## CSMC 2021 Part B - Question 3, CEMC UWaterloo

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#### Difficulty: 9

#### This problem is tagged with csmc, csmc21, highschool.

**(Canadian Senior Mathematics Contest 2021, Part B, Question 3, CEMC - UWaterloo)**

A pair of functions $f(x)$ and $g(x)$ is called a *Payneful pair* if:

$(i)$ $f(x)$ is a real number for all real numbers $x$,

$(ii)$ $g(x)$ is a real number for all real numbers $x$,

$(iii)$ $f(x+y) = f(x)g(y) + g(x)f(y)$ for all real numbers $x$ and $y$,

$(iv)$ $g(x+y) = g(x)g(y) - f(x)f(y)$ for all real numbers $x$ and $y$, and

$(v)$ $f(a) \neq 0$ for some real number $a$.

For every Payneful pair of functions $f(x)$ and $g(x)$:

a) Determine the values of $f(0)$ and $g(0)$.

b) If $h(x) = (f(x))^2 + (g(x))^2$ for all real numbers $x$, determine the value of
$h(5)h(−5)$.

c) If $-10 \le f(x) \le 10$ and $-10 \le g(x) \le 10$ for all real numbers $x$, determine the value of $h(2021)$.

*Answer Submission Note(s)*

Your answer must contain the answer for part a as "x,y" where $x$ is the value of $f(0)$ and $y$ is the value of $g(0)$. After this, the answer for part b must follow, and should be a single integer. Finally, the answer for part c must also be a single integer. Ensure answers for all parts are space-separated.

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