G
Topic
First Poster
Last Poster
Ways to Place Counters on 2mx2n board
EpicParadox 37
N
by akliu
Source: 2019 Canadian Mathematical Olympiad Problem 3
You have a 
by 
grid of squares coloured in the same way as a standard checkerboard. Find the total number of ways to place 
counters on white squares so that each square contains at most one counter and no two counters are in diagonally adjacent white squares.
by 
grid of squares coloured in the same way as a standard checkerboard. Find the total number of ways to place 
counters on white squares so that each square contains at most one counter and no two counters are in diagonally adjacent white squares.
37 replies
Number theory
Maaaaaaath 1
N
by CHESSR1DER
Let 
be a positive integer . Prove that there exists infinitely many pairs of positive integers 
such that 
and :
be a positive integer . Prove that there exists infinitely many pairs of positive integers 
such that 
and :
1 reply
Problem 4 from IMO 1997
iandrei 28
N
by akliu
Source: IMO Shortlist 1997, Q4
An 
matrix whose entries come from the set 
is called a silver matrix if, for each 
, the 
-th row and the -th column together contain all elements of 
. Show that:
(a) there is no silver matrix for
;
(b) silver matrices exist for infinitely many values of
.
matrix whose entries come from the set 
is called a silver matrix if, for each 
, the 
-th row and the -th column together contain all elements of 
. Show that:(a) there is no silver matrix for
;(b) silver matrices exist for infinitely many values of
.
28 replies
2025 Caucasus MO Seniors P8
BR1F1SZ 1
N
by sami1618
Source: Caucasus MO
Determine for which integers 
the cells of a 
table can be filled with the numbers 
such that the following conditions are satisfied:
[list=i]
[*]Each of the numbers appears exactly once.
[*]In any
rectangle, one of the numbers is the arithmetic mean of the other two.
[*]The number
is located in the middle cell of the table.
[/list]
the cells of a 
table can be filled with the numbers 
such that the following conditions are satisfied:[list=i]
[*]Each of the numbers
appears exactly once.[*]In any
rectangle, one of the numbers is the arithmetic mean of the other two.[*]The number
is located in the middle cell of the table.[/list]
1 reply
Unlimited candy in PAGMO
JuanDelPan 21
N
by akliu
Source: Pan-American Girls' Mathematical Olympiad 2021, P5
Celeste has an unlimited amount of each type of 
types of candy, numerated type 1, type 2, ... type n. Initially she takes 
candy pieces and places them in a row on a table. Then, she chooses one of the following operations (if available) and executes it:
She eats a candy of type 
, and in its position in the row she places one candy type 
followed by one candy type 
(we consider type 
to be type 1, and type 0 to be type ).
She chooses two consecutive candies which are the same type, and eats them.
Find all positive integers for which Celeste can leave the table empty for any value of and any configuration of candies on the table.
types of candy, numerated type 1, type 2, ... type n. Initially she takes 
candy pieces and places them in a row on a table. Then, she chooses one of the following operations (if available) and executes it:
She eats a candy of type 
, and in its position in the row she places one candy type 
followed by one candy type 
(we consider type 
to be type 1, and type 0 to be type ).
She chooses two consecutive candies which are the same type, and eats them.Find all positive integers
for which Celeste can leave the table empty for any value of and any configuration of candies on the table.
21 replies
set with c+2a>3b
VicKmath7 48
N
by akliu
Source: ISL 2021 A1
Let be a positive integer. Given is a subset 
of 
with 
elements. Prove that there exist three elements 
from such that 
.
Proposed by Dominik Burek and Tomasz Ciesla, Poland
be a positive integer. Given is a subset 
of 
with 
elements. Prove that there exist three elements 
from such that 
.Proposed by Dominik Burek and Tomasz Ciesla, Poland
48 replies
A property of divisors
rightways 10
N
by akliu
Source: Kazakhstan NMO 2016, P1
Prove that one can arrange all positive divisors of any given positive integer around a circle so that for any two neighboring numbers one is divisible by another.
10 replies
Famous geo configuration appears on the district MO
AndreiVila 3
N
by chirita.andrei
Source: Romanian District Olympiad 2025 10.4
Let 
be a convex hexagon with 
and 
.
[list=a]
[*] Prove that there is a unique point
which is equidistant from sides 
and 
.
[*] If
and 
are the centers of mass of 
and 
, show that 
.
be a convex hexagon with 
and 
.[list=a]
[*] Prove that there is a unique point
which is equidistant from sides 
and 
.[*] If
and 
are the centers of mass of 
and 
, show that 
.
3 replies
kind of well known?
dotscom26 2
N
by alexheinis
Source: MBL
Let 
be real numbers satisfying
Find the maximum value of
I have seen many problems with the same structure, Id really appreciate if someone could explain which approach is suitable here
be real numbers satisfying
Find the maximum value of
I have seen many problems with the same structure, Id really appreciate if someone could explain which approach is suitable here
2 replies
and 
: Prove that :
