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Difference between revisions of "1963 IMO Problems/Problem 1"

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==Solution==
 
==Solution==
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Assuming <math>x \geq 0</math>, square the equation, obtaining <math>4\sqrt {(x^2 - p)(x^2 - 1)} = p + 4 - 4x^2</math>. If we have <math>p + 4 \geq 4x^2</math>, we can square again, obtaining <math>x^2 = \frac {(p - 4)^2}{4(4 - 2p)}</math>, or <math>x = \pm\frac {p - 4}{2\sqrt {4 - 2p}}</math>. We must have <math>4 - 2p > 0 \iff p < 2</math>, so we have <math>x = \frac {4 - p}{2\sqrt {4 - 2p}}</math>. However, this is only a solution when <math>p + 4 \geq 4x^2 = \frac {(p - 4)^2}{4 - 2p} \iff (p + 4)(4 - 2p)\leq(p - 4)^2 \iff 0\leq p(3p - 4)</math>, so we have <math>p\leq 0</math> or <math>p \geq \frac {4}{3}</math>. But if <math>p < 0</math>, then <math>\sqrt {x^2 - p} > x</math>, contradiction. So we have <math>x = \frac {4 - p}{2\sqrt {4 - 2p}}</math> for <math>p = 0, \frac {4}{3}\leq p < 2</math>.
+
Assuming <math>x \geq 0</math>, square the equation, obtaining
 +
<cmath>4\sqrt {(x^2 - p)(x^2 - 1)} = p + 4 - 4x^2</cmath>.
 +
If we have <math>p + 4 \geq 4x^2</math>, we can square again, obtaining
 +
<cmath>x^2 = \frac {(p - 4)^2}{4(4 - 2p)}</cmath>, or <cmath>x = \pm\frac {p - 4}{2\sqrt {4 - 2p}}</cmath>
 +
 
 +
We must have <math>4 - 2p > 0 \iff p < 2</math>, so we have
 +
<cmath>x = \frac {4 - p}{2\sqrt {4 - 2p}}</cmath>
 +
 
 +
However, this is only a solution when <cmath>p + 4 \geq 4x^2 = \frac {(p - 4)^2}{4 - 2p} \iff (p + 4)(4 - 2p)\leq(p - 4)^2 \iff 0\leq p(3p - 4)</cmath>,
 +
 
 +
so we have <math>p\leq 0</math> or <math>p \geq \frac {4}{3}</math>.
 +
 
 +
But if <math>p < 0</math>, then <math>\sqrt {x^2 - p} > x</math>, contradiction.
 +
 
 +
So we have <math>x = \frac {4 - p}{2\sqrt {4 - 2p}}</math> for <math>p = 0, \frac {4}{3}\leq p < 2</math>.
  
 
==See Also==
 
==See Also==
  
 
{{IMO box|year=1963|before=First Question|num-a=2}}
 
{{IMO box|year=1963|before=First Question|num-a=2}}

Revision as of 14:40, 15 December 2019

Problem

Find all real roots of the equation

$\sqrt{x^2-p}+2\sqrt{x^2-1}=x$,

where $p$ is a real parameter.

Solution

Assuming $x \geq 0$, square the equation, obtaining \[4\sqrt {(x^2 - p)(x^2 - 1)} = p + 4 - 4x^2\]. If we have $p + 4 \geq 4x^2$, we can square again, obtaining \[x^2 = \frac {(p - 4)^2}{4(4 - 2p)}\], or \[x = \pm\frac {p - 4}{2\sqrt {4 - 2p}}\]

We must have $4 - 2p > 0 \iff p < 2$, so we have \[x = \frac {4 - p}{2\sqrt {4 - 2p}}\]

However, this is only a solution when \[p + 4 \geq 4x^2 = \frac {(p - 4)^2}{4 - 2p} \iff (p + 4)(4 - 2p)\leq(p - 4)^2 \iff 0\leq p(3p - 4)\],

so we have $p\leq 0$ or $p \geq \frac {4}{3}$.

But if $p < 0$, then $\sqrt {x^2 - p} > x$, contradiction.

So we have $x = \frac {4 - p}{2\sqrt {4 - 2p}}$ for $p = 0, \frac {4}{3}\leq p < 2$.

See Also

1963 IMO (Problems) • Resources
Preceded by
First Question 		1 • 2 • 3 • 4 • 5 • 6 Followed by
Problem 2
All IMO Problems and Solutions
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