((n!)!)!

Find the largest number n such that (2004!)! is divisible by ((n!)!)!

the sum of the ten numbers

Ten positive integers are arranged around a circle. Each number is one more than the
greatest common divisor of its two neighbors.
What is the sum of the ten numbers?

the moth achieve its objective

 A moth starts at vertex A of a certain cube and is trying to get to vertex B, which is
opposite A, in five or fewer “steps,” where a step consists in traveling along an edge
from one vertex to another. The moth will stop as soon as it reaches B.
How many ways can the moth achieve its objective?

f(x) = f '(x)f "(x)

A non-zero polynomial f(x) with real coefficients has the property that f(x) = f '(x)f "(x).
What is the leading coefficient of f(x)?   (1/18)

Subsets is all in one color

Given Six three element subsets of a finite set X, show that it is possible to color the elements of X  in two colors such that none of the given subsets is all in one color.

every k-element subset of {1, 2, 3, ......., 50}

find the smallest integer k such that every k-element subset of {1, 2, 3, ......., 50} contains two distinct elements a , b such that  (a + b) divides ab.

don't contain two consecutive numbers

Find the number of non-empty subsets of {1, 2, 3, .............., n} which don't contain two consecutive numbers.