MathDB

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}

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