[Solutions] William Lowell Putnam Mathematical Competition 1996

1. Find the least number $A$ such that for any two squares of combined area $1$, a rectangle of area $A$ exists such that the two squares can be packed in the rectangle (without the interiors of the squares overlapping) . You may assume the sides of the squares will be parallel to the sides of the rectangle.
2. Let $\mathcal{C}_1$ and $\mathcal{C}_2$ be circles whose centers are $10$ units apart, and whose radii are $1$ and $3$. Find, with proof, the locus of all points $M$ for which there exists points $X\in \mathcal{C}_1,Y\in \mathcal{C}_2$ such that $M$ is the midpoint of $XY$.
3. Suppose that each of $20$ students has made a choice of anywhere from $0$ to $6$ courses from a total of $6$ courses offered. Prove or disprove : there are $5$ students and $2$ courses such that all $5$ have chosen both courses or all $5$ have chosen neither course.
4. $S$ be a set of ordered triples $(a,b,c)$ of distinct elements of a finite set $A$. Suppose that
   
   • $(a,b,c)\in S\iff (b,c,a)\in S$
   • $(a,b,c)\in S\iff (c,b,a)\not\in S$
   • $(a,b,c),(c,d,a)\in S\iff (b,c,d),(d,a,b)\in S$.
   Prove there exists $g: A\to \mathbb{R}$, such that $g$ is one-one and $$g(a)<g(b)<g(c)\implies (a,b,c)\in S.$$
5. Let $p$ be a prime greater than $3$. Prove that \[ p^2\Big| \sum_{i=1}^{\left\lfloor\frac{2p}{3}\right\rfloor}\dbinom{p}{i}. \]
6. Let $c\ge 0$ be a real number. Give a complete description with proof of the set of all continuous functions $f: \mathbb{R}\to \mathbb{R}$ such that $f(x)=f(x^2+c)$ for all $x\in \mathbb{R}$.
7. Define a selfish set to be a set which has its own cardinality as its element. And a set is a minimal selfish set if none of its proper subsets are selfish. Find with proof the number of minimal selfish subsets of $\{1,2,\cdots ,n\}$.
8. Prove the inequality for all positive integer $n$ \[ \left(\frac{2n-1}{e}\right)^{\frac{2n-1}{2}}<1\cdot 3\cdot 5\cdots (2n-1)<\left(\frac{2n+1}{e}\right)^{\frac{2n+1}{2}} \]
9. Let $S_n$ be the set of all permutations of $(1,2,\ldots,n)$. Then find \[ \max_{\sigma \in S_n} \left(\sum_{i=1}^{n} \sigma(i)\sigma(i+1)\right) \] where $\sigma(n+1)=\sigma(1)$.
10. For any square matrix $\mathcal{A}$ we define $\sin {\mathcal{A}}$ by the usual power series. \[ \sin {\mathcal{A}}=\sum_{n=0}^{\infty}\frac{(-1)^n}{(2n+1)!}\mathcal{A}^{2n+1}.\] Prove or disprove $\exists 2\times 2$ matrix $A\in \mathcal{M}_2(\mathbb{R})$ such that \[ \sin{A}=\left(\begin{array}{cc}1 & 1996 \\0 & 1 \end{array}\right) \]
11. Given a finite binary string $S$ of symbols $X,O$ we define $\Delta(S)=n(X)-n(O)$ where $n(X),n(O)$ respectively denote number of $X$'s and $O$'s in a string. For example $\Delta(XOOXOOX)=3-4=-1$. We call a string $S$ balanced if every substring $T$ of $S$ has $-2\le \Delta(T)\le 2$. Find number of balanced strings of length $n$.
12. Let $(a_1,b_1),(a_2,b_2),\ldots ,(a_n,b_n)$ be the vertices of a convex polygon containing the origin in its interior. Prove that there are positive real numbers $x,y$ such that \[ (a_1,b_1)x^{a_1}y^{b_1}+(a_2,b_2)x^{a_2}y^{b_2}+\ldots +(a_n,b_n)x^{a_n}y^{b_n}=(0,0) \]

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