1970 IMO Problems/Problem 3
Problem
The real numbers satisfy the condition:
.
The numbers are defined by
(a) Prove that for all .
(b) given with , prove that there exist numbers with the above properties such that for large enough .
Solution
Let be the rectangle with the verticies: ; ; ; .
For all , the area of is . Therefore,
For all sequences and all , lies above the -axis, below the curve , and in between the lines and , Also, all such rectangles are disjoint.
Thus, as desired.
By choosing , where , is a Riemann sum for . Thus, .
Therefore, .
So for any , we can always select a small enough to form a sequence satisfying the above properties such that for large enough as desired.
See also
1970 IMO (Problems) • Resources | ||
Preceded by Problem 2 |
1 • 2 • 3 • 4 • 5 • 6 | Followed by Problem 4 |
All IMO Problems and Solutions |