Consider the set of all triangles OPQ where O | vedclass

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(C) The points $P$ and $Q$ lie on the line $5x + y = 99$. Since $x, y \ge 0$ and are integers,$x$ can take values from $0$ to $19$ (as $5(19) + y = 99

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$90$

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(C) The points $P$ and $Q$ lie on the line $5x + y = 99$. Since $x, y \ge 0$ and are integers,$x$ can take values from $0$ to $19$ (as $5(19) + y = 99 \implies y = 4$). There are $20$ such points.Let $P = (x_1, 99 - 5x_1)$ and $Q = (x_2, 99 - 5x_2)$.The area of $\Delta OPQ$ is given by $\Delta = \frac{1}{2} |x_1(99 - 5x_2) - x_2(99 - 5x_1)|$.$\Delta = \frac{1}{2} |99x_1 - 5x_1x_2 - 99x_2 + 5x_1x_2| = \frac{99}{2} |x_1 - x_2|$.For the area to be an integer,$|x_1 - x_2|$ must be even.This means $x_1$ and $x_2$ must have the same parity (both even or both odd).There are $10$ even values $(0, 2, \dots, 18)$ and $10$ odd values $(1, 3, \dots, 19)$.The number of ways to choose two distinct points is $^{10}C_2 + ^{10}C_2 = 45 + 45 = 90$.

Consider the set of all triangles $OPQ$ where $O$ is the origin and $P, Q$ are distinct points in the plane with non-negative integral coordinates $(x, y)$ such that $5x + y = 99$. The number of such distinct triangles whose area is a positive integer is:

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