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Proving a group is abelian
dragosgamer12 2
N
by RobertRogo
Source: Radu Diaconu, Gazeta Matematica seria B Nr.2/2025
Let 
be a group, 
a subgroup of 
and 
an endomorphism with the following property:
There exists a nonempty set
such that for any 
there exist 
with 
and 
, for any 
.
a)Prove that
is abelian.
b)If, additionally,
is a subgroup of , prove that 
be a group, 
a subgroup of 
and 
an endomorphism with the following property:There exists a nonempty set
such that for any 
there exist 
with 
and 
, for any 
.a)Prove that
is abelian.b)If, additionally,
is a subgroup of , prove that 
2 replies
Minimum number of points
Ecrin_eren 2
N
by Shan3t
There are 18 teams in a football league. Each team plays against every other team twice in a season—once at home and once away. A win gives 3 points, a draw gives 1 point, and a loss gives 0 points. One team became the champion by earning more points than every other team. What is the minimum number of points this team could have?
2 replies
|A/pA|<=p, finite index=> isomorphism - OIMU 2008 Problem 7
Jorge Miranda 2
N
by pi_quadrat_sechstel
Let 
be an abelian additive group such that all nonzero elements have infinite order and for each prime number 
we have the inequality 
, where 
, 
(where the sum has summands) and 
is the order of the quotient group 
(the index of the subgroup 
).
Prove that each subgroup of of finite index is isomorphic to .
be an abelian additive group such that all nonzero elements have infinite order and for each prime number 
we have the inequality 
, where 
, 
(where the sum has summands) and 
is the order of the quotient group 
(the index of the subgroup 
).Prove that each subgroup of
of finite index is isomorphic to .
2 replies
Weird locus problem
Sedro 7
N
by ReticulatedPython
Points and 
are in the coordinate plane such that 
. Let 
denote the locus of all points 
in the coordinate plane satisfying 
, and let 
be the midpoint of 
. Points 
and 
are on such that 
and 
. The value of 
can be expressed in the form 
, where 
and 
are relatively prime positive integers. Find 
.
and 
are in the coordinate plane such that 
. Let 
denote the locus of all points 
in the coordinate plane satisfying 
, and let 
be the midpoint of 
. Points 
and 
are on such that 
and 
. The value of 
can be expressed in the form 
, where 
and 
are relatively prime positive integers. Find 
.
7 replies
2024 Mock AIME 1 ** p15 (cheaters' trap) - 128 | n^{\sigma (n)} - \sigma(n^n)
parmenides51 3
N
by Sedro
Let 
be the number of positive integers such that divides 
and 
divides
where 
denotes the number of positive integers that divide , including 
and . Find the remainder when is divided by 
.
be the number of positive integers such that divides 
and 
divides
where 
denotes the number of positive integers that divide , including 
and . Find the remainder when is divided by 
.
3 replies
IOQM P23 2024
SomeonecoolLovesMaths 3
N
by lakshya2009
Consider the fourteen numbers, 
. The smallest natural numebr such that they leave distinct remainders when divided by is:
. The smallest natural numebr such that they leave distinct remainders when divided by is:
3 replies
Prove that
Pells equation
Entrepreneur 0
A Pells Equation is defined as follows 
Where 
are positive integers and 
is a non-square positive integer. If 
denotes the n-th set of solution to the equation with 
Then, prove that 
Where 
are positive integers and 
is a non-square positive integer. If 
denotes the n-th set of solution to the equation with 
Then, prove that 
0 replies
Incircle concurrency
niwobin 1
N
by niwobin
Triangle ABC with incenter I, incircle is tangent to BC, AC, and AB at D, E and F respectively.
DT is a diameter for the incircle, and AT meets the incircle again at point H.
Let DH and EF intersect at point J. Prove: AJ//BC.
DT is a diameter for the incircle, and AT meets the incircle again at point H.
Let DH and EF intersect at point J. Prove: AJ//BC.
1 reply
. Prove that
The centroid of ABC lies on ME [2023 Abel, Problem 1b]
Amir Hossein 3
N
by Captainscrubz
In the triangle 
, points 
and 
lie on the side 
, with 
. Also, is the midpoint of 
. Show that the centroid of lies on 
.
, points 
and 
lie on the side 
, with 
. Also, is the midpoint of 
. Show that the centroid of lies on 
.
3 replies
2021 SMT Guts Round 5 p17-20 - Stanford Math Tournament
parmenides51 5
N
by MATHS_ENTUSIAST
p17. Let the roots of the polynomial 
be 
, and 
. What is the sum 
?
p18. Two students are playing a game. They take a deck of five cards numbered through 
, shuffle them, and then place them in a stack facedown, turning over the top card next to the stack. They then take turns either drawing the card at the top of the stack into their hand, showing the drawn card to the other player, or drawing the card that is faceup, replacing it with the card on the top of the pile. This is repeated until all cards are drawn, and the player with the largest sum for their cards wins. What is the probability that the player who goes second wins, assuming optimal play?
p19. Compute the sum of all primes such that 
is also prime.
p20. In how many ways can one color the
vertices of an octagon each red, black, and white, such that no two adjacent sides are the same color?
PS. You should use hide for answers. Collected here.
be 
, and 
. What is the sum 
?p18. Two students are playing a game. They take a deck of five cards numbered
through 
, shuffle them, and then place them in a stack facedown, turning over the top card next to the stack. They then take turns either drawing the card at the top of the stack into their hand, showing the drawn card to the other player, or drawing the card that is faceup, replacing it with the card on the top of the pile. This is repeated until all cards are drawn, and the player with the largest sum for their cards wins. What is the probability that the player who goes second wins, assuming optimal play?p19. Compute the sum of all primes
such that 
is also prime.p20. In how many ways can one color the
vertices of an octagon each red, black, and white, such that no two adjacent sides are the same color?PS. You should use hide for answers. Collected here.
5 replies
