Докажите неравенство: x^2+y^2≥2*(x+y-1).

Ответ:

\[x² + y² \geq 2 \cdot (x + y - 1)\]

\[x^{2} + y^{2} - 2x - 2y + 2 \geq 0\]

\[\left( x^{2} - 2x + 1 \right) + \left( y^{2} - 2y + 1 \right) \geq 0\]

\[(x - 1)^{2} + (y - 1)^{2} \geq 0\]

\[При\ любом\ значении\ x\ и\ y:\]

\[(x - 1)^{2} + (y - 1)^{2} \geq 0,\ \]

\[так\ \ как\ \ (x - 1)^{2} \geq 0\ \ \ и\ \ \]

\[(y - 1)^{2} \geq 0.\]