Prove that for each positive integer $n$, there are pairwise relatively prime integers $k_0,\ldots,k_n$, all strictly greater than $1$, such that $k_0k_1 \dots k_n - 1$ is the product of two consecutive integers.
Proposed by Titu Andreescu; source discussion: https://aops.com/community/p1116186
Let $ABC$ be an acute, scalene triangle, and let $M$, $N$, and $P$ be the midpoints of $BC$, $CA$, and $AB$, respectively. Let the perpendicular bisectors of $AB$ and $AC$ intersect ray $AM$ in points $D$ and $E$ respectively, and let lines $BD$ and $CE$ intersect in point $F$, inside triangle $ABC$. Prove that points $A$, $N$, $F$, and $P$ all lie on one circle.
Proposed by Zuming Feng; source discussion: https://aops.com/community/p1116181
Let $n$ be a positive integer. Denote by $S_n$ the set of points $(x, y)$ with integer coordinates such that
$$ \left\lvert x\right\rvert + \left\lvert y + \frac12 \right\rvert < n. $$
A path is a sequence of distinct points $(x_1 , y_1), (x_2, y_2), \dots, (x_\ell, y_\ell)$ in $S_n$ such that, for $i = 2, \dots, \ell$, the distance between $(x_i , y_i)$ and $(x_{i-1} , y_{i-1} )$ is $1$. Prove that the points in $S_n$ cannot be partitioned into fewer than $n$ paths.
Proposed by Gabriel Carroll; source discussion: https://aops.com/community/p1116367
Three nonnegative real numbers $r_1$, $r_2$, $r_3$ are written on a blackboard. These numbers have the property that there exist integers $a_1$, $a_2$, $a_3$, not all zero, satisfying $a_1r_1 + a_2r_2 + a_3r_3 = 0$. We are permitted to perform the following operation: find two numbers $x$, $y$ on the blackboard with $x \le y$, then erase $y$ and write $y - x$ in its place. Prove that after a finite number of such operations, we can end up with at least one $0$ on the blackboard.
Proposed by Kiran Kedlaya; source discussion: https://aops.com/community/p1116189
At a certain mathematical conference, every pair of mathematicians are either friends or strangers. At mealtime, every participant eats in one of two large dining rooms. Each mathematician insists upon eating in a room which contains an even number of his or her friends. Prove that the number of ways that the mathematicians may be split between the two rooms is a power of two (i.e. is of the form $2^k$ for some positive integer $k$).
Proposed by Sam Vandervelde; source discussion: https://aops.com/community/p1116182