Bump this

LOL what a troll question..... The pith of this problem is noting that $2024=\binom{24}{3}$, so the number of triples of integers $0<i<j<k\leq 24$ is exactly $2024$! This shows that the number $2024$ is a chosen one and not a random one. Back to the problem, it suffices to choose $n$ so that $i\mid \tau(n+i)$ for each $1\le i\le 24$, but this is trivial because, to each $i$, assign a prime $p_i$, and let \[n\equiv -i+p_i^{i-1}\pmod{p_i^i}\]for all $1\le i\le 24$ which obviously has a solution due to CRT. This implies that $\nu_{p_i}(n+i)=i-1$ for each $1\le i\le 24$, therefore, $i\mid\tau(n+i)$ for each $1\le i\le 24$ as desired. A huge round of applause should be given to whoever proposed this. My greatest respects to him/her/them.

Comment(might be spoiler)

Students would’ve thought that the number 2024 was just given as a “random large integer”. These kind of tricks aren’t easy to figure out. It was an Interesting problem anyways haha.