and 
.![\[P=\frac{1}{5-2a}+\frac{1}{5-2b}+\frac{1}{5-2c}.\]](http://latex.artofproblemsolving.com/c/3/7/c3708768a84c2bbb1e4fae502a9bb1e92cb13d02.png)
and 
.
Y by Zuyong
Problem. Given 
be non-negative real numbers such that 
Prove that
be non-negative real numbers such that 
Prove that
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Topic
First Poster
Last Poster
Cyclic points and concurrency [1st Lemoine circle]
shobber 10
N
by Ilikeminecraft
Source: China TST 2005
Let 
be the circumcircle of acute triangle 
. Two tangents of from 
and 
intersect at 
, 
and 
intersect at 
. Point 
, 
are on 
and 
such that 
and 
.
(1) Prove that
are concyclic.
(2) Denote
the centre of the circle passing through . 
, 
are difined similarly. Prove that 
, 
, 
are concurrent.
be the circumcircle of acute triangle 
. Two tangents of from 
and 
intersect at 
, 
and 
intersect at 
. Point 
, 
are on 
and 
such that 
and 
.(1) Prove that
are concyclic.(2) Denote
the centre of the circle passing through . 
, 
are difined similarly. Prove that 
, 
, 
are concurrent.
10 replies
Hard functional equation
Jessey 4
N
by jasperE3
Source: Belarus 2005
Find all functions 
> 
that satisfy 
, for all 
€ .
> 
that satisfy 
, for all 
€ .
4 replies
Vertices of a convex polygon if and only if m(S) = f(n)
orl 12
N
by Maximilian113
Source: IMO Shortlist 2000, C3
Let 
be a fixed positive integer. Given a set 
of 
points in the plane such that no three are collinear and no four concyclic, let 
be the number of circles 
that contain 
in their interior, and let ![\[m(S)=a_1+a_2+\cdots + a_n.\]](//latex.artofproblemsolving.com/4/9/b/49b928b1f20e2d1799a1234c7f25a2bc3d62d0dd.png)
Prove that there exists a positive integer 
depending only on 
such that the points of 
are the vertices of a convex polygon if and only if 
be a fixed positive integer. Given a set 
of 
points in the plane such that no three are collinear and no four concyclic, let 
be the number of circles 
that contain 
in their interior, and let ![\[m(S)=a_1+a_2+\cdots + a_n.\]](http://latex.artofproblemsolving.com/4/9/b/49b928b1f20e2d1799a1234c7f25a2bc3d62d0dd.png)
Prove that there exists a positive integer 
depending only on 
such that the points of 
are the vertices of a convex polygon if and only if 
12 replies
Imo Shortlist Problem
Lopes 35
N
by Maximilian113
Source: IMO Shortlist 2000, Problem N4
Find all triplets of positive integers 
such that 
.
such that 
.
35 replies
Inequalities
Scientist10 2
N
by arqady
If 
, then prove that the following inequality holds:
![\[
\sum_{\text{cyc}} \sqrt{1 + \left(x\sqrt{1 + y^2} + y\sqrt{1 + x^2}\right)^2} \geq \sum_{\text{cyc}} xy + 2\sum_{\text{cyc}} x
\]](//latex.artofproblemsolving.com/9/5/5/955863a0cf8747ae45b736b0631f243d3908eb84.png)
, then prove that the following inequality holds:![\[
\sum_{\text{cyc}} \sqrt{1 + \left(x\sqrt{1 + y^2} + y\sqrt{1 + x^2}\right)^2} \geq \sum_{\text{cyc}} xy + 2\sum_{\text{cyc}} x
\]](http://latex.artofproblemsolving.com/9/5/5/955863a0cf8747ae45b736b0631f243d3908eb84.png)
2 replies
$n$ with $2000$ divisors divides $2^n+1$ (IMO 2000)
Valentin Vornicu 65
N
by ray66
Source: IMO 2000, Problem 5, IMO Shortlist 2000, Problem N3
Does there exist a positive integer such that has exactly 2000 prime divisors and divides 
?
such that has exactly 2000 prime divisors and divides 
?
65 replies
Find the smallest of sum of elements
hlminh 0
Let 
and 
is a subset of 
such that if 
then 
Find the smallest of 
and 
is a subset of 
such that if 
then 
Find the smallest of 
0 replies
Easy IMO 2023 NT
799786 133
N
by Maximilian113
Source: IMO 2023 P1
Determine all composite integers 
that satisfy the following property: if 
, 
, 
, 
are all the positive divisors of 
with 
, then 
divides 
for every 
.
that satisfy the following property: if 
, 
, 
, 
are all the positive divisors of 
with 
, then 
divides 
for every 
.
133 replies
Complicated FE
XAN4 2
N
by cazanova19921
Source: own
Find all solutions for the functional equation 
, in which 
: 
Note: the solution is actually quite obvious -
, but the proof is important.
Note 2: it is likely that the result can be generalized into a more advanced questions, potentially involving more bash.
, in which 
: 
Note: the solution is actually quite obvious -
, but the proof is important.Note 2: it is likely that the result can be generalized into a more advanced questions, potentially involving more bash.
2 replies
Something nice
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Source: own
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Y by MihaiT, Zuyong
Non sense post.
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Y by Zuyong
Something not relevant
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Y by Zuyong
Something not relevant
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Y by Zuyong
Something not relevant
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Y by Zuyong
Problem. Given be non-negative real numbers such that Prove that
be non-negative real numbers such that Prove that
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Y
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Y by Zuyong
Problem. Given be non-negative real numbers such that 
Prove that
be non-negative real numbers such that 
Prove that
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and 
.
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Y by ehuseyinyigit, Zuyong
KhuongTrang wrote:
Problem. Given be non-negative real numbers such that Prove that
be non-negative real numbers such that Prove thatProblem. Given non-negative real numbers satisfying
Prove that
Equality holds iff 
or 
and any cyclic permutations.
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Y
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#74
Jul 15, 2024, 4:54 PM
Y by
KhuongTrang wrote:
Problem. Given be non-negative real numbers such that Prove that
be non-negative real numbers such that Prove thatwhat why does that look so easy and difficult at the same time lol
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,
and 
be non-negative numbers such that 
.
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Y by Zuyong
Problem. Given be non-negative real numbers such that 
Prove that
be non-negative real numbers such that 
Prove that
Z
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Y
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#84
Aug 10, 2024, 5:59 PM
Y by
KhuongTrang wrote:
Problem. Given be non-negative real numbers such that Prove that
be non-negative real numbers such that Prove that![\[\sum a^2+8\sum ab\ge 3\sum a\sum\sqrt{ab}.\]](http://latex.artofproblemsolving.com/2/b/a/2bac168124825dbc152f3085a0d3723fa34caf47.png)
Changing 
it becomes a fourth degree inequality, so is linear in 
. Thus it remains to check only the cases 
and 
which is easy.
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Y by Zuyong
Problem. Let be non-negative real variables with 
Prove that
Equality holds iff 
or 
where 
be non-negative real variables with 
Prove that
Equality holds iff 
or 
where 
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and set 
The inequality becomes
ưhere
which ends the proofs 
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Y by Zuyong
#93 Could you please check your solution again, jokehim? I think this inequality is very hard to think of a proof in normal way.
Hope to see some ideas. Btw, it is obviously true by BW.
Hope to see some ideas. Btw, it is obviously true by BW.
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Y by
KhuongTrang wrote:
#93 Could you please check your solution again, jokehim? I think this inequality is very hard to think of a proof in normal way.
Hope to see some ideas. Btw, it is obviously true by BW.
Problem. Let be positive real variables with 
Prove that
Equality holds iff 
Hope to see some ideas. Btw, it is obviously true by BW.
Problem. Let
be positive real variables with 
Prove that
Equality holds iff 
I don't see what's wrong with my solution
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#96
Mar 27, 2025, 6:38 AM
Y by
KhuongTrang wrote:
Problem. Let be non-negative real variables with 
Prove thatEquality holds iff or where
be non-negative real variables with 
Prove thatEquality holds iff or where ![\[\frac{12a(a+2b)}{4ab+bc+ca} \geqslant \frac{32a^3+3(33b+56c)a^2+3(26b^2+102bc+13c^2)a-4(4b+c)(b-2c)^2}{11[ab(a+b)+bc(b+c)+ca(c+a)]+51abc}.\]](http://latex.artofproblemsolving.com/a/b/0/ab0f41dc6c9cf3e143885b3e7e7079048d3841c3.png)
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Y by arqady, Zuyong
Problem. Let be three non-negative real numbers with Prove that
When does equality hold?
be three non-negative real numbers with Prove that
When does equality hold?
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#111
Today at 2:08 AM
Y by
Problem. Let be three non-negative real numbers with Prove that
When does equality hold?
be three non-negative real numbers with Prove that
When does equality hold?
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