1992 USAMO Problems/Problem 2
Consider the points in the coordinate plane with origin , for integers .
Evidently, the angle between segments and is , and the length of segment is . It then follows that the area of triangle is . Therefore so as desired.
First multiply both sides of the equation by , so the right hand side is . Now by rewriting , we can derive the identity . Then the left hand side of the equation simplifies to as desired.
Multiply by . We get:
we can write this as:
This is an identity
, because of telescoping.
but since we multiplied in the beginning, we need to divide by . So we get that:
as desired. QED
Multiplying by gives
Notice that after expanding the sine, and so so
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