1973 Canadian MO Problems/Problem 7
Contents
[hide]Problem
Observe that State a general law suggested by these examples, and prove it.
Prove that for any integer greater than there exist positive integers and such that
Solution 1 (Easiest)
: We see that . By simply evaluating the left hand side using common denominators, we see that both sides are equal. So therefore we have proved the law.
: Using partial fraction decomposition, let's change the original expression to:
.
Therefore, .
Let , then we have satisfied the previous identity in part . Therefore, we have solved the problem.
~hastapasta
Solution 2
We see that:
We prove this by induction. Let Base case: Therefore, is true. Now, assume that is true for some . Then:
Thus, by induction, the formula holds for all
Incomplete
See also
1973 Canadian MO (Problems) | ||
Preceded by Problem 6 |
1 • 2 • 3 • 4 • 5 | Followed by Problem 1 |