There are two sequences of natural numbers. The first sequence consists of all natural numbers that do not contain a particular digit, in ascending order. The second sequence consists of all the remaining natural numbers in ascending order. Every natural number occurs exactly once in exactly one of the sequences.
How many terms are there in the first sequence such that each of those terms is greater than the corresponding term at the same position in the second sequence?
How many positive integers n are there such that \(10^{10^{10^{n}}} + 10^{10^{n}} + 10^{n} - 1\) is prime?
Given the number of all positive divisors of an unknown positive integer and the product of those divisors, how can one find the unknown integer?
Akio has one more coin than Bansi. They throw all of their coins and count the number of heads. If all the coins are fair, what is the probability that Akio obtains more heads than Bansi?
Two given positive integers are raised to the same power and added. The power to which they are raised is an integer greater than 1. The sum is a power of 2. What is the maximum possible difference between the two given integers?
