CMO 2022 Problem 6

For integers $0\le a\le n$ , let $f(n,a)$ denote the number of coefficients in the expansion of $(x+1)^a(x+2)^{n-a}$ that is divisible by $3.$ For example, $(x+1)^3(x+2)^1=x^4+5x^3+9x^2+7x+2$ , so $f(4,3)=1$ . For each positive integer $n$ , let $F(n)$ be the minimum of $f(n,0),f(n,1),\ldots ,f(n,n)$ .

(1) Prove that there exist infinitely many positive integer $n$ such that $F(n)\ge \frac{n-1}{3}$ .

(2) Prove that for any positive integer $n$ , $F(n)\le \frac{n-1}{3}$ .