## CSMC 2020 Part B - Question 1, CEMC UWaterloo

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#### Difficulty: 6

#### This problem is tagged with csmc, csmc20, highschool.

**(Canadian Senior Mathematics Contest 2020, Part B, Question 1, CEMC - UWaterloo)**

a. Determine the point of intersection of the lines with equations $y = 4x - 32$ and $y = -6x + 8$.

b. Suppose that $a$ is an integer. Determine the point of intersection of the lines with equations $y = -x + 3$ and $y = 2x - 3a^2$. (The coordinates of this point will be in terms of $a$.)

c. Suppose that $c$ is an integer. Show that the lines with equations $y = -c^2x + 3$ and $y = x - 3c^2$ intersect at a point with integer coordinates.

d. Determine the four integers $d$ for which the lines with equations $y = dx + 4$ and $y = 2dx + 2$ intersect at a point with integer coordinates.

**(Canadian Senior Mathematics Contest 2020, Part B, Question 1, CEMC - UWaterloo)**

a. Determine the point of intersection of the lines with equations $y = 4x - 32$ and $y = -6x + 8$.

b. Suppose that $a$ is an integer. Determine the point of intersection of the lines with equations $y = -x + 3$ and $y = 2x - 3a^2$. (The coordinates of this point will be in terms of $a$.)

c. Suppose that $c$ is an integer. Show that the lines with equations $y = -c^2x + 3$ and $y = x - 3c^2$ intersect at a point with integer coordinates.

d. Determine the four integers $d$ for which the lines with equations $y = dx + 4$ and $y = 2dx + 2$ intersect at a point with integer coordinates.

*Answer Submission Note(s)*

- For Part a, submit your answer as an ordered pair in the form: $(x,y)$.

- For Part b, submit your answer in terms of $a$, using this as a reference.

- Part c is a proof, so no answer submission is required.

- For Part d, submit the four integer values of $d$ in a sorted comma-separated list, like 1,2,3,4.

All part answers should be separated by a space.

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