HMMT 二月 2010 · GEN2 赛 · 第 1 题

英文解析

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From the picture above, we see that the union of the half-disks will be a quarter-circle with radius
2, and therefore area π . To prove that this is the case, we first prove that the boundary of everyhalf-disk intersects the quarter-circle with radius 2, and then that the half-disk is internally tangentto the quarter-circle at that point. This is sufficient because it is clear from the diagram that we neednot worry about covering the interior of the quarter-circle.
Let O be the origin. For a given half-disk D , label the vertex on the y -axis A and the vertex onythe x -axis B . Let M be the midpoint of line segment AB . Draw segment OM , and extend it until itintersects the curved boundary of D . Label the intersection point C . This construction is shown inthe diagram below.y
C2
y dMA
2O
We first prove that C lies on the quarter-circle, centered at the origin, with radius 2. Since M is the bmidpoint of AB , and A is on the y -axis, M is horizontally halfway between B and the y -axis. Since
General Test, Part 2
O and B are on the x -axis (which is perpendicular to the y -axis), segments OM and M B have the same length. Since M is the midpoint of AB , and AB = 2, OM = 1. Since D is a half-disk withradius 1, all points on its curved boundary are 1 away from its center, M . Then C is 2 away from theyorigin, and the quarter-circle consists of all points which are 2 away from the origin. Thus, C is anintersection of the half-disk D with the positive quarter-circle of radius 2.
It remains to show that the half-disk D is internally tangent to the quarter-circle. Since OC is ayyradius of the quarter-circle, it is perpendicular to the tangent of the quarter-circle at C . Since M C isa radius of the half-disk, it is perpendicular to the tangent of the half-disk at C . Then the tangentslines of the half-disk and the quarter-circle coincide, and the half-disk is tangent to the quarter-circle.
It is obvious from the diagram that the half-disk lies at least partially inside of the quarter-circle, thehalf-disk D is internally tangent to the quarter-circle.
Then the union of the half-disks is be a quarter-circle with radius 2, and has area π .y