## Factorial and power

What are the possible positive integer values of $$n$$ such that $$(n - 1)! + 1$$ is a power of $$n$$?

[SOLVED]

## Paths in an n-dimensional grid

There is an $$n$$-dimensional grid in an $$n$$-dimensional Euclidean space made of all points with integer coordinates of the form $$(x_1, x_2, \dots, x_n)$$ that satisfy the inequality $$0 \leq x_i \leq a_i$$ where $$i \in \{1, 2, \dots, n\}$$. Every pair of points in the grid that are a unit distance apart are connected by an edge. Through these edges, how many possible shortest paths are there from the point at $$(0, 0, \dots, 0)$$ to the point at $$(a_1, a_2, \dots, a_n$$)?

[SOLVED]

## Divisibility of the largest n-bit integer

What are the possible positive integer values of $$n$$ such that the largest $$n$$-bit integer is a multiple of $$n$$?

[SOLVED]

## Position in a random permutation

In a random permutation of the first $$n$$ natural numbers, what is the probability that $$k$$ appears before all the numbers greater than itself where $$1 \le k \le n$$?

[SOLVED]

## Partitioning an integer and its double

A partition of a positive integer $$n$$ is a list of positive integers, ordered from largest to smallest, such that the sum of the integers in the sequence is $$n$$. Each integer in the list is called a part.

What is the ratio of the number of possible partitions of a positive integer $$n$$ to the number of possible partitions of $$2n$$ into $$n$$ parts?

[SOLVED]