RMM 2026 Problem 6

Let $k > 1$ be an integer, and let $S$ denote the set of all $(k+1)$-tuples of integers $X = (x_1, \ldots, x_{k+1})$ such that $1 \leq x_1 < \ldots < x_{k+1} \leq k^2 + 1$. If $\sigma$ is a permutation of the numbers $1, 2, \ldots, k^2 + 1$, say that an element $X$ of $S$ is $\sigma$-nice if the sequence $\sigma(x_1), \sigma(x_2), \ldots, \sigma(x_{k+1})$ is monotone. Prove that

$$\min_{1 \leq i \leq k} \left\lfloor \frac{x_i}{i} \right\rfloor + \min_{2 \leq i \leq k+1} \left\lfloor \frac{k^2 + 2 - x_i}{k + 2 - i} \right\rfloor \geq k + 1$$

if and only if there exists a permutation $\sigma$ such that $X$ is the unique $\sigma$-nice tuple in $S$.