Sa se determine \( a,b,c\in (0,\infty) \) stiind ca :
\( \left{\begin{array}{cc}a^7=9b^4-8c\\b^7=9c^4-8a\\c^7=9a^4-8b\end{array} \)
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Deoarece sistemul este simetric in \( a,b,c \) putem presupune ca \( a=\max \{a,b,c} \). Deoarece \( 8a=9c^{4}-b^{7} \) si \( 8b=9a^{4}-c^{7} \) avem ca:
\( a\ge b\Longrightarrow9c^{4}-b^{7}\ge9a^{4}-c^{7}(1) \)
Dar in baza presupunerii facute avem:
\( 9c^{4}\le9a^{4},-b^{7}\le-c^{7}\Longrightarrow9c^{4}-b^{7}\le9a^{4}-c^{7}(2) \).
Din \( (1) \) si \( (2) \) obtinem ca \( 9c^{4}-b^{7}=9a^{4}-c^{7}\Longrightarrow a=b \).
Problema se reduce acum la rezolvarea urmatorului sistem:
\( \begin{cases}
a^{7}=9b^{4}-8c\\
b^{7}=9c^{4}-8a\end{cases}\Longleftrightarrow\begin{cases}
a^{7}=9a^{4}-8c\\
a^{7}=9c^{4}-8a\end{cases}\Longleftrightarrow9a^{4}-8c=9c^{4}-8a \)
Sa consideram functia \( f\left(x\right)=9x^{4}+8x \). Deaorece \( f(x) \) este strict crescatoare. Deducem ca \( f(x) \) este injectiva, deci pentru orice \( x \neq y \) avem ca \( f(x) \neq f(y). \)
Revenind la problema. Daca \( a\neq c \) atunci \( f(a) \neq f(c) \), deci \( 9a^{4}+8a\neq9c^{4}+8c \), situatie care evident nu convine. Obtinem deci ca \( a=b=c \).
Mai ramane de rezolvat ecuatia \( a^{7}-9a^{4}+8a=0\Longleftrightarrow a^{4}\left(a^{3}-1\right)-8a\left(a^{3}-1\right)=0\Longleftrightarrow \left(a^{3}-1\right)\left(a^{3}-8\right)=0 \) care are solutiile \( a\in\left\{ 1,2\right\} \).
Solutiile problemei sunt deci \( a=b=c=1,a=b=c=2 \).
\( a\ge b\Longrightarrow9c^{4}-b^{7}\ge9a^{4}-c^{7}(1) \)
Dar in baza presupunerii facute avem:
\( 9c^{4}\le9a^{4},-b^{7}\le-c^{7}\Longrightarrow9c^{4}-b^{7}\le9a^{4}-c^{7}(2) \).
Din \( (1) \) si \( (2) \) obtinem ca \( 9c^{4}-b^{7}=9a^{4}-c^{7}\Longrightarrow a=b \).
Problema se reduce acum la rezolvarea urmatorului sistem:
\( \begin{cases}
a^{7}=9b^{4}-8c\\
b^{7}=9c^{4}-8a\end{cases}\Longleftrightarrow\begin{cases}
a^{7}=9a^{4}-8c\\
a^{7}=9c^{4}-8a\end{cases}\Longleftrightarrow9a^{4}-8c=9c^{4}-8a \)
Sa consideram functia \( f\left(x\right)=9x^{4}+8x \). Deaorece \( f(x) \) este strict crescatoare. Deducem ca \( f(x) \) este injectiva, deci pentru orice \( x \neq y \) avem ca \( f(x) \neq f(y). \)
Revenind la problema. Daca \( a\neq c \) atunci \( f(a) \neq f(c) \), deci \( 9a^{4}+8a\neq9c^{4}+8c \), situatie care evident nu convine. Obtinem deci ca \( a=b=c \).
Mai ramane de rezolvat ecuatia \( a^{7}-9a^{4}+8a=0\Longleftrightarrow a^{4}\left(a^{3}-1\right)-8a\left(a^{3}-1\right)=0\Longleftrightarrow \left(a^{3}-1\right)\left(a^{3}-8\right)=0 \) care are solutiile \( a\in\left\{ 1,2\right\} \).
Solutiile problemei sunt deci \( a=b=c=1,a=b=c=2 \).
elev, clasa a X-a, C. N. "C-tin Carabella", Targoviste