Polygons Which Don't Fit

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Polygons Which Don't Fit

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Source: San Diego Honors Math Contest 2025 Part II, Problem 1

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somebodyyouusedtoknow

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somebodyyouusedtoknow

#1 h Apr 26, 2025, 11:28 PM

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Let $P_1,P_2,\ldots,P_n$ be polygons, no two of which are similar. Show that there are polygons $Q_1,Q_2,\ldots,Q_n$ where $Q_i$ is similar to $P_i$ so that for no $i \neq j$ does $Q_i$ contain a polygon that's congruent to $Q_j$ .

Note. Here, the word "contain" means for the construction we have, we cannot select a size for $Q_j$ so that $Q_j$ is wholly contained in $Q_i$ , and so it does not intersect the edges of $Q_i$ at all.

This post has been edited 1 time. Last edited by somebodyyouusedtoknow, Apr 26, 2025, 11:41 PM
Reason: Added clarity for the notes

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#2 h Apr 27, 2025, 12:09 PM

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Induct on $n$ . The base case $n=2$ is obvious; for fixed $Q_1=P_1$ consider the largest copy of $P_2$ inside it and slightly enlarge it to obtain $Q_2$ .

Now the step $n\to n+1$ . That is, given $Q_1, \ldots, Q_n$ with $Q_i \not\subset Q_j$ for all $i\ne j$ we seek for appropriate $Q_{n+1}$ .

For each $i\in [n]:= \{1, \ldots, n\}$ define $R_i^-$ as the largest copy of $P_{n+1}$ inside $Q_i$ . As well as define $R_i^+$ as the smallest copy of $P_{n+1}$ containing $Q_i$ .

We want to show that there is a copy of $P_{n+1}$ which is larger than $R_i^-$ and smaller than $R_i^+$ for all $i\in [n]$ . That is, we need to prove that $\max_{i\in [n]} R_i^- \subsetneq \min_{i\in [n]}R_i^+$ . Suppose not, then there are $i\ne j$ such that $Q_i \subseteq R_i^+ \subseteq R_j^- \subseteq Q_j$ . This contradiction proves that we can select $Q_{n+1}$ appropriately.

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