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

Latest revision as of 16:54, 24 March 2020

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

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

@@ Line 1: / Line 1: @@
+==Problem==
+Is <math>4\sqrt{4-2\sqrt{3}}+\sqrt{97-56\sqrt{3}}</math> an integer?
+==Solution==
 <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>.
@@ Line 7: / Line 13: @@
 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>
+Value of <math>4\sqrt{4-2\sqrt{3}} + \sqrt{97-56\sqrt{3}} = (4\sqrt{3}-4) + (7-4\sqrt{3}) = 3</math>.
+==See Also==
+{{Pan African MO box|year=2004|num-b=1|num-a=3}}
+[[Category:Introductory Number Theory Problems]]

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