Difference between revisions of "1972 IMO Problems/Problem 4"
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Find all solutions <math>(x_1, x_2, x_3, x_4, x_5)</math> of the system of inequalities | Find all solutions <math>(x_1, x_2, x_3, x_4, x_5)</math> of the system of inequalities | ||
| − | <cmath>(x_1^2 - x_3x_5)(x_2^2 - x_3x_5) \leq 0 \\ | + | <cmath> |
| + | \begin{align*} | ||
| + | (x_1^2 - x_3x_5)(x_2^2 - x_3x_5) \leq 0 \\ | ||
(x_2^2 - x_4x_1)(x_3^2 - x_4x_1) \leq 0 \\ | (x_2^2 - x_4x_1)(x_3^2 - x_4x_1) \leq 0 \\ | ||
(x_3^2 - x_5x_2)(x_4^2 - x_5x_2) \leq 0 \\ | (x_3^2 - x_5x_2)(x_4^2 - x_5x_2) \leq 0 \\ | ||
(x_4^2 - x_1x_3)(x_5^2 - x_1x_3) \leq 0 \\ | (x_4^2 - x_1x_3)(x_5^2 - x_1x_3) \leq 0 \\ | ||
| − | (x_5^2 - x_2x_4)(x_1^2 - x_2x_4) \leq 0</cmath> | + | (x_5^2 - x_2x_4)(x_1^2 - x_2x_4) \leq 0 |
| + | \end{align*}</cmath> | ||
where <math>x_1, x_2, x_3, x_4, x_5</math> are positive real numbers. | where <math>x_1, x_2, x_3, x_4, x_5</math> are positive real numbers. | ||
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Therefore, <math>x_1 = x_4 = x_2 = x_5 = x_3</math> is the only solution. | Therefore, <math>x_1 = x_4 = x_2 = x_5 = x_3</math> is the only solution. | ||
| − | Borrowed from http://www.cs.cornell.edu/~asdas/imo/imo/isoln/isoln724.html | + | Borrowed from [http://www.cs.cornell.edu/~asdas/imo/imo/isoln/isoln724.html] |
| + | |||
| + | == See Also == {{IMO box|year=1972|num-b=3|num-a=5}} | ||
Latest revision as of 14:39, 29 January 2021
Find all solutions 
of the system of inequalities
where 
are positive real numbers.
Solution
Add the five equations together to get
Expanding and combining, we get
Every term is 
, so every term must 
.
From the first term, we can deduce that 
. From the second term, 
. From the third term, 
. From the fourth term, 
.
Therefore, 
is the only solution.
Borrowed from [1]
See Also
| 1972 IMO (Problems) • Resources | ||
| Preceded by Problem 3 |
1 • 2 • 3 • 4 • 5 • 6 | Followed by Problem 5 |
| All IMO Problems and Solutions | ||
