[Solutions] William Lowell Putnam Mathematical Competition 2009

1. 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.
2. Functions $ f,g,h$ are differentiable on some open interval around $ 0$ and satisfy the equations and initial conditions $$\begin{align*}f'&=2f^2gh+\frac1{gh},\ f(0)=1,\\ g'&=fg^2h+\frac4{fh},\ g(0)=1,\\ h'&=3fgh^2+\frac1{fg},\ h(0)=1.\end{align*}$$ Find an explicit formula for $ f(x),$ valid in some open interval around $ 0.$
3. 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, $$ d_3 = \begin{vmatrix}\cos 1 & \cos2 & \cos3 \\ \cos4 & \cos5 & \cos 6 \\ \cos7 & \cos8 & \cos 9\end{vmatrix}.$$ The argument of $ \cos$ is always in radians, not degrees). Evaluate $ \lim_{n\to\infty}d_n.$
4. 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\notin\{0,1\},$ then $ \frac{1}{x(x-1)}\in S.$ Must $ S$ contain all rational numbers?
5. Is there a finite abelian group $ G$ such that the product of the orders of all its elements is $ 2^{2009}?$
6. 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 of $ (0,1)^2.$ Let $$ a=\int_0^1f(0,y)\,dy,\quad b=\int_0^1f(1,y)\,dy$$ $$c=\int_0^1f(x,0)\,dx,\quad d=\int_0^1f(x,1)\,dx.$$ Prove or disprove: There must be a point $ (x_0,y_0)$ in $ (0,1)^2$ such that $$ \frac{\partial f}{\partial x}(x_0,y_0)=b-a$$ and $$ \frac{\partial f}{\partial y}(x_0,y_0)=d-c.$$
7. 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!}.$
8. 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?$
9. 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 $$A(n+2)-2A(n+1)+A(n)=1.$$
10. Say that a polynomial with real coefficients in two variable, $ 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.$
11. Let $ f: (1,\infty)\to\mathbb{R}$ be a differentiable function such that \[ f'(x)=\frac{x^2-\left(f(x)\right)^2}{x^2\left(\left(f(x)\right)^2+1\right)}\quad\text{for all }x>1.\] Prove that $ \displaystyle\lim_{x\to\infty}f(x)=\infty.$
12. 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 nonnnegative integer $ k,$ or of the form $ b\mod{c}$ for some earlier positive terms $ b$ and $ c.$ [Here $ b\mod{c}$ denotes the remainder when $ b$ is divided by $ c,$ so $ 0\le(b\mod{c})<c.$]

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