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Number Theory
fasttrust_12-mn 13
N
by KTYC
Source: Pan African Mathematics Olympiad P1
Find all positive intgers 
and 
such that 
and 
is a prime number
and 
such that 
and 
is a prime number
13 replies
GCD of terms in a sequence
BBNoDollar 0
Determine the general term of the sequence of non-zero natural numbers (a_n)n≥1, with the property that gcd(a_m, a_n, a_p) = gcd(m^2 ,n^2 ,p^2), for any distinct non-zero natural numbers m, n, p.
Note that gcd(a,b,c) denotes the greatest common divisor of the natural numbers a,b,c .
Note that gcd(a,b,c) denotes the greatest common divisor of the natural numbers a,b,c .
0 replies
Aime type Geo
ehuseyinyigit 3
N
by sami1618
Source: Turkish First Round 2024
In a scalene triangle 
, let 
be the midpoint of side 
. Let the line perpendicular to 
at point 
intersect 
at 
. If 
is tangent to 
at 
, find 
.
, let 
be the midpoint of side 
. Let the line perpendicular to 
at point 
intersect 
at 
. If 
is tangent to 
at 
, find 
.
3 replies
be real numbers such that 
.
Equilateral Triangle inside Equilateral Triangles.
abhisruta03 2
N
by Reacheddreams
Source: ISI 2021 P6
If a given equilateral triangle 
of side length 
lies in the union of five equilateral triangles of side length 
, show that there exist four equilateral triangles of side length whose union contains .
of side length 
lies in the union of five equilateral triangles of side length 
, show that there exist four equilateral triangles of side length whose union contains .
2 replies
USAMO 1984 Problem 5 - Polynomial of degree 3n
Binomial-theorem 8
N
by Assassino9931
Source: USAMO 1984 Problem 5
is a polynomial of degree 
such that
Determine
.
8 replies
Finding positive integers with good divisors
nAalniaOMliO 2
N
by KTYC
Source: Belarusian National Olympiad 2025
For every positive integer 
write all its divisors in increasing order: 
.
Find all such that 
.
write all its divisors in increasing order: 
.Find all
such that 
.
2 replies
Balkan MO 2025 p1
Mamadi 1
N
by KevinYang2.71
Source: Balkan MO 2025
An integer 
is called good if there exists a permutation 
of the numbers 
, such that:
and 
have different parities for every 
the sum
is a quadratic residue modulo 
for every 
Prove that there exist infinitely many good numbers, as well as infinitely many positive integers which are not good.
Remark: Here an integer
is considered a quadratic residue modulo if there exists an integer 
such that 
.
is called good if there exists a permutation 
of the numbers 
, such that:
and 
have different parities for every 
the sum
is a quadratic residue modulo 
for every 
Prove that there exist infinitely many good numbers, as well as infinitely many positive integers which are not good.
Remark: Here an integer
is considered a quadratic residue modulo if there exists an integer 
such that 
.
1 reply
Random Points = Problem
kingu 4
N
by zuat.e
Source: Chinese Geometry Handout
Let be a triangle. Let 
be a circle passing through intersecting at 
and at 
. Let 
be the intersection of 
and . Further, let and be the intersections of 
and 
with the tangent to 
at 
. Now, let 
be the second intersection of 
and . Then, prove that , , , 
and are concyclic.
be a triangle. Let 
be a circle passing through intersecting at 
and at 
. Let 
be the intersection of 
and . Further, let and be the intersections of 
and 
with the tangent to 
at 
. Now, let 
be the second intersection of 
and . Then, prove that , , , 
and are concyclic.
4 replies
CooL geo
Pomegranat 2
N
by Curious_Droid
Source: Idk
In triangle
, 
is the midpoint of 
. 
is an arbitrary point on 
. Let 
be the intersection of 
and 
. The line 
intersects with the circumcircle of 
, for the second time at 
. 
is the circumcenter of triangle 
. Prove that 
.
2 replies
