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.
Here is an example of one chess-colored black and white logo.
Is chess going to be part of the 2020 Olympic Games?
Remark: I have found a geometric proof of the problem for any 3 central-symmetric convex shapes, but it is not as nice and fun as the one for circles. Also, probably the claim is true for any odd number of C-S C bodies in n-dimensional space. For the related version with squares/regular hexagons, check Black and White. It has a nice combinatorial geometric proof.
The arrows show relations between arc lengths. Color shades show relations between region sizes. Use HINT 4 to show that the relations in HINT 2 are true for well-chosen pairs of arcs.
HINT 7 (non-cryptic):
If the circles are A, B, and C, and A denotes the area only in A, BC denotes the area only in B and C, etc., then the problem is equivalent to showing that A>BC, or B>AC, or C>AB. The line drawn in HINT 1 separates both A and BC (or B and AC, or C and AB) in 2 pieces each. Can you show that the pieces from BC are smaller than their corresponding pieces from A (or possibly the same, but with BC replaced by AC/AB, and A replaced by B/C)?
There is some geometry involved, so if you have forgotten what inscribed/inside/outside angles are - http://www.regentsprep.org/Regents/math/geometry/GP15/CircleAngles.htm