Marinada '22

[lang=en] Primjer 1. Dokažite da iz $a + b + c = 0$ slijedi: $a^4 + b^4 + c^4 = 2(a^2b^2 + b^2c^2 + c^2a^2)$.\\ \\ RJEŠENJE. Za početak potrebno je prvu jednakost kvadrirati; \begin{align*} a + b + c &= 0 /()^2\\ a^2 + b^2 + c^2 + 2ab + 2bc + 2ac &= 0 \end{align*} Zatim na desnoj strani ostavimo samo kvadrate kako bismo jednakost mogli opet kvadrirati; \begin{align*} a^2 + b^2 + c^2 &= -2ab - 2bc - 2ac /()^2\\ a^4 + b^4 + c^4 + 2a^2b^2 + 2b^2c^2 + 2a^2c^2 &= 4(a^2b^2 + a^2c^2 + b^2c^2 + 2a^2bc + 2ab^2c + 2abc^2)\\ a^4 + b^4 + c^4 &= 2(a^2b^2 + a^2c^2+b^2c^2) + 4 \cdot 2abc(a + b + c)\\ a^4 + b^4 + c^4 &= 2(a^2b^2 + a^2c^2 + b^2c^2) + 8abc(a + b + c)\\ a^4 + b^4 + c^4 &= 2(a^2b^2 + a^2c^2 + b^2c^2) \\ \\ \end{align*} *Kako biste dobili 1 bod unesite 4 kao rješenje. [/lang] [lang=hr] Example 1. Proof that given $a + b + c = 0$ it holds: $a^4 + b^4 + c^4 = 2(a^2b^2 + b^2c^2 + c^2a^2)$.\\ \\ SOLUTION. For start you should square first equality; \begin{align*} a + b + c &= 0 /()^2\\ a^2 + b^2 + c^2 + 2ab + 2bc + 2ac &= 0 \end{align*} Then, on right side we just leave the squares so that we could square equality once again; \begin{align*} a^2 + b^2 + c^2 &= -2ab - 2bc - 2ac /()^2\\ a^4 + b^4 + c^4 + 2a^2b^2 + 2b^2c^2 + 2a^2c^2 &= 4(a^2b^2 + a^2c^2 + b^2c^2 + 2a^2bc + 2ab^2c + 2abc^2)\\ a^4 + b^4 + c^4 &= 2(a^2b^2 + a^2c^2+b^2c^2) + 4 \cdot 2abc(a + b + c)\\ a^4 + b^4 + c^4 &= 2(a^2b^2 + a^2c^2 + b^2c^2) + 8abc(a + b + c)\\ a^4 + b^4 + c^4 &= 2(a^2b^2 + a^2c^2 + b^2c^2) \\ \\ \end{align*} *To get 1 point, you should put 4 in the answer box. [/lang]