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- Feb 14, 2012

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Let $a<b<c$ be real numbers such that $a+b+c=6$ and $ab+bc+ca=9$. Prove that $0<a<1<b<3<c<4$.

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Remember to read the POTW submission guidelines to find out how to submit your answers!

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- Thread starter
- Admin
- #1

- Feb 14, 2012

- 3,894

-----

Let $a<b<c$ be real numbers such that $a+b+c=6$ and $ab+bc+ca=9$. Prove that $0<a<1<b<3<c<4$.

-----

Remember to read the POTW submission guidelines to find out how to submit your answers!

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- #2

- Feb 14, 2012

- 3,894

Just to let you all know that I will extend the deadline to solve last week's POTW (which actually due today) to next week, with the great hope to receive any submission of solution from the members!

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- Feb 14, 2012

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The cubic equation can be rewritten as $x(x-3)^2=k$.

Plot the graph of $f(x)=x(x-3)^2$ and the horizontal line of $y=k$ with condition $0<k<4$ on the same diagram. Since the roots are real, the line $y=k$ can only be moved in such a way that it always cuts the curve at 3 distinct points in such a way that $0<a<1<b<3<c<4$.

\begin{tikzpicture}

[scale=3]

\draw[help lines] (0,0) grid (5,4);

\draw[thick,->] (0,0) -- (4.5,0) node[anchor=north west] {x axis};

\draw[thick,->] (0,0) -- (0,4.5) node[anchor=south east] {y axis};

\draw[ domain=-0.2:4, samples=100] plot (\x,\x^3 - 6*\x^2 +9*\x);

\draw (0,3.2) -- (4,3.2);

\foreach \x in {0,1,2,3,4}

\draw (\x cm,1pt) -- (\x cm,-1pt) node[anchor=north] {$\x$};

\foreach \y in {0,1,2,3,4}

\draw (1pt,\y cm) -- (-1pt,\y cm) node[anchor=east] {$\y$};

\node at (3.6,2) {\large $y=f(x)$};

\node at (4.2,3.2) {\large $y=k$};

\draw[ densely dashed,color=blue] (0,4) -- (4,4);

\draw[ densely dashed,color=blue] (4,0) -- (4,4);

\draw[ densely dashed,color=blue] (0.52,0) -- (0.52,3.2);

\draw[ densely dashed,color=blue] (1,0) -- (1,3.2);

\draw[ densely dashed,color=blue] (1.58,0) -- (1.58,3.2);

\draw[ densely dashed,color=blue] (3.9,0) -- (3.9,3.2);

\node at (0.52,-0.08) {$a$};

\node at (1.58,-0.08) {$b$};

\node at (3.9,-0.08) {$c$};

\end{tikzpicture}

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