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 Post subject: AIME 1983 problem
PostPosted: Fri, 1 Jul 2011 18:18:54 UTC 
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Suppose that the sum of the squares of two complex numbers x and y is 7 and the sum of the cubes is 10. What is the largest real value that x + y can have?
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 Post subject: Re: AIME 1983 problem
PostPosted: Fri, 1 Jul 2011 18:49:56 UTC 
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mun wrote:
Suppose that the sum of the squares of two complex numbers x and y is 7 and the sum of the cubes is 10. What is the largest real value that x + y can have?


There are 6 points of intersection of the plane quadratic x^2+y^2=7 and the plane cubic x^3+y^3=7, and it is not that difficult to find what they are:
(x,y)=\left(2\pm\dfrac{1}{\sqrt{2}}i,2\mp\dfrac{1}{\sqrt{2}}i\right),
\left(\dfrac{-5\pm\sqrt{11}i}{2},\dfrac{-5\mp\sqrt{11}i}{2}\right),
\left(\dfrac{1\pm\sqrt{13}}{2},\dfrac{1\mp\sqrt{13}}{2}\right).
so the answer is...

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\begin{aligned}
Spin(1)&=O(1)=\mathbb{Z}/2&\quad&\text{and}\\
Spin(2)&=U(1)=SO(2)&&\text{are obvious}\\
Spin(3)&=Sp(1)=SU(2)&&\text{by }q\mapsto(\mathop{\mathrm{Im}}\mathbb{H}\ni p\mapsto qp\bar{q})\\
Spin(4)&=Sp(1)\times Sp(1)&&\text{by }(q_1,q_2)\mapsto(\mathbb{H}\ni p\mapsto q_1p\bar{q_2})\\
Spin(5)&=Sp(2)&&\text{by }\mathbb{HP}^1\cong S^4_{round}\hookrightarrow\mathbb{R}^5\\
Spin(6)&=SU(4)&&\text{by the irrep }\Lambda_+\mathbb{C}^4
\end{aligned}


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 Post subject:
PostPosted: Fri, 1 Jul 2011 22:07:05 UTC 
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This belongs in Miscellaneous. Topic moved.

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 Post subject: Re: AIME 1983 problem
PostPosted: Sat, 2 Jul 2011 13:38:27 UTC 
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Hello, mun!

I think I've solved it . . .


Quote:
Suppose that the sum of the squares of two complex numbers x and y is 7,
and the sum of the cubes is 10. .What is the largest real value that x + y can have?

\text{Let }\,x + y \:=\:a .[1]


Square [1]:
. . (x+y)^2 \:=\:a^2 \quad\Rightarrow\quad \underbrace{x^2 + y^2}_{\text{This is 7}} + 2xy \:=\:a^2 \quad\Rightarrow\quad 7 + 2xy \:=\:a^2
. . \text{Hence: }\,xy \:=\:\frac{a^2-7}{2} .[2]


Cube [1]:
. . (x+y)^3 \:=\:a^3 \quad\Rightarrow\quad x^3 + 3x^2y + 3xy^2 + y^3 \:=\:a^3

. . \underbrace{x^3 + y^3}_{\text{This is 10}} + 3\underbrace{(xy)}_{[2]}\underbrace{(x+y)}_{[1]} \:=\:a^2 \quad\Rightarrow\quad 10 + 3\left(\frac{a^2-7}{2}\right)(a) \:=\:a^3

. . a^3 - 21a + 20 \:=\:0 \quad\Rightarrow\quad (a-1)(a-4)(a+5) \:=\:0

. . \text{Hence: }\,a \;=\;-5,\:1,\:4


\text{Therefore, the maximum real value of }x+y\text{ is: }\:a \:=\:4



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