Determine all composite positive integers $n$ for which it is possible to arrange all divisors of $n$ that are greater than $1$ in a circle so that no two adjacent divisors are relatively prime.
Proposed by Zuming Feng (credited in the supplied source).
Prove that the system of equations $$\begin{aligned} x^6 + x^3 + x^3y + y &= 147^{157} \\ x^3 + x^3y + y^2 + y + z^9 &= 157^{147} \end{aligned}$$ has no integer solutions.
Proposed by Răzvan Gelca (credited in the supplied source).
Let $ABC$ be an acute-angled triangle, and let $P$ and $Q$ be two points on side $BC$. Construct a point $C_1$ in such a way that the convex quadrilateral $APBC_1$ is cyclic, $\overline{QC_1} \parallel \overline{CA}$, and $C_1$ and $Q$ lie on opposite sides of line $AB$. Construct a point $B_1$ in such a way that the convex quadrilateral $APCB_1$ is cyclic, $\overline{QB_1} \parallel \overline{BA}$, and $B_1$ and $Q$ lie on opposite sides of line $AC$. Prove that the points $B_1$, $C_1$, $P$, and $Q$ lie on a circle.
Proposed by Zuming Feng (credited in the supplied source).
Legs $L_1$, $L_2$, $L_3$, $L_4$ of a square table each have length $n$, where $n$ is a positive integer. For how many ordered $4$-tuples $(k_1, k_2, k_3, k_4)$ of nonnegative integers can we cut a piece of length $k_i$ from the end of leg $L_i$ and still have a stable table? (The table is stable if it can be placed so that all four of the leg ends touch the floor. Note that a cut leg of length $0$ is permitted.)
Proposed by Elgin Johnston (credited in the supplied source).
Let $n > 1$ be an integer. Suppose $2n$ points are given in the plane, no three of which are collinear. Suppose $n$ of the given $2n$ points are colored blue and the other $n$ colored red. A line in the plane is called a balancing line if it passes through one blue and one red point and, for each side of the line, the number of blue points on that side is equal to the number of red points on the same side. Prove that there exist at least two balancing lines.
Proposed by Kiran Kedlaya (credited in the supplied source).
For a positive integer $m$, let $s(m)$ denote the sum of the decimal digits of $m$. A set $S$ positive integers is $k$-stable if $s(\sum_{x \in X} x) = k$ for any nonempty subset $X \subseteq S$. For each integer $n \ge 2$ let $f(n)$ be the minimal $k$ for which there exists a $k$-stable set with $n$ integers. Prove that there are constants $0 < C_1 < C_2$ with $$ C_1 \log_{10} n \le f(n) \le C_2 \log_{10} n. $$
Proposed by Titu Andreescu, Gabriel Dospinescu (credited in the supplied source).