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Reflected point lies on radical axis
Mahdi_Mashayekhi 5
N
by Mahdi_Mashayekhi
Source: Iran 2025 second round P4
Given is an acute and scalene triangle 
with circumcenter 
. 
and 
intersect the altitude from 
to 
at points 
and 
respectively. 
is the circumcenter of triangle 
and 
is the reflection of over . 
is the second intersection of circumcircles of triangles 
and 
. Show that 
are collinear.
with circumcenter 
. 
and 
intersect the altitude from 
to 
at points 
and 
respectively. 
is the circumcenter of triangle 
and 
is the reflection of over . 
is the second intersection of circumcircles of triangles 
and 
. Show that 
are collinear.
5 replies
Find the value
sqing 18
N
by Yiyj
Source: 2024 China Fujian High School Mathematics Competition
Let 
and 
Find the value of 
and 
Find the value of 
18 replies
Number Theory
fasttrust_12-mn 14
N
by Namisgood
Source: Pan African Mathematics Olympiad P1
Find all positive intgers 
and 
such that 
and 
is a prime number
and 
such that 
and 
is a prime number
14 replies
find question
mathematical-forest 5
N
by Jupiterballs
Are there any contest questions that seem simple but are actually difficult? :-D
5 replies
Own made functional equation
Primeniyazidayi 10
N
by Phat_23000245
Source: own(probably)
Find all functions 
such that 
for all 
such that 
for all 
10 replies
Tough inequality
TUAN2k8 4
N
by Phat_23000245
Source: Own
Let 
be an even integer and let 
be real numbers satisfying 
.
Prove that
be an even integer and let 
be real numbers satisfying 
.Prove that
4 replies
Combi counting
Caasi_Gnow 2
N
by Justbrick
Find the number of different ways to arrange seven people around a circular meeting table if A and B must sit together and C and D cannot sit next to each other. (Note: the order for A and B might be A,B or B,A)
2 replies
Guess period of function
a1267ab 9
N
by HamstPan38825
Source: USA TST 2025
Let 
be a positive integer. Ana and Banana play a game. Banana thinks of a function 
and a prime number 
. He tells Ana that 
is nonconstant, 
, and 
for all integers 
. Ana's goal is to determine the value of . She writes down integers 
. After seeing this list, Banana writes down 
in order. Ana wins if she can determine the value of from this information. Find the smallest value of for which Ana has a winning strategy.
Anthony Wang
be a positive integer. Ana and Banana play a game. Banana thinks of a function 
and a prime number 
. He tells Ana that 
is nonconstant, 
, and 
for all integers 
. Ana's goal is to determine the value of . She writes down integers 
. After seeing this list, Banana writes down 
in order. Ana wins if she can determine the value of from this information. Find the smallest value of for which Ana has a winning strategy.Anthony Wang
9 replies
Inequality with abc=1
tenplusten 11
N
by sqing
Source: JBMO 2011 Shortlist A7
Let 
be positive reals such that 
.Prove the inequality 
11 replies
Central sequences
EeEeRUT 13
N
by v_Enhance
Source: EGMO 2025 P2
An infinite increasing sequence 
of positive integers is called central if for every positive integer , the arithmetic mean of the first 
terms of the sequence is equal to .
Show that there exists an infinite sequence
of positive integers such that for every central sequence 
there are infinitely many positive integers with 
.
of positive integers is called central if for every positive integer , the arithmetic mean of the first 
terms of the sequence is equal to .Show that there exists an infinite sequence
of positive integers such that for every central sequence 
there are infinitely many positive integers with 
.
13 replies
Prove that
