[Solutions] William Lowell Putnam Mathematical Competition 1995

1. Let $S$ be a set of real numbers which is closed under multiplication (that is $a,b\in S\implies ab\in S$). Let $T,U\subset S$ such that $T\cap U=\emptyset, T\cup U=S$. Given that for any three elements $a,b,c$ in $T$, not necessarily distinct, we have $abc\in T$ and also if $a,b,c\in U$, not necessarily distinct then $abc\in U$. Show at least one of $T$ and $U$ is closed under multiplication.
2. For what pairs of positive real numbers $(a,b)$ does the improper integral converge? $$\begin{align}\int_{b}^{\infty}\left(\sqrt{\sqrt{x+a}-\sqrt{x}}-\sqrt{\sqrt{x}-\sqrt{x-b}}\right)\,\mathrm{d}x \end{align}.$$
3. The number $d_1d_2\cdots d_9$ has nine (not necessarily distinct) decimal digits. The number $e_1e_2\cdots e_9$ is such that each of the nine $9$-digit numbers formed by replacing just one of the digits $d_i$ in $d_1d_2\cdots d_9$ by the corresponding digit $e_i \;\;(1 \le i \le 9)$ is divisible by $7$. The number $f_1f_2\cdots f_9$ is related to $e_1e_2\cdots e_9$ is the same way: that is, each of the nine numbers formed by replacing one of the $e_i$ by the corresponding $f_i$ is divisible by $7$. Show that, for each $i$, $d_i-f_i$ is divisible by $7$. [For example, if $d_1d_2\cdots d_9 = 199501996$, then $e_6$ may be $2$ or $9$, since $199502996$ and $199509996$ are multiples of $7$.]
4. Suppose we have a necklace of $n$ beads. Each bead is labelled with an integer and the sum of all these labels is $n-1$. Prove that we can cut the necklace to form a string whose consecutive labels $x_1, x_2,\cdots , x_n$ satisfy \[ \sum_{i=1}^{k}x_i\le k-1\quad \forall \;\;1\le k\le n \]
5. Let $x_1,x_2,\cdots, x_n$ be real valued differentiable functions of a variable $t$ which satisfy $$\begin{align*} & \frac{\mathrm{d}x_1}{\mathrm{d}t}=a_{11}x_1+a_{12}x_2+\cdots+a_{1n}x_n\\ & \frac{\mathrm{d}x_2}{\mathrm{d}t}=a_{21}x_1+a_{22}x_2+\cdots+a_{2n}x_n\\ & \;\qquad \vdots \\ & \frac{\mathrm{d}x_n}{\mathrm{d}t}=a_{n1}x_1+a_{n2}x_2+\cdots+a_{nn}x_n\\ \end{align*}.$$ For some constants $a_{ij}>0$. Suppose that $\lim_{t \to \infty}x_i(t)=0$ for all $1\le i \le n$. Are the functions $x_i$ necessarily linearly dependent?
6. Suppose that each of $n$ people writes down the numbers $1, 2, 3$ in random order in one column of a $3\times n$ matrix, with all orders equally likely and with the orders for different columns independent of each other. Let the row sums $a, b, c$ of the resulting matrix be rearranged (if necessary) so that $a \le b \le c$. Show that for some $n \ge 1995$ ,it is at least four times as likely that both $b = a+1$ and $c = a+2$ as that $a = b = c$.
7. For a partition $\pi$ of $\{1, 2, 3, 4, 5, 6, 7, 8, 9\}$, let $\pi(x)$ be the number of elements in the part containing $x$. Prove that for any two partitions $\pi$ and $\pi^{\prime}$, there are two distinct numbers $x$ and $y$ in $\{1, 2, 3, 4, 5, 6, 7, 8, 9\}$ such that $\pi(x) = \pi(y)$ and $\pi^{\prime}(x) = \pi^{\prime}(y)$.
8. An ellipse, whose semi-axes have length $a$ and $b$, rolls without slipping on the curve $y=c\sin{\left(\frac{x}{a}\right)}$. How are $a,b,c$ related, given that the ellipse completes one revolution when it traverses one period of the curve?
9. To each number with $n^2$ digits, we associate the $n\times n$ determinant of the matrix obtained by writing the digits of the number in order along the rows. For example : $8617\mapsto \det \left(\begin{matrix}{\;8}& 6\;\\ \;1 &{ 7\;}\end{matrix}\right)=50$. Find, as a function of $n$, the sum of all the determinants associated with $n^2$-digit integers. (Leading digits are assumed to be nonzero; for example, for $n = 2$, there are $9000$ determinants.)
10. Evaluate : \[ \sqrt[8]{2207-\frac{1}{2207-\frac{1}{2207-\cdots}}} \] Express your expression in the form $\frac{a+b\sqrt{c}}{d}$ where $a,b,c,d\in \Bbb{Z}$.
11. A game starts with four heaps of beans, containing 3, 4, 5 and 6 beans. The two players move alternately. A move consists of taking: either one bean from a heap, provided at least two beans are left behind in that heap, or a complete heap of two or three beans.
    The player who takes the last heap wins. To win the game, do you want to move first or second? Give a winning strategy.
12. For any $a>0$,set $\mathcal{S}(a)=\{\lfloor{na}\rfloor|n\in \mathbb{N}\}$. Show that there are no three positive reals $a,b,c$ such that \[ \mathcal{S}(a)\cap \mathcal{S}(b)=\mathcal{S}(b)\cap \mathcal{S}(c)=\mathcal{S}(c)\cap \mathcal{S}(a)=\emptyset \] \[ \mathcal{S}(a)\cup \mathcal{S}(b)\cup \mathcal{S}(c)=\mathbb{N} \]

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