Search found 151 matches

by Avik Roy
Tue Sep 13, 2011 7:08 pm
Forum: Algebra
Topic: Functions together
Replies: 9
Views: 2289

Re: Functions together

Edited the problem
by Avik Roy
Tue Sep 13, 2011 5:30 pm
Forum: Algebra
Topic: Functions together
Replies: 9
Views: 2289

Functions together

Define three functions $f,h:\mathbb {N} \cup \{0\} \times \mathbb {N} \to \mathbb {N} \cup \{0\}$ and $g:\mathbb {N} \times \mathbb {N} \to \mathbb {N} \cup \{0\}$ as such- for $m<n$ \[f(m,n) =0 \] otherwise \[f(m,n) = g(m,n) + f \left (m - 2^{h(m,n)}n, n \right ) \] for $m \geq 2n $ \[g(m,n) = 2g \...
by Avik Roy
Thu Sep 08, 2011 1:29 pm
Forum: Algebra
Topic: Two functions
Replies: 1
Views: 906

Two functions

Consider two functions $f,g: \mathbb {N} \to \mathbb {N}$ so that \[g(n) = |\{a|f(a) \leq n \}| \] \[f(n) = |\{a|g(a) \leq n \}| \] Find the possible ranges for the functions $f$ and $g$

I just hope I didn't make any mistake while formulating this problem :?
by Avik Roy
Wed Sep 07, 2011 7:24 pm
Forum: Algebra
Topic: Set of Natural Numbers
Replies: 13
Views: 3610

Re: Set of Natural Numbers

Shouldn't my earlier proof work in this case also?
by Avik Roy
Wed Sep 07, 2011 4:22 pm
Forum: Site Support
Topic: Like Button
Replies: 4
Views: 2672

Like Button

ফেসবুকের মতো ফোরামেও একটা লাইক বাটন থাকলে ভালই হয়!!!
by Avik Roy
Wed Sep 07, 2011 4:16 pm
Forum: Algebra
Topic: Set of Natural Numbers
Replies: 13
Views: 3610

Re: Set of Natural Numbers

Hints: Is $A$ finite? What does the well ordered property of $\mathbb {N}$ imply? Solution: Defining, \[S_n = \{a | a \in A, a \leq n\}\] This implies that $\{n\} = |S_n|$.Let, for $m \geq n$ \[R_{m,n} = \{a | a \in A, n < a \leq m\} \] Then, \[ S_m = S_n \cup R_{m,n} \] Since $S_n$ and $R_{m,n}$ ar...
by Avik Roy
Wed Sep 07, 2011 3:31 pm
Forum: Algebra
Topic: functional equation canada
Replies: 9
Views: 3137

Re: functional equation canada

Just put $x=1$ in $f(x^2)=f(0)$
by Avik Roy
Wed Sep 07, 2011 3:29 pm
Forum: College / University Level
Topic: average number of square sum representation
Replies: 2
Views: 3722

Re: average number of square sum representation

Good proof.
I actually collected the problem from a seminal article submitted by a MIT student. That article had some really nice results
by Avik Roy
Mon Sep 05, 2011 2:05 pm
Forum: Number Theory
Topic: 2003 USAMO[Nice problem]
Replies: 3
Views: 1576

Re: 2003 USAMO[Nice problem]

Nice solution!! I believe both solutions are correct.

This problem is nice, I was really charmed by the statement
by Avik Roy
Mon Sep 05, 2011 1:34 am
Forum: Number Theory
Topic: 2003 USAMO[Nice problem]
Replies: 3
Views: 1576

Re: 2003 USAMO[Nice problem]

It is only natural that the desired $n$ digit number is to end with $75$. Now, lets consider the fraction \[\frac {x_{n-2} \times 4} {5^{n-2}} + \frac {75} {5^n} \] Here, $x_{n-2}$ is an all odd digit number comprising $n-2$ digits. While dividing a number by $5^n$, the fractional part will be one o...