1993 UNCO Math Contest II Problems/Problem 6
Problem
Observe that
(a) Find integers and so that
(b) Conjecture a general rule that is being illustrated here.
(c) Prove your conjecture.
Solution
(a) We can rewrite the given equation as . Use difference of squares to obtain . Since is a prime we conclude that , giving us .
(b) It is not too hard to notice that the LHS above is and the RHS above is for . We will prove that the LHS RHS for all integers (although the proof extends to real numbers) in (c).
(c) We expand the LHS to obtain
Thus LHS = RHS and we are done.
~AK2006
See also
1993 UNCO Math Contest II (Problems • Answer Key • Resources) | ||
Preceded by Problem 5 |
Followed by Problem 7 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 | ||
All UNCO Math Contest Problems and Solutions |