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Primes p such that p and p^2+2p-8 are primes too
mhet49 44
N
by MITDragon
Source: Albanian National Math Olympiad 2012
Find all primes 
such that 
and 
are also primes.
such that 
and 
are also primes.
44 replies
Integer polynomial w factorials
Solilin 1
N
by Tkn
Source: 9th Thailand MO
Let 
be pairwise distinct integers. Show that the equation 
has at most one integral solution.
be pairwise distinct integers. Show that the equation 
has at most one integral solution.
1 reply
Combo NT
a_507_bc 4
N
by Namura
Source: Silk Road 2024 P1
Let 
be a positive integer and let 
be odd primes. Prove that the positive integers 
can be colored in 
colors, such that for any 
of the same color, 
is not divisible by and 
.
be a positive integer and let 
be odd primes. Prove that the positive integers 
can be colored in 
colors, such that for any 
of the same color, 
is not divisible by and 
.
4 replies
Old hard problem
ItzsleepyXD 2
N
by ItzsleepyXD
Source: IDK
Let 
be a triangle and let 
be its circumcenter and 
its incenter.
Let
be the radical center of its three mixtilinears and let 
be the isogonal conjugate of .
Let
be the Gergonne point of the triangle .
Prove that line
is parallel with line 
.
be a triangle and let 
be its circumcenter and 
its incenter.Let
be the radical center of its three mixtilinears and let 
be the isogonal conjugate of .Let
be the Gergonne point of the triangle .Prove that line
is parallel with line 
.
2 replies
Polynomial Factors
somebodyyouusedtoknow 1
N
by luutrongphuc
Source: San Diego Honors Math Contest 2025 Part II, Problem 2
Let 
be a polynomial with real coefficients such that 
for all 
. Prove that 
for some real constant 
and 
.
be a polynomial with real coefficients such that 
for all 
. Prove that 
for some real constant 
and 
.
1 reply
I need the technique
DievilOnlyM 15
N
by sqing
Let a,b,c be real numbers such that: 
.
FInd the minimum value of:
.FInd the minimum value of:
15 replies
Linear colorings mod 2^n
vincentwant 1
N
by vincentwant
Let be a positive integer. The ordered pairs 
where 
are integers in 
are each labeled with a positive integer less than or equal to 
such that every label is used exactly times and there exist integers 
and 
such that the following property holds: For any two lattice points 
and 
that are both labeled 
, there exists an integer 
such that 
and 
are both divisible by . How many such labelings exist?
be a positive integer. The ordered pairs 
where 
are integers in 
are each labeled with a positive integer less than or equal to 
such that every label is used exactly times and there exist integers 
and 
such that the following property holds: For any two lattice points 
and 
that are both labeled 
, there exists an integer 
such that 
and 
are both divisible by . How many such labelings exist?
1 reply
sqrt(n) or n+p (Generalized 2017 IMO/1)
vincentwant 1
N
by vincentwant
Let be an odd prime. Define 
over the positive integers as follows:
Let be chosen such that there exists an ordered pair of positive integers 
where 
such that 
. Prove that there exists at least three integers 
such that 
and 
is a perfect square.
be an odd prime. Define 
over the positive integers as follows:
Let
be chosen such that there exists an ordered pair of positive integers 
where 
such that 
. Prove that there exists at least three integers 
such that 
and 
is a perfect square.
1 reply
Flight between cities
USJL 5
N
by Photaesthesia
Source: 2025 Taiwan TST Round 1 Mock P5
A country has 2025 cites, with some pairs of cities having bidirectional flight routes between them. For any pair of the cities, the flight route between them must be operated by one of the companies 
or 
. To avoid unfairly favoring specific company, the regulation ensures that if there have three cities 
and 
, with flight routes 
and 
operated by two different companies, then there must exist flight route 
operated by the third company different from and .
Let
, 
and 
denote the number of flight routes operated by companies and , respectively. It is known that, starting from a city, we can arrive any other city through a series of flight routes (not necessary operated by the same company). Find the minimum possible value of 
.
Proposed by usjl and YaWNeeT
or 
. To avoid unfairly favoring specific company, the regulation ensures that if there have three cities 
and 
, with flight routes 
and 
operated by two different companies, then there must exist flight route 
operated by the third company different from and .Let
, 
and 
denote the number of flight routes operated by companies and , respectively. It is known that, starting from a city, we can arrive any other city through a series of flight routes (not necessary operated by the same company). Find the minimum possible value of 
.Proposed by usjl and YaWNeeT
5 replies
A problem from Le Anh Vinh book.
minhquannguyen 0
Source: LE ANH VINH, DINH HUONG BOI DUONG HOC SINH NANG KHIEU TOAN TAP 1 DAI SO
Let is a positive integer. Determine all functions 
such that
![\[f(x^{n+1}+y^{n+1})=x^nf(x)+y^nf(y),\forall x,y>1.\]](//latex.artofproblemsolving.com/a/4/7/a47d53806f2148cc3339335fb990b33641085896.png)
is a positive integer. Determine all functions 
such that![\[f(x^{n+1}+y^{n+1})=x^nf(x)+y^nf(y),\forall x,y>1.\]](http://latex.artofproblemsolving.com/a/4/7/a47d53806f2148cc3339335fb990b33641085896.png)
0 replies
