1980 USAMO Problems/Problem 3

Problem

$A + B + C$ is an integral multiple of $\pi$ . $x, y,$ and $z$ are real numbers. If $x\sin(A)+y\sin(B)+z\sin(C)=x^2\sin(2A)+y^2\sin(2B)+z^2\sin(2C)=0$ , show that $x^n\sin(nA)+y^n \sin(nB) +z^n \sin(nC)=0$ for any positive integer $n$ .

Solution

Let $a=xe^{iA}$ , $b=ye^{iB}$ , $c=ze^{iC}$ be numbers in the complex plane.

Note that $A+B+C=k\pi$ implies $abc=xyz(e^{ik\pi})=\pm xyz$ which is real. Also note that $x\sin(A), y\sin(B), z\sin(C)$ are the imaginary parts of $a, b, c$ and that $x^2\sin(2A), y^2\sin(2B), z^2\sin(2C)$ are the imaginary parts of $a^2, b^2, c^2$ by de Moivre's Theorem. Therefore, $a+b+c$ and $a^2+b^2+c^2$ are real because their imaginary parts sum to zero.

Finally, note that $\frac{1}{2}\left((a+b+c)^2-(a^2+b^2+c^2)\right)=ab+bc+ac$ is real as well.

It suffices to show that $P_n=a^n+b^n+c^n$ is real for all positive integer $n$ , which can be shown by induction.

Newton Sums gives the following relationship between sums of the form $P_k=a^k+b^k+c^k$ $P_k+S_1P_{k-1}+S_2P_{k-2}+S_3P_{k-3}=0$ Where $S_1=a+b+c$ , $S_2=ab+bc+ac$ , and $S_3=abc$ . It is given that $P_0, P_1, P_2$ are real. Note that if $P_{k-1}, P_{k-2}, P_{k-3}$ are real, then clearly $P_k$ is real because all other parts of the above equation are real, completing the induction.

1980 USAMO (Problems • Resources)
Preceded by Problem 2	Followed by Problem 4
1 • 2 • 3 • 4 • 5
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1980 USAMO Problems/Problem 3

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