Team Selection Test for IMO 2007

Using the condition $a+b+c=1$ and the inequality $a^2+b^2\ge 2ab$, we get $$\begin{array}{rl}(ab+bc+ca{)}^2& =a^2{b}^2+b^2{c}^2+c^2{a}^2+2a^{2}bc+2b^{2}ca+2c^{2}ab\\ & =a^2{b}^2+(a^2+b^2)c^2+2abc(a+b+c)\\ & \ge a^2{b}^2+2ab{c}^2+2abc.\end{array}$$ Writing this inequality in the form $$\frac{1}{ab+2c^2+2c}\ge \frac{ab}{(ab+bc+ca{)}^2},$$ and adding the inequalities obtained cyclically from this, we obtain the required inequality.