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Chess at the Olympics

For the 2020 Olympic Games, the World Chess Federation asked the Olympic committee to include chess as official sport. The committee agreed under one condition:

The Chess Federation must design a chess-colored black and white logo, consisting of three intersecting Olympic rings with areas 1 each, such that the black area - covered by odd number of rings, is less than 1. It is not necessary that each of the rings intersects all of the others.

Below you can see an example of one chess-colored black and white logo.

Is chess going to be part of the 2020 Olympic Games?


It is easy to check that if two of the circles do not intersect each other, the total black area is at least 1. Therefore now we will consider the case when each of the circles intersects both of the others.

Denote the intersection points of the circles with A, B, C, D, E, F, as shown on the diagram above. It is straightforward to check that the following 4 conditions are equivalent:

1.S(AEC)+S(BFA)+S(CDB)+S(DEF)≥1

2.S(AEC)≥S(DFB)

3.S(BFA)≥S(DCE)

4.S(CDB)≥S(AFE)

First, notice that DC+EA+BF=BD+CE+AF, regarded as arcs (we stick to this notation throughout the solution). Simple combinatoric argument shows that there are two consecutive arcs among these, touching at D, E, or F, such that each of them is at least as large as its opposite. Let without of generality DC≥AF and BD≥EA.

Now draw the line AD. Let it intersects the arcs EF and BC at points K and L.

Notice that:

•LC≥CD

•LB≥BD

•AE≥EK

•AF≥KF

•∠LCD=∠AFK

•∠DBL=∠KEA

All of these relations follow from simple symmetries and angles in the circles. For example, ∠LCD=∠DFB=∠KFA, LC=∠CKD≥CD, and AF=∠FDA≥KF.Combining the relations, we get:

1.LC≥AF, DC≥KF, ∠LCD=∠KFA

2.LB≥AE, DB≥KE, ∠LBD=∠KEA

Therefore the light green piece AFK can fully fit inside the dark green piece CDL, and the light blue piece KEA can fully fit inside the dark blue piece DBL. This implies that S(CDB)≥S(AFE), which is exactly point 4. from the conditions above.