1999 JBMO Problems
Let be five real numbers such that , and . If are all distinct numbers prove that their sum is zero.
For each nonnegative integer we define . Find the greatest common divisor of the numbers .
Let be a square with the side length 20 and let be the set of points formed with the vertices of and another 1999 points lying inside . Prove that there exists a triangle with vertices in and with area at most equal with .
Let be a triangle with . Also, let be a point such that , and let be the circumcircles of the triangles and respectively. Let and be diameters in the two circles, and let be the midpoint of . Prove that the area of the triangle is constant (i.e. it does not depend on the choice of the point ).
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