49th Mathematical Olympiad in Ukraine

Let $f(x)=(x-\alpha )(x-\beta )(x-\gamma )$ be a polynomial which satisfies the necessary conditions. Then $\alpha ,\beta ,\gamma$ are integer numbers and $|f(34)|=|(34-\alpha )(34-\beta )(34-\gamma )|$ is a prime number. Without loss of generality we have $|34-\alpha |=|34-\beta |=1$ and $|34-\gamma |$ is a prime, it follows $\alpha ,\beta \ge 33$. The nearest prime numbers for $34$ are $31$ and $37$ then $|\gamma |\ge 3$. Therefore $|c|=|\alpha \beta \gamma |\ge 33\times 33\times 3=3267>2009$. A contradiction.