G
Topic
First Poster
Last Poster
Poland Inequalities
wangzishan 6
N
by AshAuktober
Leta,b,c are postive real numbers,proof that 
6 replies
disjoint subsets
nayel 2
N
by alexanderhamilton124
Source: Taiwan 2001
Let 
be an integer and let 
be 
distinct subsets of 
. Show that there exists 
such that the n subsets 
are also disjoint.
what i have is
be an integer and let 
be 
distinct subsets of 
. Show that there exists 
such that the n subsets 
are also disjoint.what i have is
s
.2 replies
Can I find source of a geometry problem via Approach0?
xytunghoanh 1
N
by GreekIdiot
Can I find source of a geometry problem via Approach0 or AOPS search feature?
Thanks.
Thanks.
1 reply
Chinese TST 2008 P3
Fang-jh 8
N
by InterLoop
Suppose that every positve integer has been given one of the colors red, blue,arbitrarily. Prove that there exists an infinite sequence of positive integers 
such that inifinite sequence of positive integers 
has the same color.
such that inifinite sequence of positive integers 
has the same color.
8 replies
primitive polyominoes
N.T.TUAN 28
N
by quantam13
Source: USAMO 2007
An animal with cells is a connected figure consisting of equal-sized cells[1].
A dinosaur is an animal with at least
cells. It is said to be primitive it its cells cannot be partitioned into two or more dinosaurs. Find with proof the maximum number of cells in a primitive dinosaur.
(1) Animals are also called polyominoes. They can be defined inductively. Two cells are adjacent if they share a complete edge. A single cell is an animal, and given an animal with cells, one with 
cells is obtained by adjoining a new cell by making it adjacent to one or more existing cells.
cells is a connected figure consisting of equal-sized cells[1].A dinosaur is an animal with at least
cells. It is said to be primitive it its cells cannot be partitioned into two or more dinosaurs. Find with proof the maximum number of cells in a primitive dinosaur.(1) Animals are also called polyominoes. They can be defined inductively. Two cells are adjacent if they share a complete edge. A single cell is an animal, and given an animal with
cells, one with 
cells is obtained by adjoining a new cell by making it adjacent to one or more existing cells.
28 replies
Unsolved Diophantine(I think)
Nuran2010 2
N
by ohiorizzler1434
Find all solutions for the equation 
where is a positive integer and 
is a prime.(Don't get mad at me,I've used the search function and did not see a correct and complete solution anywhere.)
where is a positive integer and 
is a prime.(Don't get mad at me,I've used the search function and did not see a correct and complete solution anywhere.)
2 replies
A real sequence
Omid Hatami 14
N
by HamstPan38825
Source: Iran National Olympiad (3rd Round) 2001
Does there exist a sequence 
of positive real numbers such that for each natural 
: ![\[ b_{m}+b_{2m}+b_{3m}+\dots=\frac1m\]](//latex.artofproblemsolving.com/4/7/e/47e1c0a1d6fa937d06cc8f4577039886b3033687.png)
of positive real numbers such that for each natural 
: ![\[ b_{m}+b_{2m}+b_{3m}+\dots=\frac1m\]](http://latex.artofproblemsolving.com/4/7/e/47e1c0a1d6fa937d06cc8f4577039886b3033687.png)
14 replies
FE on R+
AshAuktober 6
N
by jasperE3
Source: 2007 MOP
(Note I couldn't find a post w/ this from AoPS search so I'm posting, please do tell if there exists a post.)
Solve over positive real numbers the functional equation
![\[ f\left( f(x) y + \frac xy \right) = xyf(x^2+y^2). \]](//latex.artofproblemsolving.com/b/6/5/b656324b6499f7457b7b9eccbf5eaa01b6ad6e59.png)
Solve over positive real numbers the functional equation
![\[ f\left( f(x) y + \frac xy \right) = xyf(x^2+y^2). \]](http://latex.artofproblemsolving.com/b/6/5/b656324b6499f7457b7b9eccbf5eaa01b6ad6e59.png)
6 replies
Perfect square preserving polynomial
Omid Hatami 34
N
by HamstPan38825
Source: Iran TST 2008
Find all polynomials 
of one variable with integer coefficients such that if 
and 
are natural numbers such that 
is a perfect square, then 
is also a perfect square.
of one variable with integer coefficients such that if 
and 
are natural numbers such that 
is a perfect square, then 
is also a perfect square.
34 replies
1/n can be written as sum of 1/T_{k}
AshAuktober 1
N
by AshAuktober
Source: 2025 BMO2/1 (I tried search fn and it yielded no results)
Prove that if is a positive integer, then 
can be written as a finite sum of reciprocals of different triangular numbers.
[A triangular number is one of the form
for some positive integer 
.]
is a positive integer, then 
can be written as a finite sum of reciprocals of different triangular numbers.[A triangular number is one of the form
for some positive integer 
.]
1 reply
