1961 IMO Problems/Problem 4
Problem
In the interior of triangle a point is given. Let be the intersections of with the opposing edges of triangle . Prove that among the ratios there exists one not larger than and one not smaller than .
Solution 1
Let denote the area of triangle .
Since triangles and share the base , we have .
Similarly, .
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.
Solution 2
Let and Note that by same base in triangles and Thus, Without loss of generality, assume Hence, and as desired.
1961 IMO (Problems) • Resources | ||
Preceded by Problem 3 |
1 • 2 • 3 • 4 • 5 • 6 | Followed by Problem 5 |
All IMO Problems and Solutions |