COMC 2020 C Problem 2

An expression like

x=1+\frac{1}{2+\frac{1}{3+\frac{1}{4}}}

is called a continued fraction.

(a) Write x given above as a reduced fraction of the form \frac{a}{b} where a and b are positive integers.

(b) Write \frac{355}{113} as a continued fraction in the form a+\frac{1}{b+\frac{1}{c}}, where a, b, c are positive integers.

(c) Let

y=8+\frac{1}{8+\frac{1}{8+\frac{1}{8+\cdots}}}

where the process continues indefinitely. Given that y can be written in the form p+\sqrt{q}, find the integers p and q.