48th Austrian Mathematical Olympiad Regional Competition (Qualifying Round)

W.l.o.g. we may assume that $x_1\ge x_2\ge x_3\ge x_4\ge x_5\ge x_6\ge x_7\ge x_8\ge x_9\ge 0$. Then it follows that $x_1{x}_2\ge x_4^2\ge x_5^2$, $x_1{x}_3\ge x_6^2\ge x_7^2$ and $x_2{x}_3\ge x_8^2\ge x_9^2$. Hence we have $$(x_1+x_2+x_3{)}^2=x_1^2+x_2^2+x_3^2+2x_1{x}_2+2x_1{x}_3+2x_2{x}_3\ge x_1^2+x_2^2+x_3^2+x_4^2+x_5^2+x_6^2+x_7^2+x_8^2+x_9^2\ge 25.$$ Now $x_1+x_2+x_3\ge 5$, which proves the assertion.