Search found 107 matches

by mutasimmim
Tue Apr 28, 2015 9:28 am
Forum: Number Theory
Topic: Sequence of integers with special property
Replies: 0
Views: 956

Sequence of integers with special property

Prove that for positive integer $n>2$ one can find $n$ positive integers $a_1<a_2<a_3...<a_n$ such that for $i=1,2,...,n-2$, the line segments with length $a_i, a_{i-1},a_{i-2}$ forms a triangle with positive integer area.
by mutasimmim
Wed Apr 08, 2015 10:56 pm
Forum: Number Theory
Topic: IMO NT Compilation
Replies: 2
Views: 1486

Re: IMO NT Compilation

There is a typo in 1984_P2. We are supposed to prove that $(a+b)^7-a^7-b^7$ is divisible by $7^7$, not just by $7$.
by mutasimmim
Fri Mar 27, 2015 8:04 am
Forum: Secondary Level
Topic: Euler's Graph Theoritic Formula
Replies: 0
Views: 877

Euler's Graph Theoritic Formula

Let there be a (simple)graph whose edges only meet at vertices. Let $V,E,F$ be the numbers of vertices,edges and faces (number of regions the graph divides the plane into) respectively. Prove that $V-E+F=2$.

Hint:
There exists at least 3 proofs with induction.
.
by mutasimmim
Fri Mar 27, 2015 7:58 am
Forum: Secondary Level
Topic: Graph Theory Fact
Replies: 1
Views: 1113

Graph Theory Fact

We have a graph without any cycle of odd length. Prove that there are not two paths one with odd length and the other with even length joining the same two vertices .
by mutasimmim
Fri Mar 27, 2015 7:51 am
Forum: Number Theory
Topic: Disibility by $n!$
Replies: 1
Views: 1040

Disibility by $n!$

Prove that for all positive integers $n$, $n!$ divides $$\prod (2^n-2^k) $$ for $1\le k \le n-1$.
by mutasimmim
Wed Feb 04, 2015 9:25 pm
Forum: Number Theory
Topic: Product of first $k$ primes
Replies: 1
Views: 1018

Re: Product of first $k$ primes

SOLUTION First note that $a$ must be equal to or greater than $p_{k+1}$, otherwise some prime divisor $q$ of $a$ would divide the RHS, but not the LHS. So, it is obvious that $n<k$. Now let us consider the smallest prime divisor $q$ of $n$. As $n<k<p_k$, $q$ is one of the primes $p_1,p_2,...,p_{k-1...
by mutasimmim
Wed Feb 04, 2015 10:01 am
Forum: Number Theory
Topic: Product of first $k$ primes
Replies: 1
Views: 1018

Product of first $k$ primes

Find all positive integer $k>1$ that $p_1 p_2 p_3 ... p_k -1=a^n$, for some positive integers $a ,n>1$, where $p_s$ denotes the $s$-th prime number.
by mutasimmim
Mon Jan 12, 2015 2:19 pm
Forum: International Mathematical Olympiad (IMO)
Topic: Hello
Replies: 1
Views: 2580

Re: Hello

Of course, great idea. But not in this sub-forum, you can do so in the 'Social Lounge'. :)
by mutasimmim
Mon Jan 12, 2015 2:16 pm
Forum: Algebra
Topic: Function material
Replies: 4
Views: 5178

Re: Function material

Check out this link. Here I have uploaded 8 pdfs on functional equations, all of them are from Math Olympiad Program of USA.

https://www.dropbox.com/sh/km87d9s8ilg3 ... tXQYa?dl=0
by mutasimmim
Wed Dec 24, 2014 7:52 pm
Forum: International Mathematical Olympiad (IMO)
Topic: IMO Shortlist 2004 N6
Replies: 1
Views: 1126

IMO Shortlist 2004 N6

For a natural number $n$, let $P_{n}$ denote the product of the natural numbers $x<n$ such that $n\mid (x^2-1).$. Find $P_{n} (mod n)$ in terms of $n$.