Difference between revisions of "Mock AIME 2 2006-2007 Problems/Problem 4"
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Two [[complex number]]s are equal if and only if their [[real part]]s and [[imaginary part]]s are equal. Thus if <math>(a + bi)^1 = (a - bi)^1</math> we have <math>b = -b</math> so <math>b = 0</math>, not a positive number. If <math>(a + bi)^2 = (a - bi)^2</math> we have <math>2ab = - 2ab</math> so <math>4ab = 0</math> so <math>a = 0</math> or <math>b = 0</math>, again violating the givens. <math>(a + bi)^3 = (a -bi)^3</math> is equivalent to <math>a^3 - 3ab^2 = a^3 - 3ab^2</math> and <math>3a^2b - b^3 = -3a^2b + b^3</math>, which are true if and only if <math>3a^2b = b^3</math> so either <math>b = 0</math> or <math>3a^2 = b^2</math>. Thus <math>n = 3</math>, where <math>\frac {b^2}{a^2} = 3</math>. | Two [[complex number]]s are equal if and only if their [[real part]]s and [[imaginary part]]s are equal. Thus if <math>(a + bi)^1 = (a - bi)^1</math> we have <math>b = -b</math> so <math>b = 0</math>, not a positive number. If <math>(a + bi)^2 = (a - bi)^2</math> we have <math>2ab = - 2ab</math> so <math>4ab = 0</math> so <math>a = 0</math> or <math>b = 0</math>, again violating the givens. <math>(a + bi)^3 = (a -bi)^3</math> is equivalent to <math>a^3 - 3ab^2 = a^3 - 3ab^2</math> and <math>3a^2b - b^3 = -3a^2b + b^3</math>, which are true if and only if <math>3a^2b = b^3</math> so either <math>b = 0</math> or <math>3a^2 = b^2</math>. Thus <math>n = 3</math>, where <math>\frac {b^2}{a^2} = 3</math>. | ||
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Revision as of 09:45, 16 September 2006
Problem
Revised statement
Let and be positive real numbers and a positive integer such that , where is as small as possible and . Compute .
Original statement
Let be the smallest positive integer for which there exist positive real numbers and such that . Compute .
Solution
Two complex numbers are equal if and only if their real parts and imaginary parts are equal. Thus if we have so , not a positive number. If we have so so or , again violating the givens. is equivalent to and , which are true if and only if so either or . Thus , where .