$$\frac{(5^4)^5 : (5^2)^4}{5 \times (5^5)^2} = \frac{5^{4 \times 5} : 5^{2 \times 4}}{5 \times 5^{5 \times 2}} = \frac{5^{20} : 5^8}{5 \times 5^{10}} = \frac{5^{20-8}}{5^{1+10}} = \frac{5^{12}}{5^{11}} = 5^{12-11} = 5^1 = 5$$
Ответ: 5