For the sake of the contradiction, let's assume that it does. Then, $a_1 \equiv a_1. a_2 \equiv a_1. a_2. a_3$ $\equiv........\equiv a_1. a_2......a_{k-2}. a_k$ $\equiv.....\equiv a_1. a_k \equiv a_k \pmod n$. So, $n$ divides $ |a_1 - a_k|$. But $0 < |a_1 - a_k| < n$, which is a contradiction. So, $...

- Tue Sep 05, 2017 12:25 am
- Forum: Algebra
- Topic: Sequence and divisibility
- Replies:
**2** - Views:
**162**

Name you'd like to be called: Tanmoy

Course you want to learn: Functional Equations and Number Theory Problem solving.

Preferred methods of communication (Forum, Messenger, Telegram, etc.):Telegram.

Do you want to take lessons through PMs or Public?: Public

Course you want to learn: Functional Equations and Number Theory Problem solving.

Preferred methods of communication (Forum, Messenger, Telegram, etc.):Telegram.

Do you want to take lessons through PMs or Public?: Public

- Wed Mar 29, 2017 7:28 pm
- Forum: National Math Camp
- Topic: The Gonit IshChool Project - Beta
- Replies:
**15** - Views:
**623**

$\text {Problem 8}$ We have $\dfrac {n(n+1)} {2}$ stones in $k$ piles. In each move we take one stone from each pile and form a new pile with these stones (if a pile has only one stone, after that stone is removed the pile vanishes). Show that regardless of the initial configuration, we always end u...

- Fri Feb 24, 2017 9:21 pm
- Forum: Combinatorics
- Topic: Combi Marathon
- Replies:
**42** - Views:
**1957**

$\text{Problem 55}$

Let $n$ be a positive integer and let $(a_1,a_2,\ldots ,a_{2n})$ be a permutation of $1,2,\ldots ,2n$ such that the numbers $|a_{i+1}-a_i|$ are pairwise distinct for $i=1,\ldots ,2n-1$.

Prove that $\{a_2,a_4,\ldots ,a_{2n}\}=\{1,2,\ldots ,n\}$ if and only if $a_1-a_{2n}=n$.

Let $n$ be a positive integer and let $(a_1,a_2,\ldots ,a_{2n})$ be a permutation of $1,2,\ldots ,2n$ such that the numbers $|a_{i+1}-a_i|$ are pairwise distinct for $i=1,\ldots ,2n-1$.

Prove that $\{a_2,a_4,\ldots ,a_{2n}\}=\{1,2,\ldots ,n\}$ if and only if $a_1-a_{2n}=n$.

- Fri Feb 24, 2017 5:09 pm
- Forum: International Mathematical Olympiad (IMO)
- Topic: IMO Marathon
- Replies:
**184** - Views:
**21836**

$\text{Problem 7}$ Elmo is drawing with colored chalk on a sidewalk outside. He first marks a set $S$ of $n>1$ collinear points. Then, for every unordered pair of points $\{X,Y\}$ in $S$, Elmo draws the circle with diameter $XY$ so that each pair of circles which intersect at two distinct points ar...

- Fri Feb 24, 2017 4:27 pm
- Forum: Combinatorics
- Topic: Combi Marathon
- Replies:
**42** - Views:
**1957**

$\text{Problem 54}$ The following operation is allowed on a finite graph: choose any cycle of length $4$ (if one exists), choose an arbitrary edge in that cycle, and delete this edge from the graph. For a fixed integer $n \ge 4$, find the least number of edges of a graph that can be obtained by rep...

- Fri Feb 24, 2017 1:25 pm
- Forum: International Mathematical Olympiad (IMO)
- Topic: IMO Marathon
- Replies:
**184** - Views:
**21836**

Problem 1 Let $n > 3$ be a fixed positive integer. Given a set $S$ of $n$ points $P_1, P_2,\cdots, P_n$ in the plane such that no three are collinear and no four concyclic, let $a_t$ be the number of circles $P_i P_j P_k$ that contain $P_t$ in their interior, and let $m(S) = a_1 + a_2 +\cdots + a_n...

- Mon Feb 20, 2017 12:05 am
- Forum: Combinatorics
- Topic: Combi Solution Writing Threadie
- Replies:
**10** - Views:
**809**

Problem 1 Let $n > 3$ be a fixed positive integer. Given a set $S$ of $n$ points $P_1, P_2,\cdots, P_n$ in the plane such that no three are collinear and no four concyclic, let $a_t$ be the number of circles $P_i P_j P_k$ that contain $P_t$ in their interior, and let $m(S) = a_1 + a_2 +\cdots + a_n...

- Sun Feb 19, 2017 10:19 pm
- Forum: Combinatorics
- Topic: Combi Marathon
- Replies:
**42** - Views:
**1957**

Problem 10:

Let $I$ be the incenter of $\triangle ABC$. The incircle touches $BC$ at $D$ and $K$ is the antipode of $D$ in $(I)$.

Let $M$ be the midpoint of $AI$. Prove that $KM$ passes through the Feuerbach Point.

Let $I$ be the incenter of $\triangle ABC$. The incircle touches $BC$ at $D$ and $K$ is the antipode of $D$ in $(I)$.

Let $M$ be the midpoint of $AI$. Prove that $KM$ passes through the Feuerbach Point.

- Sat Jan 07, 2017 10:56 pm
- Forum: Geometry
- Topic: Geometry Marathon : Season 3
- Replies:
**92** - Views:
**3572**

Problem 8: Given a cyclic quadrilateral $ABCD$ with circumcircle $(O)$. Let $AB \cap CD=E, \ AD \cap BC=F, \ AC \cap BD=G, \ AC \cap EF=P, \ BD \cap EF=Q$. Let $M, \ N$ be midpoints of $AC, \ BD$, respectively and let $MN \cap EF=H$. (i) Prove that $M, \ N, \ P, \ Q$ are concyclic. (ii) Let $K$ be ...

- Sat Jan 07, 2017 11:13 am
- Forum: Geometry
- Topic: Geometry Marathon : Season 3
- Replies:
**92** - Views:
**3572**