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Difference between revisions of "1972 IMO Problems/Problem 4"

(Created page with "==Solution== Add the five equations together to get <math>(x_1^2 - x_3 x_5)(x_2^2 - x_3 x_5) + (x_2^2 - x_4 x_1)(x_3^2 - x_4 x_1) + (x_3^2 - x_5 x_2)(x_4^2 - x_5 x_2) + (x_4^2 ...")
 
(→‎Solution)
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==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
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<cmath>(x_1^2 - x_3x_5)(x_2^2 - x_3x_5) \leq 0 \\
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(x_2^2 - x_4x_1)(x_3^2 - x_4x_1) \leq 0 \\
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(x_3^2 - x_5x_2)(x_4^2 - x_5x_2) \leq 0 \\
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(x_4^2 - x_1x_3)(x_5^2 - x_1x_3) \leq 0 \\
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(x_5^2 - x_2x_4)(x_1^2 - x_2x_4) \leq 0</cmath>
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where <math>x_1, x_2, x_3, x_4, x_5</math> are positive real numbers.
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[[1972 IMO Problems/Problem 4|Solution]]
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==Solution==
 
==Solution==
  

Revision as of 15:34, 17 October 2014

Problem 4

Find all solutions $(x_1, x_2, x_3, x_4, x_5)$ of the system of inequalities \[(x_1^2 - x_3x_5)(x_2^2 - x_3x_5) \leq 0 \\ (x_2^2 - x_4x_1)(x_3^2 - x_4x_1) \leq 0 \\ (x_3^2 - x_5x_2)(x_4^2 - x_5x_2) \leq 0 \\ (x_4^2 - x_1x_3)(x_5^2 - x_1x_3) \leq 0 \\ (x_5^2 - x_2x_4)(x_1^2 - x_2x_4) \leq 0\] where $x_1, x_2, x_3, x_4, x_5$ are positive real numbers.

Solution

Solution

Add the five equations together to get

$(x_1^2 - x_3 x_5)(x_2^2 - x_3 x_5) + (x_2^2 - x_4 x_1)(x_3^2 - x_4 x_1) + (x_3^2 - x_5 x_2)(x_4^2 - x_5 x_2) + (x_4^2 - x_1 x_3)(x_5^2 - x_1 x_3) + (x_5^2 - x_2 x_4)(x_1^2 - x_2 x_4) \leq 0$

Expanding and combining, we get

$(x_1 x_2 - x_1 x_4)^2 + (x_2 x_3 - x_2 x_5)^2 + (x_3 x_4 - x_3 x_1)^2 + (x_4 x_5 - x_4 x_2)^2 + (x_5 x_1 - x_5 x_3)^2 + (x_1 x_3 - x_1 x_5)^2 + (x_2 x_4 - x_2 x_1)^2 + (x_3 x_5 - x_3 x_2)^2 + (x_4 x_1 - x_4 x_3)^2 + (x_5 x_2 - x_5 x_4)^2 \leq 0$

Every term is $\geq 0$, so every term must $= 0$.

From the first term, we can deduce that $x_2 = x_4$. From the second term, $x_3 = x_5$. From the third term, $x_4 = x_1$. From the fourth term, $x_5 = x_2$.

Therefore, $x_1 = x_4 = x_2 = x_5 = x_3$ is the only solution.

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