1959 IMO Problems/Problem 2
Problem
For what real values of is
given (a) , (b) , (c) , where only non-negative real numbers are admitted for square roots?
Solution
Firstly, the square roots imply that a valid domain for x is .
Square both sides of the given equation:
Add the first and the last terms to get
Multiply the middle terms, and use to get: $2x + 2 \sqrt{x^2 - 2x + 1} = A^2<cmath>
Since the term inside the square root is a perfect square, and by factoring 2 out, we get </cmath>2(x + \sqrt{(x-1)^2}) = A^2<cmath> Use the property that </cmath>\sqrt{x^2}=x<cmath> to get </cmath>A^2 = 2(x+|x-1|)$ (Error compiling LaTeX. Unknown error_msg)x \le 1A^2 =2$. Otherwise, we have
<cmath>x = \frac{A^2 + 2}{4} > 1,</cmath> <cmath>A^2 > 2 </cmath>
Hence for (a) the solution is$ (Error compiling LaTeX. Unknown error_msg) x \in \left[ \frac{1}{2}, 1 \right]A^2 \ge 2 x=\frac{3}{2}$. Q.E.D.
~flamewavelight (Expanded)
Alternate solutions are always welcome. If you have a different, elegant solution to this problem, please add it to this page.
See Also
1959 IMO (Problems) • Resources | ||
Preceded by Problem 1 |
1 • 2 • 3 • 4 • 5 • 6 | Followed by Problem 3 |
All IMO Problems and Solutions |