Let 1 be a cube root of unity. Then the minimum of the set { | a + b + c 2 | 2 : a , b , c distinct non-zero integers } equals ______. [2019]

Question

Let $ω \neq 1$ be a cube root of unity. Then the minimum of the set ${| a + b ω + c ω^{2} |^{2} : a, b, c distinct non-zero integers}$ equals ______. [2019]

Smart Achievers · Accepted Answer

(3)

$a, b, c are distinct non-zero integers$

$Min. value of | a + b ω + c ω^{2} |^{2} is to be found$ $| a + b ω + c ω^{2} |^{2}$

$= | a + b (\frac{- 1 + i \sqrt{3}}{2}) + c (\frac{- 1 - i \sqrt{3}}{2}) |^{2}$

$= | \frac{1}{2} (2 a - b - c) + \frac{i \sqrt{3}}{2} (b - c) |^{2}$

$= \frac{1}{4} {(2 a - b - c)}^{2} + \frac{3}{4} {(b - c)}^{2}$

$= \frac{1}{4} (4 a^{2} + b^{2} + c^{2} - 4 a b + 2 b c - 4 a c + 3 b^{2} + 3 c^{2} - 6 b c)$

$= a^{2} + b^{2} + c^{2} - a b - b c - c a$

$= \frac{1}{2} [{(a - b)}^{2} + {(b - c)}^{2} + {(c - a)}^{2}]$

$For minimum value, let us consider a = 3, b = 2, c = 1$

$∴$ $minimum value = \frac{1}{2} [1 + 1 + 4] = \frac{6}{2} = 3$

Solution

Q.
Let $ω \neq 1$ be a cube root of unity. Then the minimum of the set ${| a + b ω + c ω^{2} |^{2} : a, b, c distinct non-zero integers}$ equals ______. [2019]

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Solution

Q. Let ω≠1 be a cube root of unity. Then the minimum of the set {|a+bω+cω2|2:a,b,c distinct non-zero integers} equals ______. [2019]

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Q.
Let $ω \neq 1$ be a cube root of unity. Then the minimum of the set ${| a + b ω + c ω^{2} |^{2} : a, b, c distinct non-zero integers}$ equals ______. [2019]