Prove that there are infinitely many positive integers $a$ such that \[a!+(a+2)!\mid (a+2\left\lfloor\sqrt{a}\right\rfloor)!.\] Proposed by Navid and the4seasons.
Source: 2023 IRN-SGP-TWN Friendly Math Competition P1
Tags: number theory
Prove that there are infinitely many positive integers $a$ such that \[a!+(a+2)!\mid (a+2\left\lfloor\sqrt{a}\right\rfloor)!.\] Proposed by Navid and the4seasons.