## CSMC 2022 Part B - Question 3, CEMC UWaterloo

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#### Difficulty: 15

#### This problem is tagged with csmc, csmc22, highschool.

**(Canadian Senior Mathematics Contest 2022, Part B, Question 3, CEMC - UWaterloo)**

For each positive integer $n$, let $S_n$ be the set that contains the integers from $1$ to $n$, inclusive; that is, $S_n = \{1,2,3, \dots, n\}$.

For each positive integer $n \ge 4$, let $f(n)$ be the number of quadruples $(a,b,c,d)$ of distinct integers from $S_n$ for which $a - b = c - d$.

For example, $f(4) = 8$ because the possibilities for $(a,b,c,d)$ are
$$(1,2,3,4),(1,3,2,4),(2,1,4,3),(2,4,1,3),$$

$$(3,1,4,2),(3,4,1,2),(4,2,3,1),(4,3,2,1)$$
a) Determine the value of $f(6)$.

b) Determine the value of $f(40)$.

c) Determine two even positive integers $n \lt 2022$ for which $2022$ is a divisor of $f(n)$.

*Answer Submission Note(s)*

Answer each part in order when submitting, and separate answers for each part with a space. Do not format any integers with commas/spaces, etc.

For the final part, answer using the format "(x,y)" (and replace x and y with the two values for $n$, in increasing order). Your final answer should look like "a b (x,y)".

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