Putnam 2009

Let $f$ be a real-valued function on the plane such that for every square $ABCD$ in the plane, $f(A)+f(B)+f(C)+f(D)=0$. Does it follow that $f(P)=0$ for all points $P$ in the plane?

Functions $f,g,h$ are differentiable on some open interval around $0$ and satisfy the equations and initial conditions

$$\begin{gathered}f' = 2f^2gh+\frac{1}{gh}, f(0)=1, \\ g'=fg^2h+\frac{4}{fh}, g(0)=1, \\ h'=3fgh^2+\frac{1}{fg}, h(0)=1.\end{gathered}$$

Find an explicit formula for $f(x)$, valid in some open interval around $0$.

Let $d_n$ be the determinant of the $n \times n$ matrix whose entries, from left to right and then from top to bottom, are $\cos 1, \cos 2, \dots, \cos n^2$. (For example, $$d_3 = \left| \begin{matrix} \cos 1 & \cos 2 & \cos 3 \\ \cos 4 & \cos 5 & \cos 6 \\ \cos 7 & \cos 8 & \cos 9 \end{matrix} \right|.$$ The argument of $\cos$ is always in radians, not degrees.) Evaluate $\lim_{n\to\infty} d_n$.

Let $S$ be a set of rational numbers such that

(a) $0 \in S$;

(b) If $x \in S$ then $x+1\in S$ and $x-1\in S$; and

(c) If $x\in S$ and $x\not\in\{0,1\}$, then $\frac{1}{x(x-1)}\in S$.

Must $S$ contain all rational numbers?

Is there a finite abelian group $G$ such that the product of the orders of all its elements is $2^{2009}$?

Let $f:[0,1]^2 \to \mathbb{R}$ be a continuous function on the closed unit square such that $\frac{\partial f}{\partial x}$ and $\frac{\partial f}{\partial y}$ exist and are continuous on the interior $(0,1)^2$. Let $a = \int_0^1 f(0,y) dy$, $b = \int_0^1 f(1,y) dy$, $c = \int_0^1 f(x,0) dx$, $d = \int_0^1 f(x,1) dx$. Prove or disprove: There must be a point $(x_0,y_0)$ in $(0,1)^2$ such that $$\frac{\partial f}{\partial x} (x_0,y_0) = b - a \text{and} \frac{\partial f}{\partial y} (x_0,y_0) = d - c.$$

Show that every positive rational number can be written as a quotient of products of factorials of (not necessarily distinct) primes. For example, $$\frac{10}{9} = \frac{2!\cdot 5!}{3!\cdot 3! \cdot 3!}.$$

A game involves jumping to the right on the real number line. If $a$ and $b$ are real numbers and $b > a$, the cost of jumping from $a$ to $b$ is $b^3-ab^2$. For what real numbers $c$ can one travel from $0$ to $1$ in a finite number of jumps with total cost exactly $c$?

Call a subset $S$ of $\{1, 2, \dots, n\}$ mediocre if it has the following property: Whenever $a$ and $b$ are elements of $S$ whose average is an integer, that average is also an element of $S$. Let $A(n)$ be the number of mediocre subsets of $\{1,2,\dots,n\}$. [For instance, every subset of $\{1,2,3\}$ except $\{1,3\}$ is mediocre, so $A(3) =7$.] Find all positive integers $n$ such that $A(n+2) - 2A(n+1) + A(n) = 1$.

Say that a polynomial with real coefficients in two variables, $x,y$, is balanced if the average value of the polynomial on each circle centered at the origin is $0$. The balanced polynomials of degree at most $2009$ form a vector space $V$ over $\mathbb{R}$. Find the dimension of $V$.

Let $f: (1, \infty) \to \mathbb{R}$ be a differentiable function such that $$f'(x) = \frac{x^2 - f(x)^2}{x^2 (f(x)^2 + 1)} \text{for all $x>1$.}$$ Prove that $\lim_{x \to \infty} f(x) = \infty$.

Prove that for every positive integer $n$, there is a sequence of integers $a_0, a_1, \dots, a_{2009}$ with $a_0 = 0$ and $a_{2009} = n$ such that each term after $a_0$ is either an earlier term plus $2^k$ for some nonnegative integer $k$, or of the form $b\,\mathrm{mod} c$ for some earlier positive terms $b$ and $c$. [Here $b\,\mathrm{mod} c$ denotes the remainder when $b$ is divided by $c$, so $0 \leq (b\,\mathrm{mod} c) < c$.]