Difference between revisions of "1960 IMO Problems/Problem 2"
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==Solution== | ==Solution== | ||
Set <math>x = -\frac{1}{2} + \frac{a^2}{2}</math>, where <math>a\ge0</math>. | Set <math>x = -\frac{1}{2} + \frac{a^2}{2}</math>, where <math>a\ge0</math>. | ||
− | <math>\frac{4(-\frac{1}{2}+\frac{a^2}{2})^2}{(1-\sqrt{1+2(-\frac{1}{2}+\frac{a^2}{2})})^2}< | + | <math>\frac{4\left(-\frac{1}{2}+\frac{a^2}{2}\right)^2}{\left(1-\sqrt{1+2\left(-\frac{1}{2}+\frac{a^2}{2}\right)}\right)^2}<2\left(-\frac{1}{2}+\frac{a^2}{2}\right)+9</math> |
After simplifying, we get | After simplifying, we get | ||
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<math>a^2+2a+1<a^2+8</math> | <math>a^2+2a+1<a^2+8</math> | ||
− | Which gives <math>a<\frac{7}{2}</math> and hence <math>x<\frac{45}{8}</math>. | + | Which gives <math>a<\frac{7}{2}</math> and hence <math>-\frac{1}{2} \le x<\frac{45}{8}</math>. |
− | But <math>x=0</math> makes the | + | But <math>x=0</math> makes the LHS indeterminate. |
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+ | So, answer: <math>-\frac{1}{2} \le x<\frac{45}{8}</math>, except <math>x=0</math>. | ||
==See Also== | ==See Also== | ||
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[[Category:Olympiad Algebra Problems]] | [[Category:Olympiad Algebra Problems]] | ||
+ | [[Category:Olympiad Inequality Problems]] |
Revision as of 11:41, 19 June 2017
Problem
For what values of the variable does the following inequality hold:
Solution
Set , where .
After simplifying, we get
So
Which gives and hence .
But makes the LHS indeterminate.
So, answer: , except .
See Also
1960 IMO (Problems) • Resources | ||
Preceded by Problem 1 |
1 • 2 • 3 • 4 • 5 • 6 | Followed by Problem 3 |
All IMO Problems and Solutions |