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Two times derivable real function
Valentin Vornicu 13
N
by solyaris
Source: RMO 2008, 11th Grade, Problem 3
Let 
be a function, two times derivable on 
for which there exist 
such that
![\[ \frac { f(b)-f(a) }{b-a} \neq f'(c) ,\]](//latex.artofproblemsolving.com/a/3/1/a31861ef909a6e17ed8b495b9b9b16d887ba573f.png)
for all 
.
Prove that
.
be a function, two times derivable on 
for which there exist 
such that![\[ \frac { f(b)-f(a) }{b-a} \neq f'(c) ,\]](http://latex.artofproblemsolving.com/a/3/1/a31861ef909a6e17ed8b495b9b9b16d887ba573f.png)
for all 
.Prove that
.
13 replies
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
and 
.
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
