1961 IMO Problems/Problem 4
Let denote the area of triangle .
Since triangles and share the base , we have .
Adding all of these gives .
We see that we must have at least one of the three fractions not greater than , and at least one not less than . These correspond to ratios being less than or equal to , and greater than or equal to , respectively, so we are done.
Let and Note that by same base in triangles and Thus, Without loss of generality, assume Hence, and as desired.
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