Tessellate S.T.E.M.S. (2019) - Mathematics - Category A (School) - Set 2 - Subjective Problem 1

Find all polynomials $f(x,y,z)$ with real coefficients such that $f(a+\frac{1}{a}, b+\frac{1}{b}, c+\frac{1}{c}) = 0$ whenever $abc=1$ .

This problem is a part of Tessellate S.T.E.M.S. (2019)

#Algebra

Easy Math Editor

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Math	Appears as
Remember to wrap math in `$` ... `$` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$

Comments

I believe this should be the set of multiples of the polynomial $p(x,y,z) = x^2+y^2+z^2-xyz-4.$ Not sure about a proof, but I imagine it should be doable since the polynomial is only quadratic in each variable separately...

(I found this polynomial by setting $a = e^q, b = e^r, c = e^s.$ Then the arguments become $2\cosh(q), 2\cosh(r), 2\cosh(s),$ and $abc=1$ is equivalent to $q+r+s=0,$ and then I used the $\cosh$ addition formula on $2\cosh(q+r).$ )

Patrick Corn - 2 years, 5 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$