10.2. What is the smallest positive number that the leading coefficient of a quadratic trinomial $P(x)$, which takes integer values for all integer $x$, can equal?
Answer: $\frac{1}{2}$. Hint. Let $P(x)=a x^{2}+b x+c$. Then $c=P(0)$ is an integer. Substituting $x= \pm 1$, we get that $P(-1)+P(1)=2 a+2 c$. Therefore, $2 a-$ is an integer, and thus $|a| \geq \frac{1}{2}$. The value $a=\frac{1}{2}$ is achieved, for example, for the polynomial $P(x)=\frac{x(x+1)}{2}$.