[Solutions] William Lowell Putnam Mathematical Competition 2010

1. Given a positive integer $n,$ what is the largest $k$ such that the numbers $1,2,\dots,n$ can be put into $k$ boxes so that the sum of the numbers in each box is the same? [When $n=8,$ the example $\{1,2,3,6\},\{4,8\},\{5,7\}$ shows that the largest $k$ is at least 3.]
2. Find all differentiable functions $f:\mathbb{R}\to\mathbb{R}$ such that \[f'(x)=\frac{f(x+n)-f(x)}n\] for all real numbers $x$ and all positive integers $n.$
3. Suppose that the function $h:\mathbb{R}^2\to\mathbb{R}$ has continuous partial derivatives and satisfies the equation \[h(x,y)=a\frac{\partial h}{\partial x}(x,y)+b\frac{\partial h}{\partial y}(x,y)\] for some constants $a,b.$ Prove that if there is a constant $M$ such that $|h(x,y)|\le M$ for all $(x,y)$ in $\mathbb{R}^2,$ then $h$ is identically zero.
4. Prove that for each positive integer $n,$ the number $10^{10^{10^n}}+10^{10^n}+10^n-1$ is not prime.
5. Let $G$ be a group, with operation $*$. Suppose that
   (i) $G$ is a subset of $\mathbb{R}^3$ (but $*$ need not be related to addition of vectors);
   (ii) For each $\mathbf{a},\mathbf{b}\in G,$ either $\mathbf{a}\times\mathbf{b}=\mathbf{a}*\mathbf{b}$ or $\mathbf{a}\times\mathbf{b}=\mathbf{0}$ (or both), where $\times$ is the usual cross product in $\mathbb{R}^3.$
   Prove that $\mathbf{a}\times\mathbf{b}=\mathbf{0}$ for all $\mathbf{a},\mathbf{b}\in G.$
6. Let $f:[0,\infty)\to\mathbb{R}$ be a strictly decreasing continuous function such that $$\lim_{x\to\infty}f(x)=0.$$ Prove that $$\int_0^{\infty}\frac{f(x)-f(x+1)}{f(x)}\,dx$$ diverges.
7. Is there an infinite sequence of real numbers $a_1,a_2,a_3,\dots$ such that \[a_1^m+a_2^m+a_3^m+\cdots=m\] for every positive integer $m?$
8. Given that $A,B,$ and $C$ are noncollinear points in the plane with integer coordinates such that the distances $AB,AC,$ and $BC$ are integers, what is the smallest possible value of $AB?$
9. There are 2010 boxes labeled $B_1,B_2,\dots,B_{2010},$ and $2010n$ balls have been distributed among them, for some positive integer $n.$ You may redistribute the balls by a sequence of moves, each of which consists of choosing an $i$ and moving exactly $i$ balls from box $B_i$ into any one other box. For which values of $n$ is it possible to reach the distribution with exactly $n$ balls in each box, regardless of the initial distribution of balls?
10. Find all pairs of polynomials $p(x)$ and $q(x)$ with real coefficients for which \[p(x)q(x+1)-p(x+1)q(x)=1.\]
11. Is there a strictly increasing function $f:\mathbb{R}\to\mathbb{R}$ such that $f'(x)=f(f(x))$ for all $x?$
12. Let $A$ be an $n\times n$ matrix of real numbers for some $n\ge 1.$ For each positive integer $k,$ let $A^{[k]}$ be the matrix obtained by raising each entry to the $k$th power. Show that if $A^k=A^{[k]}$ for $k=1,2,\cdots,n+1,$ then $A^k=A^{[k]}$ for all $k\ge 1.$

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