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 $\dfrac{a}{b},$ where $a$ and $b$ are relatively prime positive integers. What is $a+b?$

$31$

$47$

$62$

$72$

$85$

Solution:

For slope $m,$ column $x$ (with $1\le x\le 30$ ) contributes $\min(30,\lfloor mx\rfloor)$ points on or below $y=mx,$ and we need the total to equal $300.$

The count is a step function of $m$ that jumps at fractions $\tfrac{y}{x}.$ Sweeping through these breakpoints, the count equals $300$ for $m$ in a single interval whose endpoints are consecutive such slopes.

That interval runs from $m=\dfrac{2}{3}$ up to $m=\dfrac{19}{28},$ of length $\dfrac{19}{28}-\dfrac{2}{3}=\dfrac{57-56}{84}=\dfrac{1}{84}.$

Since $\gcd(1,84)=1,$ $a+b=1+84=85.$

Thus, the correct answer is E.

2021 AMC 12B Spring Problem 25

Solution:

Problem 25 in Other Years