2024 AIME II Problem 11

Find the number of triples of nonnegative integers (a, b, c) satisfying a + b + c = 300 and \[ \begin{aligned} &a^2 b + a^2 c \\ &\quad {}+ b^2 a + b^2 c \\ &\quad {}+ c^2 a + c^2 b = 6{,}000{,}000. \end{aligned} \]

Leading zeroes must be inputted, so if your answer is 34, then input 034

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.