TopsOJ Summer 2025 - Guts Round - Problem 12

Let \mathcal{S} represent the set of all cows on planet Earth (you may assume that |\mathcal{S}| = 1.5\cdot10^9). Define the \mathcal{S}-\mathcal{C}-generating function as \varphi: \mathcal{S} \rightarrow \oplus, such that \varphi(x) returns a spherical cow (uniform mass density \sigma = 510 \;\mathrm{kg/m}^3 can be assumed). Let \varphi^\prime represent the partial derivative \frac{\partial \varphi}{\partial \oplus}, which represents the rate of morphological distortion per unit of spherical abstraction under heavily aerodynamic conditions. Let \Gamma(x) return the mass of such a spherical cow. Given that the cows exhibit supersonic and transonic flow, with supercritical fluids in their veins and an average volume of 1.00\;\mathrm{m}^3, compute \sum_{\alpha\in\mathcal{S}}\Gamma(\varphi(\alpha)).

*Note: please carefully enter your answer as an integer without commas or scientific notation.*