[x]$, both having $ n$ real roots, and \[ R(x) = 2\cdot P(x) - Q(x)\] Let $ x_1< x_2 < \dots < x_n$ be arbitrary real numbers, and suppose that $ a_i = P(x_i)$, and let's take some real numbers $ b_i$, such that for all odd $ i$: \[ 2\cdot a_i < b_i, b_i > 0\] and for all even $ i$ we have: \[ 2\cdot a_i > b_i, b_i < 0\] According to the Interpolational Lagrange Formula, there does exist the polynomial, such that $ Q(x_i) = b_i$. As we know, $ Q(x_1), Q(x_3), \dots > 0$ and $ Q(x_2),Q(x_4),\dots < 0$, so $ Q$ has at least $ n - 1$ real roots, consequently it has precisely $ n$ real roots. On the other hand, $ R(x_1) = 2a_1 - b_1 < 0$, $ R(x_2) = 2a_2 - b_2 > 0$ and so on. Therefore, $ R(x)$ has exactly $ n$ real roots either. So the only problem left is to both of the polynomials to have leading coefficients equal $ 1$. Note that when using Interpolational Formula, we used only $ n$ points, $ x_1,x_2,\dots,x_n$ so we may choose another point $ x$, such that $ Q$ is monic. Hence, $ R$ is monic either, seeing as $ R(x) = 2P(x) - Q(x)$. P.S I managed to come up with this solution after thinkin' for 30 minutes, so it may contain mistakes.

Let $\mathbb{R}$ be the set of real numbers. Determine all functions $f: \mathbb{R} \to \mathbb{R}$ such that \[ f(x^2 - y^2) = x f(x) - y f(y) \] for all pairs of real numbers $x$ and $y$.

Let $a,b$ be integers greater than 2. Prove that there exists a positive integer $k$ and a finite sequence $n_1, n_2, \dots, n_k$ of positive integers such that $n_1 = a$, $n_k = b$, and $n_i n_{i+1}$ is divisible by $n_i + n_{i+1}$ for each $i$ ($1 \leq i < k$).

I have an $n \times n$ sheet of stamps, from which I've been asked to tear out blocks of three adjacent stamps in a single row or column. (I can only tear along the perforations separating adjacent stamps, and each block must come out of the sheet in one piece.) Let $b(n)$ be the smallest number of blocks I can tear out and make it impossible to tear out any more blocks. Prove that there are real constants $c$ and $d$ such that \[ \dfrac{1}{7} n^2 - cn \leq b(n) \leq \dfrac{1}{5} n^2 + dn \] for all $n > 0$.

These problems are copyright $\copyright$ Mathematical Association of America.