G
Topic
First Poster
Last Poster
Arrange marbles
FunGuy1 3
N
by FunGuy1
Source: Own?
Anna has 
marbles in 
colors such that there are exactly 
marbles of each color. She wants to arrange them on 
shelves, 
marbles on each shelf such that for any 
colors there is a shelf that has marbles of those colors.
Can Anna achieve her goal?
marbles in 
colors such that there are exactly 
marbles of each color. She wants to arrange them on 
shelves, 
marbles on each shelf such that for any 
colors there is a shelf that has marbles of those colors.Can Anna achieve her goal?
3 replies
Projective geometry
definite_denny 1
N
by Funcshun840
Source: IDK
Let ABC be a triangle and let DEF be the tangency point of incircirle with sides BC,CA,AB. Points P,Q are chosen on sides AB,AC such that PQ is parallel to BC and PQ is tangent to the incircle. Let M denote the midpoint of PQ. Let EF intersect BC at T. Prove that TM is tangent to the incircle
1 reply
1 viewing
Nice problem of concurrency
deraxenrovalo 1
N
by Funcshun840
Let 
be an inscribed circle of 
and touching 
, 
, 
at 
, 
, 
respectively. Let 
and 
be diameters of . Let 
and 
be the pole of 
and 
with respect to , respectively. 
cuts again at 
. 
cuts again at 
. The tangent at of cuts 
at 
. The tangent at of cuts 
at 
. Let 
and 
be midpoint of 
and 
, respectively.
Show that :
, 
and perpendicular bisector of 
are concurrent.
be an inscribed circle of 
and touching 
, 
, 
at 
, 
, 
respectively. Let 
and 
be diameters of . Let 
and 
be the pole of 
and 
with respect to , respectively. 
cuts again at 
. 
cuts again at 
. The tangent at of cuts 
at 
. The tangent at of cuts 
at 
. Let 
and 
be midpoint of 
and 
, respectively.Show that :
, 
and perpendicular bisector of 
are concurrent.1 reply
Prove that
Let 
Prove that
D1036 : Composition of polynomials
Dattier 0
Source: les dattes à Dattier
Find all ![$A \in \mathbb Q[x]$](//latex.artofproblemsolving.com/d/1/5/d158f344cf30f8beb5f2199da53272bd8e304704.png)
with ![$\exists Q \in \mathbb Q[x], Q(A(x))= x^{2025!+2}+x^2+x+1$](//latex.artofproblemsolving.com/1/1/9/119a7b617e09ba16bb266b44ab3c85acc529f88f.png)
and 
.
![$A \in \mathbb Q[x]$](http://latex.artofproblemsolving.com/d/1/5/d158f344cf30f8beb5f2199da53272bd8e304704.png)
with ![$\exists Q \in \mathbb Q[x], Q(A(x))= x^{2025!+2}+x^2+x+1$](http://latex.artofproblemsolving.com/1/1/9/119a7b617e09ba16bb266b44ab3c85acc529f88f.png)
and 
.
0 replies
inequality
NTssu 4
N
by Oksutok
Source: Peking University Mathematics Autumn Camp
For given real number 
, prove there exists positive integer 
and positive real number 
, such that 
, for any 
, 
, 
holds.
, prove there exists positive integer 
and positive real number 
, such that 
, for any 
, 
, 
holds.
4 replies
Nice geometry
gggzul 0
Let be a acute triangle with 
. 
are the orthocenter and excenter. Let be a point on the same side of 
as 
, such that 
is equilateral. Let 
be a point on the same side of 
as , such that 
is equilateral. Let 
be a point on the same side of 
as , such that 
is equilateral. Let be the midpoint of 
. Prove that 
are collinear.
be a acute triangle with 
. 
are the orthocenter and excenter. Let be a point on the same side of 
as 
, such that 
is equilateral. Let 
be a point on the same side of 
as , such that 
is equilateral. Let 
be a point on the same side of 
as , such that 
is equilateral. Let be the midpoint of 
. Prove that 
are collinear.
0 replies
Inspired by 2025 KMO
sqing 3
N
by sqing
Source: Own
Let 
be real numbers satisfying 
and 
Prove that 
Let be real numbers satisfying 
and 
Prove that 
be real numbers satisfying 
and 
Prove that 
Let be real numbers satisfying 
and 
Prove that 
3 replies
Reflections and midpoints in triangle
TUAN2k8 0
Source: Own
Given an triangle and a line 
in the plane.Let 
be reflections of 
across the line , respectively.Let 
be the midpoints of 
, respectively.Let 
be the reflections of across , respectively.Prove that the points lie on a line parallel to .
and a line 
in the plane.Let 
be reflections of 
across the line , respectively.Let 
be the midpoints of 
, respectively.Let 
be the reflections of across , respectively.Prove that the points lie on a line parallel to .
0 replies
a exhaustive question
shrayagarwal 19
N
by SomeonecoolLovesMaths
Source: number theory
If 
and 
are natural numbers such that 
is divisible by 
and 
is divisible by 
, then find the least possible value of 
.
and 
are natural numbers such that 
is divisible by 
and 
is divisible by 
, then find the least possible value of 
.
19 replies
