<s>[INST] how to prove $\sum_{cyc}(1-a_{1}+a_{1}a_{2})^2\ge\frac{n}{2}$ let $a_{i}\in [0,1]$,prove or disprove
$$f_{n}=(1-a_{1}+a_{1}a_{2})^2+(1-a_{2}+a_{2}a_{3})^2+\cdots+(1-a_{n}+a_{n}a_{1})^2\ge\dfrac{n}{2}$$
I can only prove $n=3$.
$$f_{3}=\sum_{cyc}[1-a_{1}(1-a_{2})]^2=3-2\sum_{cyc}a_{1}(1-a_{2})+\sum_{cyc}a^2_{1}(1-a_{2})^2$$
let $$t=a_{1}(1-a_{2})+a_{2}(1-a_{3})+a_{3}(1-a_{1})=1-(1-a_{1})(1-a_{2})(1-a_{3})-a_{1}a_{2}a_{3}\le 1$$
and
$$\sum_{cyc}a^2_{1}(1-a_{2})^2=(\sum_{cyc}a_{1}(1-a_{2}))^2-2\sum_{cyc}a_{1}a_{2}(1-a_{1})(1-a_{3})\ge t^2-\dfrac{1}{4}\sum_{cyc}a_{2}(1-a_{3})=t^2-\dfrac{1}{4}t$$
so we have
$$f_{3}\ge 3-2t+t^2-\dfrac{1}{4}t\ge\dfrac{3}{2}$$
But for $n$ I can't,maybe use induction to prove it?
 [/INST] I would say that you can not prove it using induction.
Firstly, let us rewrite $f(n)$ as $$f_{n}=(1+a_{1}(a_{2}-1))^2+(1+a_{2}(a_{3}-1))^2+\cdots+(1+a_{n}(a_{1}-1))^2$$
By inductive hypothesis, $$(1+a_{1}(a_{2}-1))^2+(1+a_{2}(a_{3}-1))^2+\cdots+(1+a_{n}(a_{1}-1))^2\ge\dfrac{n}{2}$$
Therefore, we have that $$(1+a_{1}(a_{2}-1))^2+(1+a_{2}(a_{3}-1))^2+\cdots+(1+a_{n}(a_{n+1}-1))^2+(1+a_{n+1}(a_{1}-1))^2\geq$$
$$\dfrac{n}{2}-(1+a_{n}(a_{1}-1))^2+(1+a_{n}(a_{n+1}-1))^2+(1+a_{n+1}(a_{1}-1))^2$$
We have to prove that $$(1+a_{n}(a_{n+1}-1))^2+(1+a_{n+1}(a_{1}-1))^2-(1+a_{n}(a_{1}-1))^2\geq\frac{1}{2}$$
Unfortunately, this inequality is false (take for instance $a_n=0$, $a_{n+1}=1$, and $a_1$ whatever value you want lesser than $\frac{3}{4}$).
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