Let $n$ and $m$ be positive integers. The daycare nanny uses $n \times m$ square floor mats to construct an $n \times m$ rectangular area, with a baby on each of the mats. Each baby initially faces toward one side of the rectangle. When the nanny claps, all babies crawl one mat forward in the direction it is facing at, and then turn 90 degrees clockwise. If a baby crawls outside of the rectangle, it cries. If two babies simultaneously crawl onto the same mat, they bump into each other and cry. Suppose that it is possible for the nanny to arrange the initial direction of each baby so that, no matter how many times she claps, no baby would cry. Find all possible values of $n$ and $m$. Proposed by Chu-Lan Kao

Find all positive integers $n$ satisfying the following conditions simultaneously: (a) the number of positive divisors of $n$ is not a multiple of $8$; (b) for all integers $x$, we have \[x^n \equiv x \mod n.\] Proposed by usjl

Let $O$ be the center of circle $\Gamma$, and $A$, $B$ be two points on $\Gamma$ so that $O, A$ and $B$ are not collinear. Let $M$ be the midpoint of $AB$. Let $P$ and $Q$ be points on $OA$ and $OB$, respectively, so that $P \neq A$ and $P, M, Q$ are collinear. Let $X$ be the intersection of the line passing through $P$ and parallel to $AB$ and the line passing through $Q$ and parallel to $OM$. Let $Y$ be the intersection of the line passing through $X$ and parallel to $OA$ and the line passing through $B$ and orthogonal to $OX$. Prove that: if $X$ is on $\Gamma$, then $Y$ is also on $\Gamma$. Proposed by usjl

Let $n$ and $k$ be positive integers. Let $A$ be a set of $2n$ distinct points on the Euclidean plane such that no three points in $A$ are collinear. Some pairs of points in $A$ are linked with a segment so that there are $n^2 + k$ distinct segments on the plane. Prove that there exists at least $\frac{4}{3}k^{3/2}$ distinct triangles on the plane with vertices in $A$ and sides as the aforementioned segments. Proposed by Ho-Chien Chen

Let $m$ be a positive integer, and real numbers $a_1, a_2,\ldots , a_m$ satisfy \[\frac{1}{m}\sum_{i=1}^{m}a_i = 1,\]\[\frac{1}{m}\sum_{i=1}^{m}a_i ^2= 11,\]\[\frac{1}{m}\sum_{i=1}^{m}a_i ^3= 1,\]\[\frac{1}{m}\sum_{i=1}^{m}a_i ^4= 131.\]Prove that $m$ is a multiple of $7$. Proposed by usjl