G
Topic
First Poster
Last Poster
Find all p(x) such that p(p) is a power of 2
truongphatt2668 5
N
by tom-nowy
Source: ???
Find all polynomial ![$P(x) \in \mathbb{R}[x]$](//latex.artofproblemsolving.com/4/5/3/453a624c3b002c1b0e78e0023b24dd22ddd03557.png)
such that:
with 
is an 
th prime and 
is an arbitrary positive integer.
![$P(x) \in \mathbb{R}[x]$](http://latex.artofproblemsolving.com/4/5/3/453a624c3b002c1b0e78e0023b24dd22ddd03557.png)
such that:
with 
is an 
th prime and 
is an arbitrary positive integer.
5 replies
1 viewing
Interesting problem from a friend
v4913 10
N
by OronSH
Source: I'm not sure...
Let the incircle 
of 
touch 
at 
, 
, let 
denote the line tangent to through 
. Define 
such that 
. Prove that the circumcenter 
of lies on 
.
of 
touch 
at 
, 
, let 
denote the line tangent to through 
. Define 
such that 
. Prove that the circumcenter 
of lies on 
.
10 replies
IMO ShortList 2002, algebra problem 3
orl 25
N
by Mathandski
Source: IMO ShortList 2002, algebra problem 3
Let 
be a cubic polynomial given by 
, where 
are integers and 
. Suppose that 
for infinitely many pairs 
of integers with 
. Prove that the equation 
has an integer root.
be a cubic polynomial given by 
, where 
are integers and 
. Suppose that 
for infinitely many pairs 
of integers with 
. Prove that the equation 
has an integer root.
25 replies
1 viewing
Inequality on APMO P5
Jalil_Huseynov 41
N
by Mathandski
Source: APMO 2022 P5
Let be real numbers such that 
. Determine the minimum value of 
and determine all values of 
such that the minimum value is achived.
be real numbers such that 
. Determine the minimum value of 
and determine all values of 
such that the minimum value is achived.
41 replies
APMO 2016: one-way flights between cities
shinichiman 18
N
by Mathandski
Source: APMO 2016, problem 4
The country Dreamland consists of 
cities. The airline Starways wants to establish some one-way flights between pairs of cities in such a way that each city has exactly one flight out of it. Find the smallest positive integer 
such that no matter how Starways establishes its flights, the cities can always be partitioned into groups so that from any city it is not possible to reach another city in the same group by using at most 
flights.
Warut Suksompong, Thailand
cities. The airline Starways wants to establish some one-way flights between pairs of cities in such a way that each city has exactly one flight out of it. Find the smallest positive integer 
such that no matter how Starways establishes its flights, the cities can always be partitioned into groups so that from any city it is not possible to reach another city in the same group by using at most 
flights.Warut Suksompong, Thailand
18 replies
Circles intersecting each other
rkm0959 9
N
by Mathandski
Source: 2015 Final Korean Mathematical Olympiad Day 2 Problem 6
There are 
distinct circles in a plane, with radius 
.
Prove that you can select
circles, which form a set 
, which satisfy the following.
For two arbitrary circles in, they intersect with each other or
For two arbitrary circles in, they don't intersect with each other.
distinct circles in a plane, with radius 
.Prove that you can select
circles, which form a set 
, which satisfy the following.For two arbitrary circles in
, they intersect with each other orFor two arbitrary circles in
, they don't intersect with each other.
9 replies
Max value
Hip1zzzil 0
Source: KMO 2025 Round 1 P12
Three distinct nonzero real numbers 
satisfy:
(i)
(ii)
Find the maximum value of
.
satisfy:(i)
(ii)
Find the maximum value of
.0 replies
2018 Hong Kong TST2 problem 4
YanYau 4
N
by Mathandski
Source: 2018HKTST2P4
In triangle 
with incentre 
, let 
and 
by the midpoints of 
and 
respectively, and 
and 
be the feet of the altitudes from 
and to the respective sides. Denote by 
the line being tangent tot he circumcircle of triangle and passing through 
, and denote by 
the reflection of in 
. Let 
by the intersection of 
and , and let 
be the intersection of 
and . Defined 
analogously. If 
is the intersection of 
and 
, prove that 
.
with incentre 
, let 
and 
by the midpoints of 
and 
respectively, and 
and 
be the feet of the altitudes from 
and to the respective sides. Denote by 
the line being tangent tot he circumcircle of triangle and passing through 
, and denote by 
the reflection of in 
. Let 
by the intersection of 
and , and let 
be the intersection of 
and . Defined 
analogously. If 
is the intersection of 
and 
, prove that 
.
4 replies
Prove that the triangle is isosceles.
TUAN2k8 4
N
by JARP091
Source: My book
Given acute triangle with two altitudes 
and 
.Let be the point on the line such that 
.The lines 
and 
intersect at point 
, and 
is the point on segment 
such that 
.Suppose that bisects .Prove that triangle 
is isosceles.
with two altitudes 
and 
.Let be the point on the line such that 
.The lines 
and 
intersect at point 
, and 
is the point on segment 
such that 
.Suppose that bisects .Prove that triangle 
is isosceles.
4 replies
Pythagoras...
Hip1zzzil 0
Source: KMO 2025 Round 1 P20
Find the sum of all s such that:
There exists two odd positive integers
such that 
s such that:There exists two odd positive integers
such that 
0 replies
