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# x, y, z are real numbers for which x + y + z = 1, xa² yb² + zc² = w², a, b, c and w being distinct real numbers.?

w, a, b, c are distinct real numbers and x, y, z are real numbers for which

x + y + z = 1

xa² yb² + zc² = w²

xa³ + yb³ + zc³ = w³

xa⁴ + yb⁴ + zc⁴ =w⁴.

Prove that w = abc/(ab + bc + ca).

Update:

w, a, b, c are distinct real numbers and x, y, z are real numbers for which

x + y + z = 1

xa² yb² + zc² = w²

xa³ + yb³ + zc³ = w³

xa⁴ + yb⁴ + zc⁴ =w⁴.

Prove that w = -abc/(ab + bc + ca).

### 1 Answer

Relevance

- IndicaLv 75 years agoFavourite answer
Let a,b,c be roots of f(μ)=μ³+Aμ²+Bμ+C=0 so a+b+c=−A, ab+bc+ca=B and abc=−C

Setting s=xa+yb+zc and using stated conditions gives

0 = xf(a)+yf(b)+zf(c) = w³+Aw²+Bs+C = f(w)+B(s−w) … (i)

0 = xaf(a)+ybf(b)+zcf(c) = wf(w)+C(s−w) … (ii)

B(ii)−C(i) gives (Bw−C)f(w) = 0

Since w≠a,b,c and so f(w)≠0 you must have w = C/B = −abc/(ab+bc+ca)

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