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k Funcional equation problem
khongphaiwminh 1
N
by jasperE3
Determine all functions 
that satisfy the equation
for any 
. Note that 
.
that satisfy the equation
for any 
. Note that 
.
1 reply
AO and KI meet on $\Gamma$
Kayak 29
N
by Mathandski
Source: Indian TST 3 P2
Let 
be an acute-angled scalene triangle with circumcircle 
and circumcenter 
. Suppose 
. Let 
be the orthocenter and 
be the incenter of triangle . Let 
be the midpoint of the arc 
of the circumcircle of triangle 
, containing .
Let
be a point on the arc 
of not containing 
, such that 
. Let 
be the circumcenter of triangle 
. Prove that the lines 
and 
meet on .
Proposed by Anant Mudgal
be an acute-angled scalene triangle with circumcircle 
and circumcenter 
. Suppose 
. Let 
be the orthocenter and 
be the incenter of triangle . Let 
be the midpoint of the arc 
of the circumcircle of triangle 
, containing .Let
be a point on the arc 
of not containing 
, such that 
. Let 
be the circumcenter of triangle 
. Prove that the lines 
and 
meet on .Proposed by Anant Mudgal
29 replies
4 Variables Cyclic Ineq
nataliaonline75 1
N
by NO_SQUARES
Prove that for every 
non-negative real numbers, then we have:
non-negative real numbers, then we have:
1 reply
IMO Genre Predictions
ohiorizzler1434 56
N
by Theoryman007
Everybody, with IMO upcoming, what are you predictions for the problem genres?
Personally I predict:
Personally I predict:
56 replies
Number theory
MathsII-enjoy 2
N
by SimplisticFormulas
Prove that when 
| 
with 
are positive integers and 
is prime (
), we get: 
| 
with 
are positive integers and 
is prime (
), we get: 
2 replies
National diophantine equation
KAME06 2
N
by damyan
Source: OMEC Ecuador National Olympiad Final Round 2024 N3 P5 day 2
Find all triples of non-negative integer numbers 
such that 
and:
such that 
and:
2 replies
IMO ShortList 1998, number theory problem 1
orl 55
N
by reni_wee
Source: IMO ShortList 1998, number theory problem 1
Determine all pairs 
of positive integers such that 
is divisible by 
.
of positive integers such that 
is divisible by 
.
55 replies
Number of sets S
Jackson0423 2
N
by Jackson0423
Let 
be a set consisting of non-negative integers such that:
1.
,
2. For any
, both 
and 
.
Find the number of such sets.
be a set consisting of non-negative integers such that:1.
,2. For any
, both 
and 
.Find the number of such sets
.
2 replies
F.E....can you solve it?
Jackson0423 16
N
by jasperE3
Find all functions 
such that
![\[
f\left(\frac{x^2 - f(x)}{f(x) - 1}\right) = x
\]](//latex.artofproblemsolving.com/8/0/0/8009a4709ffd6708023d67db2b544e605d1fbe68.png)
for all real numbers 
satisfying 
.
such that![\[
f\left(\frac{x^2 - f(x)}{f(x) - 1}\right) = x
\]](http://latex.artofproblemsolving.com/8/0/0/8009a4709ffd6708023d67db2b544e605d1fbe68.png)
for all real numbers 
satisfying 
.
16 replies
Find all positive a,b
shobber 14
N
by reni_wee
Source: APMO 2002
Find all positive integers 
and 
such that
![\[ {a^2+b\over b^2-a}\quad\mbox{and}\quad{b^2+a\over a^2-b} \]](//latex.artofproblemsolving.com/e/3/b/e3bc96af9f635e1c10e21dc91894c8afbf113a6d.png)
are both integers.
and 
such that![\[ {a^2+b\over b^2-a}\quad\mbox{and}\quad{b^2+a\over a^2-b} \]](http://latex.artofproblemsolving.com/e/3/b/e3bc96af9f635e1c10e21dc91894c8afbf113a6d.png)
are both integers.
14 replies
