USAMO 1998 — Days 1 and 2

Suppose that $\{1,2,\ldots,1998\}$ has been partitioned into disjoint pairs $\{a_i,b_i\}$, $1\le i\le999$, so that every $|a_i-b_i|$ equals $1$ or $6$. Prove that $\sum_{i=1}^{999}|a_i-b_i|$ ends in the digit $9$.

Let $\mathcal C_1$ and $\mathcal C_2$ be concentric circles, with $\mathcal C_2$ inside $\mathcal C_1$. From $A\in\mathcal C_1$, draw the tangent $AB$ to $\mathcal C_2$, with $B\in\mathcal C_2$. Let $C$ be the second intersection of ray $AB$ with $\mathcal C_1$, and let $D$ be the midpoint of $AB$. A line through $A$ meets $\mathcal C_2$ at $E,F$ so that the perpendicular bisectors of $DE$ and $CF$ intersect at a point $M$ on line $AB$. Find, with proof, $AM/MC$.

Let $a_0,a_1,\ldots,a_n\in(0,\pi/2)$ satisfy
\[\sum_{i=0}^n\tan(a_i-\pi/4)\ge n-1.\]
Prove that $\prod_{i=0}^n\tan a_i\ge n^{n+1}$.

A computer screen shows a $98\times98$ chessboard with the usual alternating colors. A mouse click selects any rectangle whose sides lie on board gridlines and switches every color inside it. Find, with proof, the minimum number of clicks needed to make the board all one color.

Prove that for each integer $n\ge2$, there exists a set $S$ of $n$ integers such that $(a-b)^2$ divides $ab$ for every pair of distinct $a,b\in S$.

Let $n\ge5$ be an integer. Find the largest integer $k$, as a function of $n$, for which there exists a convex $n$-gon $A_1A_2\cdots A_n$ with exactly $k$ of the quadrilaterals $A_iA_{i+1}A_{i+2}A_{i+3}$ having an inscribed circle. Indices are taken modulo $n$.