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Solution needed ASAP
UglyScientist 9
N
by MathsII-enjoy
is acute triangle. 
is orthocenter, 
is the midpoint of 
, 
is the midpoint of smaller arc . Point 
is on 
such that, 
is perpendicular to 
. Prove that: 
is paralelogram(Synthetic sol needed).
9 replies
AC, BF, DE concurrent
a1267ab 75
N
by NicoN9
Source: APMO 2020 Problem 1
Let 
be the circumcircle of 
. Let 
be a point on the side 
. The tangent to at 
intersects the parallel line to 
through at point 
. The segment 
intersects again at 
. Suppose 
, , , are concyclic. Prove that 
, 
, 
are concurrent.
be the circumcircle of 
. Let 
be a point on the side . The tangent to at 
intersects the parallel line to 
through at point 
. The segment 
intersects again at 
. Suppose 
, , , are concyclic. Prove that 
, 
, 
are concurrent.75 replies
FE on R+
AshAuktober 7
N
by GingerMan
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)
7 replies
2-adic Valuation Unbounded
tigerzhang 14
N
by GingerMan
Source: Own
For any nonzero integer, define 
as the largest integer 
such that 
. Find all integers 
that are not powers of 
such that the sequence 
is unbounded.
as the largest integer 
such that 
. Find all integers 
that are not powers of 
such that the sequence 
is unbounded.
14 replies
IMO 90/3 and IMO 00/5 cross-up
v_Enhance 60
N
by GingerMan
Source: USA TSTST 2018 Problem 8
For which positive integers 
do there exist infinitely many positive integers such that 
divides 
?
Evan Chen and Ankan Bhattacharya
do there exist infinitely many positive integers such that 
divides 
?Evan Chen and Ankan Bhattacharya
60 replies
gcd (a^n+b,b^n+a) is constant
EthanWYX2009 81
N
by GingerMan
Source: 2024 IMO P2
Determine all pairs 
of positive integers for which there exist positive integers 
and 
such that
holds for all integers 
(Note that 
denotes the greatest common divisor of integers 
and 
)
Proposed by Valentio Iverson, Indonesia
of positive integers for which there exist positive integers 
and 
such that
holds for all integers 
(Note that 
denotes the greatest common divisor of integers 
and 
)Proposed by Valentio Iverson, Indonesia
81 replies
Orthocenter
jayme 5
N
by Ianis
Dear Mathlinkers,
1. ABC an acuatangle triangle
2. H the orthcenter of ABC
3. DEF the orthic triangle of ABC
4. A* the midpoint of AH
5. X the point of intersection of AH and EF.
Prove : X is the orthocenter of A*BC.
Sincerely
Jean-Louis
1. ABC an acuatangle triangle
2. H the orthcenter of ABC
3. DEF the orthic triangle of ABC
4. A* the midpoint of AH
5. X the point of intersection of AH and EF.
Prove : X is the orthocenter of A*BC.
Sincerely
Jean-Louis
5 replies
positive integers forming a perfect square
cielblue 1
N
by aaravdodhia
Find all positive integers such that 
is a perfect square.
such that 
is a perfect square.
1 reply
