## Towers of Hanoi problem

Introduction :

The Tower of Hanoi (also called the Tower of Brahma or Lucas’ Tower, and sometimes pluralised) is a mathematical game or puzzle. It consists of three rods, and a number of disks of different sizes which can slide onto any rod. The puzzle starts with the disks in a neat stack in ascending order of size on one rod, the smallest at the top, thus making a conical shape.

The objective of the puzzle is to move the entire stack to another rod, obeying the following rules:

• Only one disk may be moved at a time.
• Each move consists of taking the upper disk from one of the rods and sliding it onto another rod, on top of the other disks that may already be present on that rod.
• No disk may be placed on top of a smaller disk.

With three disks, the puzzle can be solved in seven moves.

Our Lemma:

for   disks and  towers, the number of steps required to transfer the disks from   tower to another is .

Proof of our lemma:

Let  denote the minimum number of moves required for transferring  discs from one peg to another.Suppose that there are  discs on peg  . We will first transfer the top  discs to peg  . This can be done in  moves.
Now we transfer the th disc to peg 2 . One move will be required for this.
Finally we will transfer  discs which are on peg  to peg  . This will require another  moves. Thus the transfer will be done in  moves which will be equal to  movesTherefore we have formed a recurrence relation that

Now we will obtain a general formula for

We will multiply both sides of the above relations by
we get

Solving this gives us

so

Since  ( one move will be required to move )
so

THIS FINALLY GIVES

Sources :
1. wikepedia ( for the introduction) : http://en.wikipedia.org/wiki/Tower_of_Hanoi

## NMTC 2012 ( CLASS 9th and 10th) Question paper

The AMTI is a pioneer organisation in promoting and conducting Maths Talent Tests in India. Last year (43rd TC Data) (in the 43rd National level tests) 54058 students from 332 institutions spread all over India, participated at the screening level; 10% of them insitutionwise were selected for the final test. For the benefit of final level contestants and the chosen few for INMO, special orientation camps were conducted. Merit certificates and prizes were awarded to the deserving students.

NMTC Question Paper

## CGMO-2012 (China Girls Math Olympiad 2012) Problem 8

Find the number of integers $k$ in the set $\{0, 1, 2,\cdots, 2012\}$   such that  $\binom{2012}{k}$ is a multiple of $2012$

## CGMO – 2012 ( China Girls Math Olympiad 2012 ) Problem 7

Let  $\{a_n\}$ be a sequence of nondecreasing positive integers such that  $\frac{r}{a_{r}}= k+1$ for some positive integers $k$ and $r$. Prove that there exists a positive integer $s$ such that  $\frac{s}{a_s} = k$

## CGMO 2012 ( China Girls Math Olympiad 2012) Problem 6

There are $n$ cities, $2$ airline companies in a country. Between any two cities, there is exactly one $2$-way flight connecting them which is operated by one of the two companies. A female mathematician plans a travel route, so that it starts and ends at the same city, passes through at least two other cities, and each city in the route is visited once. She finds out that wherever she starts and whatever route she chooses, she must take flights of both companies. Find the maximum value of  $n$

## CGMO 2012 ( China Girls Math Olympiad 2012) Problem 5

As shown in the figure below, the in-circle of  is tangent to sides  and  at  and  respectively, and  is the circumcenter of . Prove that .

## CGMO-2012 (China Girls Math Olympiad 2012) Problem 4

There is a stone at each vertex of a given regular $13$-gon, and the color of each stone is black or white. Prove that we may exchange the position of two stones such that the coloring of these stones are symmetric with respect to some symmetric axis of the $13$-gon

## CGMO-2012 (China Girls Math Olympiad 2012) Problem-3

Find all pairs $(a,b)$ of integers satisfying: there exists an integer  $d \ge 2$ such that $a^n + b^n+1$  is divisible by $d$ for all positive integers $n$

## CGMO 2012 (China Girls Math Olympiad 2012)- Problem 2

Circles  and  are tangent to each other externally at . Points  and  are on , lines  and  are tangent to  at  and , respectively, lines  and  meet at point
Prove that

(1) ;
(2) .

## CGMO 2012 (China Girls Math Olympiad 2012) Problem 1

Let  $a_{1}, a_{2},\ldots, a_{n}$ be non-negative real numbers. Prove that                                                                          $\LARGE \frac{1}{1+a_{1}}+\frac{ a_{1}}{(1+a_{1})(1+a_{2})}+\frac{ a_{1}a_{2}}{(1+a_{1})(1+a_{2})(1+a_{3})}\cdots+\frac{ a_{1}a_{2}\cdots a_{n-1}}{(1+a_{1})(1+a_{2})\cdots (1+a_{n})}\le 1.$