isi-entrance 2019 Q7

isi-entrance · India · UGB Number Theory Congruence Reasoning and Parity Arguments

Let $f$ be a polynomial with integer coefficients. Define $$a_{1} = f(0),\quad a_{2} = f(a_{1}) = f(f(0)),$$ and $$a_{n} = f(a_{n-1}) \quad \text{for } n \geq 3$$ If there exists a natural number $k \geq 3$ such that $a_{k} = 0$, then prove that either $a_{1} = 0$ or $a_{2} = 0$.

Let $f$ be a polynomial with integer coefficients. Define
$$a_{1} = f(0),\quad a_{2} = f(a_{1}) = f(f(0)),$$
and
$$a_{n} = f(a_{n-1}) \quad \text{for } n \geq 3$$
If there exists a natural number $k \geq 3$ such that $a_{k} = 0$, then prove that either $a_{1} = 0$ or $a_{2} = 0$.

Paper Questions

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