Difference between revisions of "2004 Pan African MO Problems/Problem 2"

(Created page with "<math>\sqrt{4-2\sqrt{3}} = a\sqrt{3}-b</math>. Through guess and check with small numbers, <math>a = 1</math> and <math>b = 1</math>. So <math>\sqrt{4-2\sqrt{3}} = \sqrt{3}-...")
 
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==Problem==
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Is <math>4\sqrt{4-2\sqrt{3}}+\sqrt{97-56\sqrt{3}}</math> an integer?
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==Solution==
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<math>\sqrt{4-2\sqrt{3}} = a\sqrt{3}-b</math>.  
 
<math>\sqrt{4-2\sqrt{3}} = a\sqrt{3}-b</math>.  
 
Through guess and check with small numbers, <math>a = 1</math> and <math>b = 1</math>.  
 
Through guess and check with small numbers, <math>a = 1</math> and <math>b = 1</math>.  
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So <math>\sqrt{97-56\sqrt{3}} = 7-4\sqrt{3}</math>.
 
So <math>\sqrt{97-56\sqrt{3}} = 7-4\sqrt{3}</math>.
  
Value of <math>4\sqrt{4-2\sqrt{3}} + \sqrt{97-56\sqrt{3}} = (4\sqrt{3}-4) + (7-4\sqrt{3}) = 3</math>
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Value of <math>4\sqrt{4-2\sqrt{3}} + \sqrt{97-56\sqrt{3}} = (4\sqrt{3}-4) + (7-4\sqrt{3}) = 3</math>.
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==See Also==
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{{Pan African MO box|year=2004|num-b=1|num-a=3}}
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[[Category:Introductory Number Theory Problems]]

Latest revision as of 15:54, 24 March 2020

Problem

Is $4\sqrt{4-2\sqrt{3}}+\sqrt{97-56\sqrt{3}}$ an integer?

Solution

$\sqrt{4-2\sqrt{3}} = a\sqrt{3}-b$. Through guess and check with small numbers, $a = 1$ and $b = 1$. So $\sqrt{4-2\sqrt{3}} = \sqrt{3}-1$.

$\sqrt{97-56\sqrt{3}} = a-b\sqrt{3}$. Through prime factorization, $a = 7$ and $b = -4$. So $\sqrt{97-56\sqrt{3}} = 7-4\sqrt{3}$.

Value of $4\sqrt{4-2\sqrt{3}} + \sqrt{97-56\sqrt{3}} = (4\sqrt{3}-4) + (7-4\sqrt{3}) = 3$.

See Also

2004 Pan African MO (Problems)
Preceded by
Problem 1
1 2 3 4 5 6 Followed by
Problem 3
All Pan African MO Problems and Solutions
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