1. By how many zeros does the number \(460^{1500} - 216000^{500}\) end in decimal notation?
2. What is the largest value of \(k\) such that \(2^k\) divides \(23^{2^{2016}} + 7^{2^{2016}}\)?
3. How many pairs \((a, b)\) of strictly positive integers satisfy \(a > b\) and \((a - b)^9 = a^8 - b^8\)?
4. What is the difference between the quadratic mean and the harmonic mean of 1, 10, 100, and 1000?
5. Consider three numbers \(a, b, c > 0\) with an arithmetic mean of 14. Suppose the geometric mean of \(a\) and \(b\) is 6, the geometric mean of \(a\) and \(c\) is 9, and the geometric mean of \(b\) and \(c\) is 18. What is the quadratic mean of the three numbers?
6. Find the number of zeros at the end of \(10^{2000} - 5^{1999}\) in decimal representation.
7. What is the largest integer \(k\) such that \(3^k\) divides \(7^{2^{2015}} + 5^{2^{2015}}\)?
8. How many positive integer solutions \((x, y)\) satisfy \(x^3 - y^3 = 8\)?
9. What is the value of \(\sqrt{a^2 + b^2}\) if \(a\) and \(b\) are such that \(a^2 + b^2 = 10^4\) and \(a \geq b\)?
10. Find the number of positive integers \(n\) such that \(2^n + 1\) is divisible by \(n\).
11. Determine the largest integer \(k\) for which \(4^k\) divides \(5^{2^{2017}} - 3^{2^{2017}}\).
12. How many positive integers \(m\) satisfy \(m^3 - m = 2^k\) for some integer \(k\)?
13. What is the difference between the arithmetic mean and the geometric mean of \(2, 3, 7, 21\)?
14. Find the largest integer \(k\) such that \(5^k\) divides \(6^{2^{2018}} + 4^{2^{2018}}\).
15. Determine the number of distinct solutions to the equation \(x^4 - 4x^2 + 4 = 0\).