In each square of a chessboard with $a$ rows and $b$ columns, a $0$ or $1$ is written satisfying the following conditions. If a row and a column intersect in a square with a $0$, then that row and column have the same number of $0$s. If a row and a column intersect in a square with a $1$, then that row and column have the same number of $1$s. Find all pairs $(a,b)$ for which this is possible.

On a line, there are $n$ closed intervals (none of which is a single point) whose union we denote by $S$. It's known that for every real number $d$, $0<d\le 1$, there are two points in $S$ that are a distance $d$ from each other. (a) Show that the sum of the lengths of the $n$ closed intervals is larger than $\frac{1}{n}$. (b) Prove that, for each positive integer $n$, the $\frac{1}{n}$ in the statement of part (a) cannot be replaced with a larger number.

Find all integers $k\ge 2$ such that for all integers $n\ge 2$, $n$ does not divide the greatest odd divisor of $k^n+1$.

Can the positive integers be partitioned into $12$ subsets such that for each positive integer $k$, the numbers $k, 2k,\ldots,12k$ belong to different subsets?

In triangle $ABC$, where $AB<AC$, let $X$, $Y$, $Z$ denote the points where the incircle is tangent to $BC$, $CA$, $AB$, respectively. On the circumcircle of $ABC$, let $U$ denote the midpoint of the arc $BC$ that contains the point $A$. The line $UX$ meets the circumcircle again at the point $K$. Let $T$ denote the point of intersection of $AK$ and $YZ$. Prove that $XT$ is perpendicular to $YZ$.

Consider a collection of stones whose total weight is $65$ pounds and each of whose stones is at most $w$ pounds. Find the largest number $w$ for which any such collection of stones can be divided into two groups whose total weights differ by at most one pound. Note: The weights of the stones are not necessarily integers.