Mein gott what have you done to my coat

by cjquines0, Nov 3, 2016, 1:19 PM

A warm-up problem first:
Engel, PSS wrote:
There are several circles of total circumference $10$ inside a square of side length $1$. Prove that there is a line that intersects at least $4$ of the circles.

Solution

Now for the meat:
Engel, PSS wrote:
A tramp has a coat of area $1$ with $5$ patches. Each patch has area at least $\frac{1}{2}$. Prove that two patches exist with common area at least $\frac{1}{5}$.

Solution

Tidbit

Tidbit 2

Comment

5 Comments

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Hello both of these were in Interm CP as well. The second problem can also be solved with PIE.

by shiningsunnyday, Nov 3, 2016, 1:37 PM

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What is "Engel" and what is "PSS"? :maybe:

by zephyrcrush78, Nov 3, 2016, 11:10 PM

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@above

PSS is the book Problem Solving Strategies written by Arthur Engel. It is quite useful for olympiad level math.

by MathAwesome123, Nov 3, 2016, 11:23 PM

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wth is the second problem's problem statement :|

by MathStudent2002, Nov 4, 2016, 1:37 AM

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I am not understanding part of your solution to the coat patches problem. You write
Quote:
If all ten pairs of patches have a common area less than $\frac{1}{5}$, the total area would be less $2$ and thus there must be some point on the coat that lies on only one patch.

However, if you are intending to count the total area of the patches as 5/2 (i.e. each patch individually),then each common area of up to 1/5 applies to two patches, so you cover a total area of up to 4, i.e. no point on the coat need lie on only one patch (and indeed having each of the 10 area shared by patches be up to 1/5 would more than cover the total coat-area of 1)

It's also easy to see by considering a single patch: if the area it shares with each of the other four is up to 1/5, those four shares could easily cover its total area of 1/5.

by TMA, Feb 20, 2020, 8:14 PM

To share with readers my favorite problem I came across today :) (Shout for contrib.)

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