2013 UMO Problems/Problem 1

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Problem

Consider the following diagram.

[asy] path S=polygon(4); D(S,black+linewidth(.75)); D(shift(sqrt(2),0)*S,black+linewidth(.75)); D(shift(2*sqrt(2),0)*S,black+linewidth(.75)); D(shift(sqrt(2),sqrt(2))*S,black+linewidth(.75)); D(shift(2*sqrt(2),sqrt(2))*S,black+linewidth(.75)); D(shift(2*sqrt(2),2*sqrt(2))*S,black+linewidth(.75)); [/asy]

(a) Show that you can retrace the diagram without lifting up your pencil using exactly nine (possibly overlapping) line segments.

(b) Show that you cannot retrace the diagram in the same way using eight or fewer segments.


Solution

See Also

2013 UMO (ProblemsAnswer KeyResources)
Preceded by
First Problem
Followed by
Problem 2
1 2 3 4 5 6
All UMO Problems and Solutions
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