Difference between revisions of "1960 IMO Problems/Problem 2"
Pianoman24 (talk | contribs) m (→Solution) |
|||
(5 intermediate revisions by 4 users not shown) | |||
Line 5: | Line 5: | ||
==Solution== | ==Solution== | ||
− | {{ | + | Set <math>x = -\frac{1}{2} + \frac{a^2}{2}</math>, where <math>a\ge0</math>. |
+ | <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 | ||
+ | <math>(a+1)^2<a^2+8</math> | ||
+ | |||
+ | So | ||
+ | <math>a^2+2a+1<a^2+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 LHS indeterminate. | ||
+ | |||
+ | So, answer: <math>-\frac{1}{2} \le x<\frac{45}{8}</math>, except <math>x=0</math>. | ||
==See Also== | ==See Also== | ||
Line 11: | Line 24: | ||
{{IMO box|year=1960|num-b=1|num-a=3}} | {{IMO box|year=1960|num-b=1|num-a=3}} | ||
− | [[Category:Olympiad | + | [[Category:Olympiad Algebra Problems]] |
+ | [[Category:Olympiad Inequality Problems]] |
Latest 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 |