Tessellate S.T.E.M.S. (2019) - Mathematics - Category A (School) - Set 4 - Subjective Problem 2

Given prime number $p>2$, let $m, n$ be integers such that $ m \equiv n \equiv 0 \pmod{p} $, $n \equiv 1 \pmod{2} $. Call a function $f: \{1, \cdots, m \} \longrightarrow \{ 1, \cdots , n \} $ "sweet" if it satisfies $ \sum_{k=1}^m f(k) \equiv 0 \pmod{p}$. Then show that the sum of the products $ \prod_{k=1}^m f(k) $ over all sweet functions $ f $ is a multiple of $ (\frac{n}{p})^m $.

This problem is a part of Tessellate S.T.E.M.S. (2019)

Easy Math Editor

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Math	Appears as
Remember to wrap math in `$` ... `$` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$

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Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$