Difference between revisions of "2011 AIME II Problems/Problem 9"
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Peterungar (talk | contribs) (→Solution: A rational solution) |
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x_1 &= x_2 = \frac{5 - \sqrt{20}}{30} \\ | x_1 &= x_2 = \frac{5 - \sqrt{20}}{30} \\ | ||
x_5 &= x_4 = \frac{5 + \sqrt{20}}{30} | x_5 &= x_4 = \frac{5 + \sqrt{20}}{30} | ||
+ | \end{align*} | ||
+ | </cmath> | ||
+ | |||
+ | Another example is | ||
+ | <cmath> | ||
+ | \begin{align*} | ||
+ | x_1 = x_3 = \frac{1}{3} \\ | ||
+ | x_2 = \frac{19}{60}, \ x_5 = \frac{1}{60} \\ | ||
+ | x_4 &= x_6 = 0 | ||
\end{align*} | \end{align*} | ||
</cmath> | </cmath> |
Revision as of 15:27, 3 June 2011
Problem 9
Let be non-negative real numbers such that
, and
. Let
and
be positive relatively prime integers such that
is the maximum possible value of
. Find
.
Solution
Note that neither the constraint nor the expression we need to maximize involves products with
. Factoring out say
and
we see that the constraint is
, while the expression we want to maximize is
. Adding the left side of the constraint to the expression we get:
. This new expression is the product of three non-negative terms whose sum is equal to 1. By AM-GM this product is at most
. Since we have added at least
the desired maximum is at most
. It is easy to see that this upper bound can in fact be achieved by ensuring that the constraint expression is equal to
with
—for example, by choosing
and
small enough—so our answer is
An example is:
Another example is