Let $ABC$ be a right triangle with $\angle B=90^\circ$. A point $D$ lies on line $CB$ with $B$ between $D,C$. Let $E$ be the midpoint of $AD$, and let $F$ be the second intersection of the circumcircles of $ACD$ and $BDE$. Prove that, as $D$ varies, line $EF$ passes through a fixed point.
Editorial clarification: $F$ is a distinct second intersection, so exclude $DB=BC$, when the two circles are tangent at $D$. The fixed-point assertion applies to every well-defined position. The source wording is retained above; the limiting line at the excluded position is described in the solution.
Find all positive integers $k\lt 202$ for which some positive integer $n$ satisfies
$$\left\{\frac n{202}\right\}+\left\{\frac{2n}{202}\right\}+\cdots+\left\{\frac{kn}{202}\right\}=\frac k2.$$
Here $\{x\}$ is the fractional part: the unique real number in $[0,1)$ whose difference from $x$ is an integer.
Let $n,k$ be positive integers. There are $n$ marbles labelled $1,\ldots,n$ and $k$ boxes. Initially all marbles are in one box. On each turn Cathy chooses a box and moves the marble with smallest label in it, say $i$, either to an empty box or to the box containing marble $i+1$. Cathy wins if some box contains only marble $n$. Determine all pairs $(n,k)$ for which she can win.
