Question 3
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
$$ \left\lvert x\right\rvert + \left\lvert y + \frac12 \right\rvert < n. $$
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