Each square of a chessboard contains a number. The sum of the two largest numbers in each row is the same. The sum of the two largest numbers in each column is also the same. What is the maximum possible difference between the two sums?

What is the largest integer \(n\) such that \(n!\) can not be expressed as the sum of \(n\) distinct factors of itself?

For a given even positive integer, we create two lists of integers such that every positive integer less than or equal to the given integer belongs to exactly one of the two lists and both lists contain an equal number of integers. There is a third list which is initially empty. Then, at every step, we remove the smallest number from the first list, the largest number from the second list and insert the positive difference between the two numbers into the third list.

What do we get if we divide the square of the given even positive integer by the sum of all the numbers in the third list when the first list becomes empty?

For a given perfect square, there is a set of five integers such that the sum of the squares of the integers in the set is ten times the given perfect square and the smallest positive difference between any pair of integers in the set is maximum. What do we get when we subtract the square of this difference from the given perfect square?

There is an infinite grid of numbers. The first row contains all the natural numbers in ascending order. Any other number in this grid is the sum of those numbers from the row above it which do not exceed the number above it.

Given two positive integers n and k such that n ≥ k, where in the grid can we find the binomial coefficient C(n, k)?

There is a 200-storey building. You are given 5 identical glass marbles. You are allowed to drop any marble from any floor to the ground. A marble either breaks or remains intact after a drop. If it remains intact, the marble can be reused.

In the worst case, what is the minimum number of drops needed to find the highest floor in the building from which you can drop the marbles without breaking them?

Albert is opening a new library. He wants to keep a special rack of books facing the entrance to give the visitors a glimpse of the kinds of books it has. This special rack can hold only 10 books at a time. Every time a new book arrives at the library, he needs to decide whether to add it to the special rack or just place it in one of the regular racks in the library. A book is removed from this rack only if Albert decides to add a new book to it.

He wants to ensure that at any point in time, each book that has ever arrived at the library will be equally likely to be on the special rack. How is he going to do this?

If a_n represents the largest n-bit integer for a positive integer n, how many bits are 1 in a₁ + a₂ + … + a_1000000?

There is an empty list initially. First, 2 is inserted into the list. Then, at every step, every number in the list is replaced with (2 + √n) and (2 − √n) where n is the number being replaced.

What is the product of all numbers in the list when this step has been performed 1000 times?