Question

Let $S$ be the set of lattice points in the coordinate plane, both of whose coordinates are integers between $1$ and $30,$ inclusive. Exactly $300$ points in $S$ lie on or below a line with equation $y=mx.$ The possible values of $m$ lie in an interval of length $\frac ab,$ where $a$ and $b$ are relatively prime positive integers. What is $a+b?$

$31$

$47$

$62$

$72$

$85$

Answer: E

Concepts:lattice point floor and ceiling functions

Difficulty rating: 2390

Solution:

For a fixed slope $m,$ the number of points in $S$ on or below $y=mx$ is

$\sum_{x=1}^{30}\lfloor mx\rfloor,$

for the slopes near the answer.

At $m=\frac23,$ grouping $x=3k+1,3k+2,3k+3$ for $k=0,1,\ldots,9$ gives

$\lfloor 2x/3\rfloor=2k,\ 2k+1, \ 2k+2,$

whose sum over each block is $6k+3.$ Thus the total is

$\sum_{k=0}^9(6k+3)=270+30=300.$

If $m<\frac23,$ the ten points with ratios $y/x=2/3$ are no longer counted, so the count is less than $300.$ Therefore the lower end is $\frac23.$

The next possible ratio $y/x$ greater than $\frac23,$ with $1\le x,y\le30,$ is minimized by checking $x$ modulo $3.$ The best candidates are

$\frac{19}{28},\qquad \frac{20}{29},\qquad \frac{21}{30}=\frac{7}{10},$

and the smallest is $\frac{19}{28}.$ Hence the interval length is

$\frac{19}{28}-\frac23=\frac1{84}.$

Thus $a+b=1+84=85.$

Thus, the answer is E .

2021 AMC 10B Spring Problem 25

Answer: E

Difficulty rating: 2390

Solution:

Problem 25 in Other Years