11. Archimedes already knew that $\pi \approx 3.1416$ can be approximated from above by the fraction $22 / 7 \approx$ 3.1429 and that at least the first two decimal places are correct.

For how many pairs of integers $(m, n)$, with $1<n<100$, does the decimal representation of the fraction $\frac{m}{n}$ start precisely with 3.14?
(A) 14
(B) 21
(C) 48
(D) 50
(E) None of the above
11. The answer is $\mathbf{( C )}$. It is necessary to count the pairs $(m, n)$ such that 314/100 $\leq m / n$, $0<n<100$.

The points that satisfy these conditions form a triangle with vertices $O=(0,0), A=(314,100)$, $B=(315,100)$, including the segment $O A$, but excluding the other two. The points with integer coordinates inside a segment that has one vertex at the origin are the multiples of the coordinate pair of the other vertex, once reduced to their lowest terms. Thus for $O A$ they are $\operatorname{MCD}(314,100)-1=$

1, for $A B$ they are $\operatorname{MCD}(315-314,100-100)-1=0$, for $O B$ they are $\operatorname{MCD}(315,100)-1=4$. For the triangle $O A B$ by Pick's theorem,

$$
a=i+\frac{b}{2}-1
$$

where $a=100 \cdot(315-314) / 2$ is the area of $O A B, i$ is the number of points with integer coordinates inside $O A B$ and $b=3+1+0+4$ is the number of points with integer coordinates on the boundary of $O A B$; therefore $i=47$. The answer is given by the points inside $O A B$ and those inside the segment $O A$, that is $i+1=48$.