Inspired by Humberto_Filho
by sqing, Apr 24, 2025, 3:39 AM
Let 
and 
. Prove that
and 
. Prove that
Find the smallest of sum of elements
by hlminh, Apr 24, 2025, 3:37 AM
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 
Inequalities
by Scientist10, Apr 23, 2025, 6:36 PM
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)
Complicated FE
by XAN4, Apr 23, 2025, 11:53 AM
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.
Easy IMO 2023 NT
by 799786, Jul 8, 2023, 4:53 AM
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 
.This post has been edited 7 times. Last edited by v_Enhance, Sep 18, 2023, 12:33 AM
Reason: minor latex pet peeve
Reason: minor latex pet peeve
Hard functional equation
by Jessey, Mar 11, 2020, 1:10 PM
Find all functions 
> 
that satisfy 
, for all 
€ .
> 
that satisfy 
, for all 
€ .Vertices of a convex polygon if and only if m(S) = f(n)
by orl, Aug 10, 2008, 12:26 AM
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 
This post has been edited 1 time. Last edited by djmathman, Oct 3, 2016, 3:25 AM
Reason: changed formatting to match imo compendium
Reason: changed formatting to match imo compendium
Cyclic points and concurrency [1st Lemoine circle]
by shobber, Jun 27, 2006, 7:46 AM
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.$n$ with $2000$ divisors divides $2^n+1$ (IMO 2000)
by Valentin Vornicu, Oct 24, 2005, 10:23 AM
Does there exist a positive integer such that has exactly 2000 prime divisors and divides 
?
such that has exactly 2000 prime divisors and divides 
?This post has been edited 1 time. Last edited by Amir Hossein, Mar 21, 2016, 7:33 PM
Imo Shortlist Problem
by Lopes, Feb 27, 2005, 7:13 PM
Find all triplets of positive integers 
such that 
.
such that 
.The oldest, shortest words — "yes" and "no" — are those which require the most thought.
April 2021
February 2021
March 2020
February 2018
January 2018
November 2017
September 2017
July 2017
June 2017
May 2017
April 2017
March 2017
February 2017
December 2016
April 2016