A New Year, A New Hard Problem! 1 of 5

Geometry Level 2

The dissection of the 3 - 4 - 5 triangle shown below has a diameter 5 2 \frac{5}{2} Find the least diameter of a dissection of this triangle into four parts. (The diameter of a dissection is the least upper bound of the distances between pairs of points belonging to the same part.)

25/13 It is impossible to find the diameter. 25/14 300/49

Vishruth Bharath by Vishruth Bharath , 17, USA

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1 solution

Vishruth Bharath
Dec 31, 2017

Place the triangle on the cartesian plane so that its vertices are A = (4, 0), B = (0, 0), C = (0, 3). The five points A, B, C, D = (27/13, 0), E = (20/13, 24/13) lie at distance at least 25/13 apart from each other (note that AD = DE = EC = 25/13). By the Pigeonhole Principle, any dissection of thetriangle into four parts has a part containing at least two of these points, and hence has diameter at least 25 13 \boxed{\frac{25}{13}} .

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