be real numbers such that 
.
Y by Exponent11
Have you a nice solution for this problem?
Thank you very much
Thank you very much
G
Topic
First Poster
Last Poster
Easy divisibility
a_507_bc 2
N
by TUAN2k8
Source: ARO Regional stage 2023 9.4~10.4
Let 
be positive integers such that no number divides some other number. If 
, prove that 
.
be positive integers such that no number divides some other number. If 
, prove that 
.
2 replies
integer functional equation
ABCDE 149
N
by ezpotd
Source: 2015 IMO Shortlist A2
Determine all functions 
with the property that ![\[f(x-f(y))=f(f(x))-f(y)-1\]](//latex.artofproblemsolving.com/f/2/5/f25cc1e8ae0be1fd02b347fd94be4fab88af1d46.png)
holds for all 
.
with the property that ![\[f(x-f(y))=f(f(x))-f(y)-1\]](http://latex.artofproblemsolving.com/f/2/5/f25cc1e8ae0be1fd02b347fd94be4fab88af1d46.png)
holds for all 
.
149 replies
A geometry problem involving 2 circles
Ujiandsd 0
Source: L
Point M is the midpoint of side BC of triangle ABC. The length of the radius of the outer circle of triangle ABM, triangle ACM
is 5 and 7 respectively find the distance between the center of their outer circles
is 5 and 7 respectively find the distance between the center of their outer circles
0 replies
Inequality, inequality, inequality...
Assassino9931 10
N
by sqing
Source: Al-Khwarizmi Junior International Olympiad 2025 P6
Let 
be real numbers such that ![\[ab^2+bc^2+ca^2=6\sqrt{3}+ac^2+cb^2+ba^2.\]](//latex.artofproblemsolving.com/3/d/0/3d015292c249a69a44d5f923092201d7a271b896.png)
Find the smallest possible value of 
.
Binh Luan and Nhan Xet, Vietnam
be real numbers such that ![\[ab^2+bc^2+ca^2=6\sqrt{3}+ac^2+cb^2+ba^2.\]](http://latex.artofproblemsolving.com/3/d/0/3d015292c249a69a44d5f923092201d7a271b896.png)
Find the smallest possible value of 
.Binh Luan and Nhan Xet, Vietnam
10 replies
Grid with rooks
a_507_bc 3
N
by TUAN2k8
Source: ARO Regional stage 2022 9.3
Given is a positive integer 
. There are 
mutually non-attacking rooks placed on a grid 
. The grid is splitted into two connected parts, symmetric with respect to the center of the grid. What is the largest number of rooks that could lie in the same part?
. There are 
mutually non-attacking rooks placed on a grid 
. The grid is splitted into two connected parts, symmetric with respect to the center of the grid. What is the largest number of rooks that could lie in the same part?
3 replies
IMO Shortlist 2013, Number Theory #3
lyukhson 47
N
by cursed_tangent1434
Source: IMO Shortlist 2013, Number Theory #3
Prove that there exist infinitely many positive integers such that the largest prime divisor of 
is equal to the largest prime divisor of 
.
such that the largest prime divisor of 
is equal to the largest prime divisor of 
.
47 replies
Darboux cubic
srirampanchapakesan 1
N
by srirampanchapakesan
Source: Own
Let P be a point on the Darboux cubic (or the McCay Cubic ) of triangle ABC.
P1P2P3 is the circumcevian or pedal triangle of P wrt ABC.
Prove that P also lie on the Darboux cubic ( or the McCay Cubic) of P1P2P3 .
P1P2P3 is the circumcevian or pedal triangle of P wrt ABC.
Prove that P also lie on the Darboux cubic ( or the McCay Cubic) of P1P2P3 .
1 reply
IMO Shortlist 2011, Algebra 2
orl 43
N
by ezpotd
Source: IMO Shortlist 2011, Algebra 2
Determine all sequences 
of positive integers, such that for every positive integer there exists an integer 
with ![\[\sum^{2011}_{j=1} j x^n_j = a^{n+1} + 1\]](//latex.artofproblemsolving.com/e/8/6/e866c59c4603d1fe6a92f12cd2de89c0f34c6845.png)
Proposed by Warut Suksompong, Thailand
of positive integers, such that for every positive integer there exists an integer 
with ![\[\sum^{2011}_{j=1} j x^n_j = a^{n+1} + 1\]](http://latex.artofproblemsolving.com/e/8/6/e866c59c4603d1fe6a92f12cd2de89c0f34c6845.png)
Proposed by Warut Suksompong, Thailand
43 replies
Sequence inequality
BR1F1SZ 1
N
by IndoMathXdZ
Source: 2025 Francophone MO Seniors P1
Let 
be a sequence of positive integers satisfying the following property: for all positive integers 
, for all distinct integers 
and for all distinct integers 
,
![\[
a_{m_1} + a_{m_2} + \cdots + a_{m_k} \leqslant a_{n_1} + a_{n_2} + \cdots + a_{n_\ell}.
\]](//latex.artofproblemsolving.com/d/1/6/d1632ff7e5e55319cdbf56f45dca9ed84343aa47.png)
Prove that there exist two integers 
and 
such that 
for all 
.
be a sequence of positive integers satisfying the following property: for all positive integers 
, for all distinct integers 
and for all distinct integers 
,![\[
a_{m_1} + a_{m_2} + \cdots + a_{m_k} \leqslant a_{n_1} + a_{n_2} + \cdots + a_{n_\ell}.
\]](http://latex.artofproblemsolving.com/d/1/6/d1632ff7e5e55319cdbf56f45dca9ed84343aa47.png)
Prove that there exist two integers 
and 
such that 
for all 
.
1 reply
Nice problem about a trapezoid
G
H
G
H
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
#2
Apr 21, 2025, 7:11 PM
Y by
This problem is solved in the article "Quadrilateral gratings" by Nikolai Ivanov Beluhov in Kvant №10 2017 (page 19, problem 2, in Russian).
Z
K
Y
N Quick Reply
G
H
