## CSMC 2023 Part B - Question 2, CEMC UWaterloo

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#### Difficulty: 8

#### This problem is tagged with csmc, csmc23, highschool.

**(Canadian Senior Mathematics Contest 2023, Part B, Question 2, CEMC - UWaterloo)**

The rectangular region $A$ is in the first quadrant. The bottom side and top side of $A$ are formed by the lines with equations $y = 0.5$ and $y = 99.5$, respectively. The left
side and right side of $A$ are formed by the lines with equations $x = 0.5$ and $x = 99.5$, respectively.

a) Determine the number of lattice points that are on the line with equation $y = 2x + 5$ and are inside the region $A$. (A point with coordinates $(r, s)$ is called a lattice point if $r$ and $s$ are both integers.)

b) For some integer $b$, the number of lattice points on the line with equation $y=\frac{5}{3}x+b$ and inside the region $A$ is at least $15$. Determine the greatest possible value of $b$.

c) For some real numbers $m$, there are no lattice points that lie on the line with equation $y = mx + 1$ and inside the region $A$. Determine the greatest possible real number $n$ that has the property that, for all real numbers $m$ with
$\frac{2}{7} \lt m \lt n$, there are no lattice points on the line with equation $y = mx + 1$ and inside the region $A$.

*Answer Submission Note(s)*

The answer for the first part should be a single integer, the answer for the second part should be a single integer, and the third part's answer should be a fraction of the form "x/y". Separate part answers with commas (no spaces).

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