Difference between revisions of "1959 IMO Problems/Problem 2"
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Since the term inside the square root is a perfect square, and by factoring 2 out, we get | 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> | <cmath>2(x + \sqrt{(x-1)^2}) = A^2</cmath> | ||
− | Use the property that < | + | Use the property that <math>\sqrt{x^2}=x</math> to get |
<cmath>A^2 = 2(x+|x-1|)</cmath> | <cmath>A^2 = 2(x+|x-1|)</cmath> | ||
Revision as of 13:47, 15 December 2019
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:
Since the term inside the square root is a perfect square, and by factoring 2 out, we get Use the property that to get
If , then we must clearly have . Otherwise, we have
Hence for (a) the solution is , for (b) there is no solution, since we must have , and for (c), the only solution is . 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 |