This is from one of the Finnish national Math competitions MAOL http://www.maol.fi/fileadmin/users/K...vastaukset.pdf
prove that
I have a question
1) The sample answer splits 17 into 9 and 8. Yet it moves the eight to the "ends" of the equation
Why is this?
Thank you!
Hello, Coach!
Dan has the best approach . . .
In case you can't "eyeball" those radicals, here's a tip.Prove that: .
Since the difference of those two radicals in an integer,
. . we might suspect that the radicands are perfect squares.
We conjecture: . . . . . where is positive and rational
The left side is: .
So we have: .
Equate rational and irrational components:
. . .[1]
. . .[2]
and solve the system of equations.
Solve [2] for
Substitute into [1]: .
. . and we have: .
. . which factors: .
. . and has roots: .
Using , we have: .
Therefore: . . . . . ta-DAA!