Putnam 1984

Let $A$ be a solid $a\times b\times c$ rectangular brick, where $a,b,c>0$ . Let $B$ be the set of all points which are a distance of at most one from some point of $A$ . Express the volume of $B$ as a polynomial in $a,b,$ and $c$ .

Express $\sum_{k=1}^\infty\frac{6^k}{(3^{k+1}-2^{k+1})(3^k-2^k)}$ as a rational number.

A convex pentagon $P=ABCDE$ is inscribed in a circle of radius $1$ . Find the maximum area of $P$ subject to the condition that the chords $AC$ and $BD$ are perpendicular.

Let $n$ be a positive integer, and let $f(n)$ denote the last nonzero digit in the decimal expansion of $n!$ . $(\text a)$ Show that if $a_1,a_2,\ldots,a_k$ are distinct nonnegative integers, then $f(5^{a_1}+5^{a_2}+\ldots+5^{a_k})$ depends only on the sum $a_1+a_2+\ldots+a_k$ . $(\text b)$ Assuming part $(\text a)$ , we can define $$g(s)=f(5^{a_1}+5^{a_2}+\ldots+5^{a_k}),$$ where $s=a_1+a_2+\ldots+a_k$ . Find the least positive integer $p$ for which $$g(s)=g(s+p),\enspace\text{for all }s\ge1,$$ or show that no such $p$ exists.

Let $n$ be a positive integer, and define $f(n)=1!+2!+\ldots+n!$ . Find polynomials $P$ and $Q$ such that $$f(n+2)=P(n)f(n+1)+Q(n)f(n)$$ for all $n\ge1$ .

Find the minimum value of$$(u-v)^2+\left(\sqrt{2-u^2}-\frac{9}{v}\right)^2$$for $0<u<\sqrt{2}$ and $v>0$

Prove or disprove the following statement: If $F$ is a finite set with two or more elements, then there exists a binary operation $*$ on $F$ such that for all $x,y,z$ in $F$ , $(\text i)$ $x*z=y*z$ implies $x=y$ $(\text{ii})$ $x*(y*z)\ne(x*y)*z$

Find, with proof, all real-valued functions $y=g(x)$ defined and continuous on $[0,\infty)$ , positive on $(0,\infty)$ , such that for all $x>0$ the $y$ -coordinate of the centroid of the region $$R_x=\{(s,t)\mid0\le s\le x,\enspace0\le t\le g(s)\}$$ is the same as the average value of $g$ on $[0,x]$ .

For each nonnegative integer $k$ , let $d(k)$ denote the number of $1$ 's in the binary expansion of $k$ . Let $m$ be a positive integer. Express $$\sum_{k=0}^{2^m-1}(-1)^{d(k)}k^m$$ in the form $(-1)^ma^{f(m)}g(m)!$ , where $a$ is an integer and $f$ and $g$ are polynomials.

A sequence of convex polygons $(P_n),n\ge0,$ is defined inductively as follows. $P_0$ is an equilateral triangle with side length $1$ . Once $P_n$ has been determined, its sides are trisected; the vertices of $P_{n+1}$ are the interior trisection points of the sides of $P_n$ . Express $\lim_{n\to\infty}[P_n]$ in the form $\frac{\sqrt a}b$ , where $a,b$ are integers.