What is the largest number of steps that this process can take?

A bookshelf contains n volumes, labelled 1 to n, in some order.
The librarian wishes to put them in the correct order as follows:-
The librarian selects a volume that is too far to the right, say
the volume with label k, takes it out, and inserts it in the k-th position.
For example, if the bookshelf contains the volumes 1, 3, 2, 4 in that order,
the librarian could take out volume 2 and place it in the second position.
The books will then be in the correct order 1, 2, 3, 4.
(a) Show that if this process is repeated, then, however the librarian makes the
selections, all the volumes will eventually be in the correct order.
(b) What is the largest number of steps that this process can take?

for any positive integer n & k, let L(n,k)....

For any positive integers n and k, let L(n , k) be the least common multiple of the k
consecutive integers n, n + 1, . . . , (n + k - 1).
Show that for any integer b, there exist integers n and k such that L(n , k) > b L(n + 1; k)

a positive integer N such that for all integers a > N

Show that there exists a positive integer N such that for all integers a > N, there exists
a contiguous sub-string of the decimal expansion of a that is divisible by 2011.
(For instance, if a = 153204, then 15, 532, and 0 are all contiguous substrings of a.
 Note that 0 is divisible by 2011.)

Radhey has divided a square up into finitely many white and red rectangles

Radhey has divided a square up into finitely many white and red rectangles, each with sides
parallel to the sides of the square. Within each white rectangle, she writes down its width
divided by its height. Within each red rectangle, she writes down its height divided by
its width. Finally, she calculates x, the sum of these numbers. If the total area of the
white rectangles equals the total area of the red rectangles, what is the smallest possible
value of x?

no number of this form can divide another number of this form

Consider 70-digit numbers n, with the property that each of the digits 1, 2, 3, . . . , 7
appears in the decimal expansion of n ten times (and 8, 9, and 0 do not appear).
Show that no number of this form can divide another number of this form.

a, b, c, d, e, f, g, h, i, j are all distinct

Is there a 14-digit perfect square of the form 2013abcdefghij, 

where a, b, c, d, e, f, g, h, i, j are all distinct?

Divisible by 2013.

Find the least n such that 1 + 2 + . . . + n is divisible by 2013.

the largest power of 5

Find the largest power of 5 that can divide a number all whose digits are distinct.

all arithmetic sequences a, b, c, d

Find all arithmetic sequences a, b, c, d for which
a − 1, b − 5, c − 6, d − 1 is a geometric sequence.

the probability that the entire disk is on the top of the table

Point P is chosen at random atop a circular table of diameter 1 meter. A circular disk of
diameter 1cm. is placed on the table with its center on point P.
What is the probability that the entire disk is on the top of the table, that is,
no part of the disk hangs over the table?

U(n)/L(n) = 3

For every integer n > 2 let L(n) denote the sum of the integers from 1 through
[n/2] which are relatively prime to n, and let U(n) denote the sum of the integers
from [n/2] + 1 through n which are relatively prime to n. Prove that if n is
divisible by 4, then U(n)/L(n) = 3. ([ . ] is the greatest integer function.)

at least one pair of relatively prime numbers

Show that every set of n+1 positive integers, chosen from a set of 2n consecutive
integers, contains at least one pair of relatively prime numbers.

the probability that after the sixth roll the walker is back at its starting point (0, 0, 0)

A random walk on the three dimensional integer lattice is defined as follows.
The walker starts at (0, 0, 0). A standard six sided die is rolled six times. After
each roll the walker moves to one of its six nearest neighbors, according to the
following protocol: if the die rolls 1, 2, 3, 4, 5, or 6 dots the walker jumps one
unit in the +x, −x, +y, −y,z, −z direction respectively.
Find the probability that after the sixth roll the walker is back at its starting
point (0, 0, 0)

Start with an even number of points (at least four points) in the plane

Start with an even number of points (at least four points) in the plane, no three
on the same straight line, half colored blue and half colored yellow.
Show there is a straight line, which does not meet any of the points,
which divides the points into two non-empty sets of points, both sets being
half blue and half yellow.

the probability there is only one winner is at least 2/3

Start with some coins. Flip each coin until a head comes up on that coin.
The winner(s) are the coin(s) which were flipped the most times.
Prove that the probability there is only one winner is at least 2/3.

S (n) = n E(n) / 2

Let E denote the Euler totient; E(n) is the number of integers  r =1,2,3,....., n
such that (r, n) = 1. Define S (n) as the sum of those E(n) integers.
Show that for n > 2, S (n) = n E(n) / 2.

D, H, F, G are concyclic

Let ABCD be a cyclic quadrilateral whose diagonals AC and BD meet at E. The extensions
of the sides AD and BC beyond A and B meet at F. Let G be the point such that ECGD
is a parallelogram, and let H be the image of E under reflection in AD.
Prove that D, H, F, G are concyclic.

M is the midpoint of ST

In the triangle ABC the point J is the center of the ex-circle opposite to A. This ex-circle
is tangent to the side BC at M, and to the lines AB and AC at K and L respectively. The
lines LM and BJ meet at F, and the lines KM and CJ meet at G. Let S be the point of
intersection of the lines AF and BC, and let T be the point of intersection of the lines AG
and BC. Prove that M is the midpoint of ST.

100 sums of the pairs of numbers at the endpoints of the chosen chords are equal

There are given N = (2.2.2.......500 times) points on a circle labeled 1, 2, . . . , N in some order.
Prove that one can choose 100 pairwise disjoint chords joining some of these points so that the
100 sums of the pairs of numbers at the endpoints of the chosen chords are equal.

Find the least N that enables her to succeed

Players A and B play a game with N ≥ 2012 coins and 2012 boxes arranged around a
circle. Initially A distributes the coins among the boxes so that there is at least 1 coin in each
box. Then the two of them make moves in the order B, A, B, A, . . . by the following rules:
• On every move of his B passes 1 coin from every box to an adjacent box.
• On every move of hers A chooses several coins that were not involved in B’s previous
move and are in different boxes. She passes every chosen coin to an adjacent box.
Player A’s goal is to ensure at least 1 coin in each box after every move of hers, regardless of
how B plays and how many moves are made. Find the least N that enables her to succeed.

f has a rational root

Let f and g be two nonzero polynomials with integer coefficients and deg f  > deg g.
Suppose that for infinitely many primes p the polynomial pf + g has a rational root. Prove
that f has a rational root.