Floor and Ceiling (AIME / USAJMO / AMO)

Proof-style AIME / USAJMO / AMO statements — develop a full argument; type only the final answer.

Proof track: treat each problem as a full olympiad writeup on paper. Here you only enter the final answer (integer or short expression).

Floor and Ceiling: core idea

Contest floors.

**(Proof track.)** Give a complete justification. Find $\lfloor\sqrt{999}\rfloor$ .

Hint: Replace the floor or ceiling with its sharp bounding inequalities. Answer: 31.

No proof submission required — answers only.

1. **(Proof track.)** Give a complete justification. How many integers satisfy $\lceil x/7\rceil=12$ ?
2. **(Proof track.)** Give a complete justification. Find $\sum_{k=1}^{20}\lceil k/6\rceil$ .
3. **(Proof track.)** Give a complete justification. Find $\lfloor\pi^2\rfloor+\lceil e^2\rceil$ .
4. **(Proof track.)** Give a complete justification. How many positive integers $n$ have $\lfloor100/n\rfloor=4$ ?
5. **(Proof track.)** Give a complete justification. Find $\left\lfloor\frac{\sqrt5+1}{2}\cdot10\right\rfloor$ .
6. **(Proof track.)** Give a complete justification. Find the greatest integer less than $1/(\sqrt3-1)$ .
7. **(Proof track.)** Give a complete justification. How many pairs $(m,n)$ of positive integers satisfy $\lfloor m/n\rfloor=2$ and $m\le10,n\le10$ ?
8. **(Proof track.)** Give a complete justification. Find $\sum_{k=1}^{9}\lfloor\sqrt k\rfloor$ .
9. **(Proof track.)** Give a complete justification. Find $\lceil\log_2 1000\rceil$ .