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Peru IMO TST 2023
diegoca1 4
N
by grupyorum
Source: Peru IMO TST 2023 pre-selection P1
Let 
be non-negative real numbers such that 
. Prove the inequality
![\[
6xyz \leq x(1 - x) + y(1 - y) + z(1 - z),
\]](//latex.artofproblemsolving.com/f/5/b/f5b208ea6439994cb40f6b7f0806c216d9868e86.png)
and determine when equality holds.
be non-negative real numbers such that 
. Prove the inequality![\[
6xyz \leq x(1 - x) + y(1 - y) + z(1 - z),
\]](http://latex.artofproblemsolving.com/f/5/b/f5b208ea6439994cb40f6b7f0806c216d9868e86.png)
and determine when equality holds.
4 replies
IMO Shortlist 2017 A1
math90 83
N
by prMoLeGend42
Source: IMO Shortlist 2017
Let 
, and 
be positive integers such that
If 
, prove that the polynomial
has no positive roots.
, and 
be positive integers such that
If 
, prove that the polynomial
has no positive roots.
83 replies
,
.
Inequality
SunnyEvan 7
N
by SunnyEvan
Source: Own
Let 
, such that: 
Prove that:
When does the equality hold ?
, such that: 
Prove that:
When does the equality hold ?
7 replies
k tokens in nxn unit squares board, game conditions, min and max wanted
parmenides51 1
N
by wasd-
Source: 47th Austrian Mathematical Olympiad National Competition (Final Round, part 2 ) May 26, 2016 p5
Consider a board consisting of 
unit squares where 
. Two cells are called neighbors if they share a horizontal or vertical border. In the beginning, all cells together contain 
tokens. Each cell may contain one or several tokens or none. In each turn, choose one of the cells that contains at least one token for each of its neighbors and move one of those to each of its neighbors. The game ends if no such cell exists.
(a) Find the minimal such that the game does not end for any starting configuration and choice of cells during the game.
(b) Find the maximal such that the game ends for any starting configuration and choice of cells during the game.
Proposed by Theresia Eisenkölbl
unit squares where 
. Two cells are called neighbors if they share a horizontal or vertical border. In the beginning, all cells together contain 
tokens. Each cell may contain one or several tokens or none. In each turn, choose one of the cells that contains at least one token for each of its neighbors and move one of those to each of its neighbors. The game ends if no such cell exists.(a) Find the minimal
such that the game does not end for any starting configuration and choice of cells during the game.(b) Find the maximal
such that the game ends for any starting configuration and choice of cells during the game.Proposed by Theresia Eisenkölbl
1 reply
Number Theory Chain!
JetFire008 75
N
by whwlqkd
I will post a question and someone has to answer it. Then they have to post a question and someone else will answer it and so on. We can only post questions related to Number Theory and each problem should be more difficult than the previous. Let's start!
?75 replies
Set or graph or well known
ItzsleepyXD 1
N
by maromex
Source: IDK
Let ![$[ n ]$](//latex.artofproblemsolving.com/4/c/a/4caca647174f974de170d214409efd015799f144.png)
be set of 
. Consider 
be 
subset of ![$[n]$](//latex.artofproblemsolving.com/0/d/e/0deff9086fd7840ef23860f5bc924f1d4d2d2310.png)
such that ![$S_1 \cup S_2 \cup \cdots \cup S_{n-1} = [n]$](//latex.artofproblemsolving.com/b/3/e/b3e4063c64e12ae5c645732475ffde30e26cd011.png)
and 
for 
. Prove that there exist set ![$T \subset [n]$](//latex.artofproblemsolving.com/d/a/b/dab13038bebce879b3ad3cb99320c0906964384b.png)
such that 
for all .
![$[ n ]$](http://latex.artofproblemsolving.com/4/c/a/4caca647174f974de170d214409efd015799f144.png)
be set of 
. Consider 
be 
subset of ![$[n]$](http://latex.artofproblemsolving.com/0/d/e/0deff9086fd7840ef23860f5bc924f1d4d2d2310.png)
such that ![$S_1 \cup S_2 \cup \cdots \cup S_{n-1} = [n]$](http://latex.artofproblemsolving.com/b/3/e/b3e4063c64e12ae5c645732475ffde30e26cd011.png)
and 
for 
. Prove that there exist set ![$T \subset [n]$](http://latex.artofproblemsolving.com/d/a/b/dab13038bebce879b3ad3cb99320c0906964384b.png)
such that 
for all .
1 reply
1 viewing
IMO Shortlist 2012, Geometry 3
lyukhson 78
N
by blueprimes
Source: IMO Shortlist 2012, Geometry 3
In an acute triangle 
the points 
and 
are the feet of the altitudes through 
and 
respectively. The incenters of the triangles 
and 
are 
and 
respectively; the circumcenters of the triangles 
and 
are 
and 
respectively. Prove that 
and 
are parallel.
the points 
and 
are the feet of the altitudes through 
and 
respectively. The incenters of the triangles 
and 
are 
and 
respectively; the circumcenters of the triangles 
and 
are 
and 
respectively. Prove that 
and 
are parallel.
78 replies
equal segments concerning circumcircle
parmenides51 5
N
by Fly_into_the_sky
Source: IGO Elementary 2016 2
Let 
be the circumcircle of triangle with 
. Let 
be a point on 
and 
be a point on the circle , such that 
. (The points 
and lie on different sides of the line 
). The line 
intersects for the second time in point 
. Show that 
.
by Iman Maghsoudi
be the circumcircle of triangle with 
. Let 
be a point on 
and 
be a point on the circle , such that 
. (The points 
and lie on different sides of the line 
). The line 
intersects for the second time in point 
. Show that 
.by Iman Maghsoudi
5 replies
Peru IMO TST 2024
diegoca1 2
N
by RagvaloD
Source: Peru IMO TST 2024 D2 P2
Consider the system of equations:
![\[
\begin{cases}
b^2 + 1 = ac, \\
c^2 + 1 = bd,
\end{cases}
\qquad (1)
\]](//latex.artofproblemsolving.com/f/0/b/f0b02d1f56bfc12a2f72618b6a12822b1ad5fe0b.png)
where 
are positive integers.
a) Prove that there are infinitely many positive integer solutions to system (1).
b) Prove that if
is a solution of (1), then
![\[
a = 3b - c, \quad d = 3c - b.
\]](//latex.artofproblemsolving.com/9/b/d/9bd0e3d39592062f1cc5620b5d13729843d0c805.png)
![\[
\begin{cases}
b^2 + 1 = ac, \\
c^2 + 1 = bd,
\end{cases}
\qquad (1)
\]](http://latex.artofproblemsolving.com/f/0/b/f0b02d1f56bfc12a2f72618b6a12822b1ad5fe0b.png)
where 
are positive integers.a) Prove that there are infinitely many positive integer solutions to system (1).
b) Prove that if
is a solution of (1), then![\[
a = 3b - c, \quad d = 3c - b.
\]](http://latex.artofproblemsolving.com/9/b/d/9bd0e3d39592062f1cc5620b5d13729843d0c805.png)
2 replies
