2025 SSMO Team Round Problems/Problem 1

Problem

There are $N$ solutions $(a,b,c)$ to $a + b^2 + c = 2025,$ where $a$ and $b$ are positive integers and $c$ is a nonnegative integer. Find the number of positive factors of $N$ .

Solution

Rearrange the given equation as $a+c = 2025-b^2$ and fix $b$ . The possible values of $a$ are $1, 2, \dots, 2025-b^2$ , and the corresponding possible values of $c$ are $2025-b^2-1, 2025-b^2-2, \dots, 0$ . Therefore, there are $2025-b^2$ solutions $(a,c)$ to $a+c = 2025-b^2$ . The possible values of $b$ are $1, 2, \dots, 44$ , so \begin{align*} N &= \sum_{b=1}^{44} (2025 - b^2) \\ &= \sum_{b=1}^{44} 2025 - \sum_{b=1}^{44} b^2 \\ &= 44\cdot 2025 - \tfrac{44\cdot45\cdot89}{6} \\ &= 59730. \end{align*} Since $59730 = 2\cdot 3 \cdot 5\cdot 11 \cdot 181$ , the number of positive factors of $N$ is $2^5 = \boxed{32}$ .

~Sedro

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2025 SSMO Team Round Problems/Problem 1

Problem

Solution