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A final attempt to make a combinatorics problem
JARP091 4
N
by JARP091
Source: At the time of posting the problem I do not know the source if any
Let 
be a positive integer and consider the set 
.
Two players alternate moves. On each turn, the current player must select a nonempty subset
of numbers not previously chosen such that for every distinct 
, neither 
divides 
nor divides .
After selecting
, all multiples of every element in , including those in itself, are removed from 
.
The game continues with the reduced set until no moves are possible.
Determine, for each, which player has a winning strategy if any
Note: It might be wrong or maybe too easy.
be a positive integer and consider the set 
.Two players alternate moves. On each turn, the current player must select a nonempty subset
of numbers not previously chosen such that for every distinct 
, neither 
divides 
nor divides .After selecting
, all multiples of every element in , including those in itself, are removed from 
.The game continues with the reduced set
until no moves are possible.Determine, for each
, which player has a winning strategy if anyNote: It might be wrong or maybe too easy.
4 replies
Less than or equal to 30°
orl 11
N
by Twan
Source: IMO 1991, Day 2, Problem 4, IMO ShortList 1991, Problem 24 (FRA 2)
Let 
be a triangle and 
an interior point of . Show that at least one of the angles 
is less than or equal to 
.
be a triangle and 
an interior point of . Show that at least one of the angles 
is less than or equal to 
.
11 replies
Simple Geometry
AbdulWaheed 4
N
by shanelin-sigma
Source: EGMO
Try to avoid Directed angles
Let ABC be an acute triangle inscribed in circle
. Let 
be the midpoint of the arc 
not containing 
and define 
similarly. Show that the orthocenter of 
is the incenter 
of 
.
Let ABC be an acute triangle inscribed in circle
. Let 
be the midpoint of the arc 
not containing 
and define 
similarly. Show that the orthocenter of 
is the incenter 
of 
.
4 replies
Serbian selection contest for the IMO 2025 - P6
OgnjenTesic 1
N
by JARP091
Source: Serbian selection contest for the IMO 2025
For an 
table filled with natural numbers, we say it is a divisor table if:
- the numbers in the
-th row are exactly all the divisors of some natural number 
,
- the numbers in the
-th column are exactly all the divisors of some natural number 
,
-
for every 
.
A prime number
is given. Determine the smallest natural number 
, divisible by , such that there exists an divisor table, or prove that such does not exist.
Proposed by Pavle Martinović
table filled with natural numbers, we say it is a divisor table if:- the numbers in the
-th row are exactly all the divisors of some natural number 
,- the numbers in the
-th column are exactly all the divisors of some natural number 
,-
for every 
.A prime number
is given. Determine the smallest natural number 
, divisible by , such that there exists an divisor table, or prove that such does not exist.Proposed by Pavle Martinović
1 reply
FE with conditions on $x,y$
Adywastaken 2
N
by Adywastaken
Source: OAO
Find all functions 
such that 
,
![\[
f(x^2+f(y))=f(xf(x))+y
\]](//latex.artofproblemsolving.com/d/e/d/dedd2338779fcc6723077ce586713fb4a68b9931.png)
such that 
,![\[
f(x^2+f(y))=f(xf(x))+y
\]](http://latex.artofproblemsolving.com/d/e/d/dedd2338779fcc6723077ce586713fb4a68b9931.png)
2 replies
2021 EGMO P4: Reflection of A over EF lies on BC
anser 48
N
by cursed_tangent1434
Source: EGMO 2021 P4
Let be a triangle with incenter and let 
be an arbitrary point on the side 
. Let the line through perpendicular to 
intersect 
at 
. Let the line through perpendicular to intersect at 
. Prove that the reflection of across the line 
lies on the line .
be a triangle with incenter and let 
be an arbitrary point on the side 
. Let the line through perpendicular to 
intersect 
at 
. Let the line through perpendicular to intersect at 
. Prove that the reflection of across the line 
lies on the line .
48 replies
Serbian selection contest for the IMO 2025 - P2
OgnjenTesic 10
N
by Mahdi_Mashayekhi
Source: Serbian selection contest for the IMO 2025
Let be an acute triangle. Let 
be the reflection of point over the line . Let 
and 
be the circumcenter and the orthocenter of triangle , respectively, and let be the midpoint of segment 
. Let and 
be the points where the reflection of line 
with respect to line 
intersects the circumcircle of triangle , where point lies on the arc not containing . If 
is a point on the line 
such that 
, prove that 
.
Proposed by Strahinja Gvozdić
be an acute triangle. Let 
be the reflection of point over the line . Let 
and 
be the circumcenter and the orthocenter of triangle , respectively, and let be the midpoint of segment 
. Let and 
be the points where the reflection of line 
with respect to line 
intersects the circumcircle of triangle , where point lies on the arc not containing . If 
is a point on the line 
such that 
, prove that 
.Proposed by Strahinja Gvozdić
10 replies
find all Polynomials
andria 9
N
by A.H.H
Source: Iranian third round 2015 algebra problem 5
Find all polynomials ![$p(x)\in\mathbb{R}[x]$](//latex.artofproblemsolving.com/9/a/e/9aecb18edc17f7391a44db6e4f9d1502ab1d92f7.png)
such that for all 
:
such that:
![$p(x)\in\mathbb{R}[x]$](http://latex.artofproblemsolving.com/9/a/e/9aecb18edc17f7391a44db6e4f9d1502ab1d92f7.png)
such that for all 
:
such that:
9 replies
AT // BC wanted
parmenides51 105
N
by Adywastaken
Source: IMO 2019 SL G1
Let be a triangle. Circle 
passes through , meets segments 
and 
again at points and respectively, and intersects segment at and 
such that lies between 
and . The tangent to circle 
at and the tangent to circle 
at meet at point 
. Suppose that points and are distinct. Prove that line 
is parallel to .
(Nigeria)
be a triangle. Circle 
passes through , meets segments 
and 
again at points and respectively, and intersects segment at and 
such that lies between 
and . The tangent to circle 
at and the tangent to circle 
at meet at point 
. Suppose that points and are distinct. Prove that line 
is parallel to .(Nigeria)
105 replies
Two lines concur on (ABC)
amar_04 19
N
by Giant_PT
Source: XVII Sharygin Corespondnce Round P13
In triangle with circumcircle and incenter , point 
bisects arc 
and line 
meets at 
. The excircle opposite to touches 
at point . Point 
on the circumcircle of 
is such that 
. Prove that the lines 
and 
meet on .
with circumcircle and incenter , point 
bisects arc 
and line 
meets at 
. The excircle opposite to touches 
at point . Point 
on the circumcircle of 
is such that 
. Prove that the lines 
and 
meet on .
19 replies
