Consider the set of all triangles OPQ where ' | vedclass

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(C) The points $P(x_1, y_1)$ and $Q(x_2, y_2)$ lie on the line $5x + y = 99$. Since $y = 99 - 5x$ and $y \ge 0$,we have $5x \le 99$,so $x \in \{0, 1,

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$90$

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(C) The points $P(x_1, y_1)$ and $Q(x_2, y_2)$ lie on the line $5x + y = 99$. Since $y = 99 - 5x$ and $y \ge 0$,we have $5x \le 99$,so $x \in \{0, 1, 2, \dots, 19\}$. There are $20$ such points.The area of triangle $OPQ$ with vertices $(0,0)$,$(x_1, 99-5x_1)$,and $(x_2, 99-5x_2)$ is given by:$\Delta = \frac{1}{2} |x_1(99-5x_2) - x_2(99-5x_1)|$$\Delta = \frac{1}{2} |99x_1 - 5x_1x_2 - 99x_2 + 5x_1x_2|$$\Delta = \frac{99}{2} |x_1 - x_2|$For the area to be an integer,$|x_1 - x_2|$ must be an even number. 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 with the same parity 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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