Question 6
a) Find all positive integers $a$ for which
$$
\frac{1}{4} \lt \frac{1}{a+1} + \frac{1}{a+2} + \frac{1}{a+3} \lt \frac{1}{3}.
$$
b) Prove that for any integer $p \ge 2$ there exist $p$ consecutive positive integers $a_1, a_2, \dots, a_p$ such that
$$
\frac{1}{p+1} \lt \frac{1}{a_1} + \frac{1}{a_2} + \dots + \frac{1}{a_p} \lt \frac{1}{p}.
$$
$$
\frac{1}{4} \lt \frac{1}{a+1} + \frac{1}{a+2} + \frac{1}{a+3} \lt \frac{1}{3}.
$$
b) Prove that for any integer $p \ge 2$ there exist $p$ consecutive positive integers $a_1, a_2, \dots, a_p$ such that
$$
\frac{1}{p+1} \lt \frac{1}{a_1} + \frac{1}{a_2} + \dots + \frac{1}{a_p} \lt \frac{1}{p}.
$$