Difference between revisions of "1995 IMO Problems/Problem 4"
(→Solution) |
m |
||
(2 intermediate revisions by one other user not shown) | |||
Line 42: | Line 42: | ||
<math>x_{i}=\begin{cases} \frac{x_{0}}{2^{i}} & 1 \le i \le 1994 \\ \frac{1}{x_{i-1}} & i=1995\end{cases}</math> | <math>x_{i}=\begin{cases} \frac{x_{0}}{2^{i}} & 1 \le i \le 1994 \\ \frac{1}{x_{i-1}} & i=1995\end{cases}</math> | ||
− | Then <math>x_{1994}=\frac{x_0}{2^{1994}}</math>, <math>x_{1995}=\frac{1}{x_{1994}}=\frac{2^{1994}}{x_0}={x_0}</math> | + | Then <math>x_{1994}=\frac{x_0}{2^{1994}}</math>, |
+ | |||
+ | thus <math>x_{1995}=\frac{1}{x_{1994}}=\frac{2^{1994}}{x_0}={x_0}</math> | ||
Solving for <math>{x_0}</math> we get: | Solving for <math>{x_0}</math> we get: | ||
Line 48: | Line 50: | ||
<math>{x_0}^{2}=2^{1994}</math> | <math>{x_0}^{2}=2^{1994}</math> | ||
− | <math>{x_0}=\pm\sqrt{2^{1994}}=\pm2^{997}</math> | + | <math>{x_0}=\pm\sqrt{2^{1994}}=\pm2^{997}</math>. Since all <math>{x_i}</math> are defined as positive, <math>{x_0}=2^{997}</math>. |
− | + | Therefore, the maximum value that <math>x_0</math> can have is <math>{x_0}=2^{997}</math> | |
~ Tomas Diaz. orders@tomasdiaz.com | ~ Tomas Diaz. orders@tomasdiaz.com | ||
Line 59: | Line 61: | ||
− | == See Also == {{IMO box|year=1995|num-b=3|num-a=5}} | + | == See Also == |
+ | {{IMO box|year=1995|num-b=3|num-a=5}} |
Latest revision as of 21:24, 4 July 2024
Problem
The positive real numbers satisfy the relations
and
for
Find the maximum value that can have.
Solution
First we start by solving for in the recursive relation
or
So we have two recursive properties to chose from.
If we want to maximize then we can use
from
to
. This will make
the largest and
the smallest.
Then we can simply use to get
since the reciprocal will make it very large.
Then we use and solve for
This means that we can write as:
Then ,
thus
Solving for we get:
. Since all
are defined as positive,
.
Therefore, the maximum value that can have is
~ Tomas Diaz. orders@tomasdiaz.com
Alternate solutions are always welcome. If you have a different, elegant solution to this problem, please add it to this page.
See Also
1995 IMO (Problems) • Resources | ||
Preceded by Problem 3 |
1 • 2 • 3 • 4 • 5 • 6 | Followed by Problem 5 |
All IMO Problems and Solutions |