Question Number 20430 by Tinkutara last updated on 26/Aug/17
$$\mathrm{Let}\:{a},\:{b}\:\mathrm{and}\:{c}\:\mathrm{be}\:\mathrm{such}\:\mathrm{that}\:{a}\:+\:{b}\:+\:{c}\:=\:\mathrm{0} \\ $$$$\mathrm{and} \\ $$$${P}\:=\:\frac{{a}^{\mathrm{2}} }{\mathrm{2}{a}^{\mathrm{2}} \:+\:{bc}}\:+\:\frac{{b}^{\mathrm{2}} }{\mathrm{2}{b}^{\mathrm{2}} \:+\:{ca}}\:+\:\frac{{c}^{\mathrm{2}} }{\mathrm{2}{c}^{\mathrm{2}} \:+\:{ab}} \\ $$$$\mathrm{is}\:\mathrm{defined}.\:\mathrm{What}\:\mathrm{is}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:{P}? \\ $$
Commented by dioph last updated on 27/Aug/17
$${P}\:=\:\frac{\mathrm{12}{a}^{\mathrm{2}} {b}^{\mathrm{2}} {c}^{\mathrm{2}} +\mathrm{4}{a}^{\mathrm{3}} {b}^{\mathrm{3}} +\mathrm{4}{a}^{\mathrm{3}} {c}^{\mathrm{3}} +\mathrm{4}{b}^{\mathrm{3}} {c}^{\mathrm{3}} +{a}^{\mathrm{4}} {bc}+{ab}^{\mathrm{4}} {c}+{abc}^{\mathrm{4}} }{\mathrm{9}{a}^{\mathrm{2}} {b}^{\mathrm{2}} {c}^{\mathrm{2}} +\mathrm{4}{a}^{\mathrm{3}} {b}^{\mathrm{3}} +\mathrm{4}{a}^{\mathrm{3}} {c}^{\mathrm{3}} +\mathrm{4}{b}^{\mathrm{3}} {c}^{\mathrm{3}} +\mathrm{2}{a}^{\mathrm{4}} {bc}+\mathrm{2}{ab}^{\mathrm{4}} {c}+\mathrm{2}{abc}^{\mathrm{4}} } \\ $$$$\left({a}+{b}+{c}\right)^{\mathrm{3}} ={a}^{\mathrm{3}} +{b}^{\mathrm{3}} +{c}^{\mathrm{3}} +\mathrm{3}{a}^{\mathrm{2}} {b}+\mathrm{3}{ab}^{\mathrm{2}} +\mathrm{3}{ac}^{\mathrm{2}} +\mathrm{3}{bc}^{\mathrm{2}} +\mathrm{3}{a}^{\mathrm{2}} {c}+\mathrm{3}{b}^{\mathrm{2}} {c}+\mathrm{6}{abc} \\ $$$${a}^{\mathrm{3}} +{b}^{\mathrm{3}} +{c}^{\mathrm{3}} +\mathrm{6}{abc}=−\mathrm{3}{ab}\left({a}+{b}\right)−\mathrm{3}{ac}\left({a}+{c}\right)−\mathrm{3}{bc}\left({b}+{c}\right) \\ $$$${a}^{\mathrm{3}} +{b}^{\mathrm{3}} +{c}^{\mathrm{3}} +\mathrm{6}{abc}=\mathrm{3}{abc}+\mathrm{3}{abc}+\mathrm{3}{abc} \\ $$$${a}^{\mathrm{3}} +{b}^{\mathrm{3}} +{c}^{\mathrm{3}} +\mathrm{6}{abc}=\mathrm{9}{abc} \\ $$$${a}^{\mathrm{3}} +{b}^{\mathrm{3}} +{c}^{\mathrm{3}} =\mathrm{3}{abc}\:\left(\ast\right) \\ $$$${P}\:=\:\frac{\mathrm{4}\left({a}^{\mathrm{3}} {b}^{\mathrm{3}} +{a}^{\mathrm{3}} {c}^{\mathrm{3}} +{b}^{\mathrm{3}} {c}^{\mathrm{3}} \right)+{abc}\left({a}^{\mathrm{3}} +{b}^{\mathrm{3}} +{c}^{\mathrm{3}} +\mathrm{12}{abc}\right)}{\mathrm{4}\left({a}^{\mathrm{3}} {b}^{\mathrm{3}} +{a}^{\mathrm{3}} {c}^{\mathrm{3}} +{b}^{\mathrm{3}} {c}^{\mathrm{3}} \right)+{abc}\left(\mathrm{2}{a}^{\mathrm{3}} +\mathrm{2}{b}^{\mathrm{3}} +\mathrm{2}{c}^{\mathrm{3}} +\mathrm{9}{abc}\right)} \\ $$$${P}\:=\:\frac{\mathrm{4}\left({a}^{\mathrm{3}} {b}^{\mathrm{3}} +{a}^{\mathrm{3}} {c}^{\mathrm{3}} +{b}^{\mathrm{3}} {c}^{\mathrm{3}} \right)+{abc}\left(\mathrm{15}{abc}\right)}{\mathrm{4}\left({a}^{\mathrm{3}} {b}^{\mathrm{3}} +{a}^{\mathrm{3}} {c}^{\mathrm{3}} +{b}^{\mathrm{3}} {c}^{\mathrm{3}} \right)+{abc}\left(\mathrm{15}{abc}\right)} \\ $$$${P}\:=\:\mathrm{1} \\ $$
Answered by Tinkutara last updated on 27/Aug/17
$$\mathrm{Since}\:{a}\:+\:{b}\:+\:{c}\:=\:\mathrm{0}, \\ $$$$\therefore\:{a}\:=\:−\:{b}\:−\:{c}\:\Rightarrow\:{a}^{\mathrm{2}} \:=\:−\:{ab}\:−\:{ac} \\ $$$$\mathrm{2}{a}^{\mathrm{2}} \:+\:{bc}\:=\:{a}^{\mathrm{2}} \:−\:{ab}\:−\:{ac}\:+\:{bc}\:=\:\left({a}\:−\:{b}\right)\left({a}\:−\:{c}\right) \\ $$$$\mathrm{Similarly},\:\mathrm{2}{b}^{\mathrm{2}} \:+\:{ca}\:=\:\left({b}\:−\:{a}\right)\left({b}\:−\:{c}\right), \\ $$$$\mathrm{2}{c}^{\mathrm{2}} \:+\:{ab}\:=\:\left({c}\:−\:{a}\right)\left({c}\:−\:{b}\right) \\ $$$$\mathrm{Hence}, \\ $$$${P}\:=\:\frac{−{a}^{\mathrm{2}} }{\left({a}\:−\:{b}\right)\left({c}\:−\:{a}\right)}\:+\:\frac{−{b}^{\mathrm{2}} }{\left({a}\:−\:{b}\right)\left({b}\:−\:{c}\right)}\:+\:\frac{−{c}^{\mathrm{2}} }{\left({c}\:−\:{a}\right)\left({b}\:−\:{c}\right)} \\ $$$$=\:\frac{−\underset{{cyc}} {\sum}{a}^{\mathrm{2}} \left({b}\:−\:{c}\right)}{\left({a}\:−\:{b}\right)\left({b}\:−\:{c}\right)\left({c}\:−\:{a}\right)} \\ $$$$\mathrm{After}\:\mathrm{expanding}\:\mathrm{the}\:\mathrm{denominator},\:\mathrm{we}\:\mathrm{will}\:\mathrm{find}\:\mathrm{this} \\ $$$$\mathrm{value}\:\mathrm{just}\:\mathrm{the}\:\mathrm{same}\:\mathrm{as}\:\mathrm{numerator}. \\ $$$$\mathrm{So},\:{P}\:=\:\mathrm{1}. \\ $$
Answered by ajfour last updated on 27/Aug/17
$${If}\:{a}+{b}+{c}=\mathrm{0}, \\ $$$${Let}\:{a}={pc},\:\:{b}={qc}\: \\ $$$${a}+{b}=−{c}\:\Rightarrow\:\:\:{p}+{q}=−\mathrm{1} \\ $$$${P}=\frac{{p}^{\mathrm{2}} {c}^{\mathrm{2}} }{\mathrm{2}{p}^{\mathrm{2}} {c}^{\mathrm{2}} +{qc}^{\mathrm{2}} }+\frac{{q}^{\mathrm{2}} {c}^{\mathrm{2}} }{\mathrm{2}{q}^{\mathrm{2}} {c}^{\mathrm{2}} +{pc}^{\mathrm{2}} }+\frac{{c}^{\mathrm{2}} }{\mathrm{2}{c}^{\mathrm{2}} +{pqc}^{\mathrm{2}} } \\ $$$$\:=\frac{{p}^{\mathrm{2}} }{\mathrm{2}{p}^{\mathrm{2}} +{q}}+\frac{{q}^{\mathrm{2}} }{\mathrm{2}{q}^{\mathrm{2}} +{p}}+\frac{\mathrm{1}}{\mathrm{2}+{pq}} \\ $$$$\:=\frac{\mathrm{4}{p}^{\mathrm{2}} {q}^{\mathrm{2}} +{p}^{\mathrm{3}} +{q}^{\mathrm{3}} }{\mathrm{4}{p}^{\mathrm{2}} {q}^{\mathrm{2}} +\mathrm{2}\left({p}^{\mathrm{3}} +{q}^{\mathrm{3}} \right)+{pq}}+\frac{\mathrm{1}}{\mathrm{2}+{pq}} \\ $$$$\:=\frac{\mathrm{4}{p}^{\mathrm{2}} {q}^{\mathrm{2}} −\left({p}^{\mathrm{2}} +{q}^{\mathrm{2}} −{pq}\right)}{\mathrm{4}{p}^{\mathrm{2}} {q}^{\mathrm{2}} −\mathrm{2}\left({p}^{\mathrm{2}} +{q}^{\mathrm{2}} −{pq}\right)+{pq}}+\frac{\mathrm{1}}{\mathrm{2}+{pq}} \\ $$$$\:=\frac{\mathrm{4}{p}^{\mathrm{2}} {q}^{\mathrm{2}} −\left(\mathrm{1}−\mathrm{3}{pq}\right)}{\mathrm{4}{p}^{\mathrm{2}} {q}^{\mathrm{2}} −\mathrm{2}\left(\mathrm{1}−\mathrm{3}{pq}\right)+{pq}}+\frac{\mathrm{1}}{\mathrm{2}+{pq}} \\ $$$$\:{let}\:\:\:\:\boldsymbol{{pq}}=\boldsymbol{{r}} \\ $$$$\:\boldsymbol{{P}}\:=\:\frac{\mathrm{4}{r}^{\mathrm{2}} +\mathrm{3}{r}−\mathrm{1}}{\mathrm{4}{r}^{\mathrm{2}} +\mathrm{7}{r}−\mathrm{2}}+\frac{\mathrm{1}}{\mathrm{2}+{r}} \\ $$$$\:\:\:\:\:=\mathrm{1}+\frac{\left(\mathrm{1}−\mathrm{4}{r}\right)}{\mathrm{4}{r}^{\mathrm{2}} +\mathrm{7}{r}−\mathrm{2}}+\frac{\mathrm{1}}{\mathrm{2}+{r}} \\ $$$$\:\:\:\:\:=\mathrm{1}+\frac{\left(\mathrm{2}−\mathrm{7}{r}−\mathrm{4}{r}^{\mathrm{2}} \right)+\left(\mathrm{4}{r}^{\mathrm{2}} +\mathrm{7}{r}−\mathrm{2}\right)}{\left(\mathrm{4}{r}^{\mathrm{2}} +\mathrm{7}{r}−\mathrm{2}\right)\left(\mathrm{2}+{r}\right)} \\ $$$$\Rightarrow\:\boldsymbol{{P}}\:=\mathrm{1}+\mathrm{0}\:. \\ $$
Commented by ajfour last updated on 27/Aug/17
$${good}\:{question},\:{thanks}. \\ $$
Commented by Tinkutara last updated on 27/Aug/17
$$\mathrm{Thank}\:\mathrm{you}\:\mathrm{very}\:\mathrm{much}\:\mathrm{Sir}! \\ $$