Tessellate S.T.E.M.S. (2019) - Mathematics - Category B - Set 2 - Subjective Problem 1

Let $m$ be a positive integer and $n = 2^m+1$ . Consider $f_1 , f_2, \cdots f_n$ $[0, 1] \mapsto [0, 1]$ to be increasing functions such that $f_i(0) = 0$ and $|f_i(x) — f_i(y)| \leq |x-y|$ for all $1 \leq i \leq n$ and all $x, y \in [0,1]$ . Prove that there exist $1 \leq i \leq j \leq n$ such that $|f_i (x)— f_j (x)| \leq \frac{1}{m+1}$ for all $x \in [0, 1]$

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

#Algebra

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`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`	$\sum_{i=1}^3$
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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}`	$\boxed{123}$