## Search found 107 matches

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.
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\$.
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:
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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 .
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\$.
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...
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.
Mon Jan 12, 2015 2:19 pm
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'.
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
Wed Dec 24, 2014 7:52 pm