How do I start?
Probability
Level
4
Of all the possible triangles that are inscribed in a fixed circle, find the probability that the triangle is acute.
This is a problem of IMO.
The answer is 0.25.
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1 solution
One can start by picking two points on a circle and drawing the diameters through them. It is easy to verify that if the third point is on the side of a diameter closer to the point that doesn't go through the diameter then the triangle is obtuse, otherwise it is acute (the third point could technically be on a diameter, but that chance is measure 0, so we may ignore that case).
Thus, we can break the circle into 4 regions: between the two points picked originally, between one of the points and one of the intersections between the circle and the diameter, and finally between the two intersections of the circle and the diameters. Clearly, any acute triangles must have their third point in the last region. This region is equal to the minor arc between the two originally picked points because sum of the areas of the regions on the correct side of each diameter is the total circumference of the triangle, with the only area being over counted being the arc between the two originally picked points.
Thus, the problem is equivalent to asking what the average measure of the minor arc is between any two randomly picked points in a circle, which can be trivially shown to be 0.25.