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and 
Prove that
Two permutations
Nima Ahmadi Pour 12
N
by Zhaom
Source: Iran prepration exam
Suppose that 
, 
, 
, 
are integers such that 
.
Prove that there exist two permutations
and 
of 
such that for each integer 
with 
, we have
![\[ n\mid a_i - b_i - c_i
\]](//latex.artofproblemsolving.com/1/a/e/1ae8d54f068b646ac7a1e43d75e7f26510cfdc95.png)
Proposed by Ricky Liu & Zuming Feng, USA
, 
, 
, 
are integers such that 
.Prove that there exist two permutations
and 
of 
such that for each integer 
with 
, we have![\[ n\mid a_i - b_i - c_i
\]](http://latex.artofproblemsolving.com/1/a/e/1ae8d54f068b646ac7a1e43d75e7f26510cfdc95.png)
Proposed by Ricky Liu & Zuming Feng, USA
12 replies
Easy Number Theory
math_comb01 37
N
by John_Mgr
Source: INMO 2024/3
Let 
be an odd prime and 
be integers so that the integers 
are divisible by .
Prove that divides each of .
Proposed by Navilarekallu Tejaswi
be an odd prime and 
be integers so that the integers 
are divisible by .Prove that
divides each of .
Proposed by Navilarekallu Tejaswi
37 replies
Prove that 
Inspired by hlminh
sqing 3
N
by sqing
Source: Own
Let 
be real numbers such that 
Prove that 
Where 
be real numbers such that 
Prove that 
Where 
3 replies
A Familiar Point
v4913 51
N
by xeroxia
Source: EGMO 2023/6
Let 
be a triangle with circumcircle 
. Let 
and 
respectively denote the midpoints of the arcs 
and 
that do not contain the third vertex. Let 
denote the midpoint of arc 
(the arc 
including 
). Let 
be the incenter of . Let 
be the circle that is tangent to and internally tangent to at , and let 
be the circle that is tangent to and internally tangent to at . Show that the line 
, and the lines through the intersections of and , meet on .
be a triangle with circumcircle 
. Let 
and 
respectively denote the midpoints of the arcs 
and 
that do not contain the third vertex. Let 
denote the midpoint of arc 
(the arc 
including 
). Let 
be the incenter of . Let 
be the circle that is tangent to and internally tangent to at , and let 
be the circle that is tangent to and internally tangent to at . Show that the line 
, and the lines through the intersections of and , meet on .
51 replies
Apple sharing in Iran
mojyla222 3
N
by math-helli
Source: Iran 2025 second round p6
Ali is hosting a large party. Together with his 
friends, 
people are seated around a circular table in a fixed order. Ali places apples for serving directly in front of himself and wants to distribute them among everyone. Since Ali and his friends dislike eating alone and won't start unless everyone receives an apple at the same time, in each step, each person who has at least one apple passes one apple to the first person to their right who doesn't have an apple (in the clockwise direction).
Find all values of such that after some number of steps, the situation reaches a point where each person has exactly one apple.
friends, 
people are seated around a circular table in a fixed order. Ali places apples for serving directly in front of himself and wants to distribute them among everyone. Since Ali and his friends dislike eating alone and won't start unless everyone receives an apple at the same time, in each step, each person who has at least one apple passes one apple to the first person to their right who doesn't have an apple (in the clockwise direction).Find all values of
such that after some number of steps, the situation reaches a point where each person has exactly one apple.
3 replies
Iran second round 2025-q1
mohsen 5
N
by math-helli
Find all positive integers n>2 such that sum of n and any of its prime divisors is a perfect square.
5 replies
2500th post
Solocraftsolo 33
N
by Solocraftsolo
2500 is cool.
i am not very sentimental so im not going to post a math story or anything.
here are some problems though
?
students want to share 
pencils. If every student gets at least one pencil, how many ways are there to distribute the pencils?33 replies
Iran Team Selection Test 2016
MRF2017 9
N
by SimplisticFormulas
Source: TST3,day1,P2
Let be an arbitrary triangle and 
is the circumcenter of 
.Points 
lie on 
,respectively such that the reflection of WRT 
is tangent to circumcircle of 
.Prove that the circumcircle of triangle 
is tangent to circumcircle of triangle 
.
be an arbitrary triangle and 
is the circumcenter of 
.Points 
lie on 
,respectively such that the reflection of WRT 
is tangent to circumcircle of 
.Prove that the circumcircle of triangle 
is tangent to circumcircle of triangle 
.
9 replies
Combo problem
soryn 3
N
by soryn
The school A has m1 boys and m2 girls, and ,the school B has n1 boys and n2 girls. Each school is represented by one team formed by p students,boys and girls. If f(k) is the number of cases for which,the twice schools has,togheter k girls, fund f(k) and the valute of k, for which f(k) is maximum.
3 replies
