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Algebra inequalities
TUAN2k8 1
N
by lbh_qys
Source: Own
Is that true?
Let
be real numbers such that 
for all 
.
Prove that:
.
Let
be real numbers such that 
for all 
.Prove that:
.
1 reply
Quadrilateral with Congruent Diagonals
v_Enhance 37
N
by Ilikeminecraft
Source: USA TSTST 2012, Problem 2
Let 
be a quadrilateral with 
. Diagonals 
and 
meet at 
. Let 
and 
denote the circumcircle and the circumcenter of triangle 
. Let 
and 
denote the circumcircle and circumcenter of triangle 
. Segment 
meets and again at 
and 
(other than 
and 
), respectively. Let 
and 
be the midpoints of minor arcs 
(not including ) and 
(not including ). Prove that 
.
be a quadrilateral with 
. Diagonals 
and 
meet at 
. Let 
and 
denote the circumcircle and the circumcenter of triangle 
. Let 
and 
denote the circumcircle and circumcenter of triangle 
. Segment 
meets and again at 
and 
(other than 
and 
), respectively. Let 
and 
be the midpoints of minor arcs 
(not including ) and 
(not including ). Prove that 
.
37 replies
Oh my god
EeEeRUT 0
Source: TMO 2025 P5
In a class, there are 
students and a teacher with marbles. The teacher then play a Marble distribution according to the following rules. At the start, each student receives at least 
marbles from the teacher. Then, the teacher chooses a student , who has never been chosen before, such that the number of marbles that he owns in a multiple of 
. That chosen student then equally distribute half of his marbles to 
other students. The same goes on until the teacher is not able to choose anymore student.
Find all integer, such that for some initial numbers of marbles that the students receive, the teacher can choose all the student(according to the rule above), so that each student receiving equal amount of marbles at the end.
students and a teacher with marbles. The teacher then play a Marble distribution according to the following rules. At the start, each student receives at least 
marbles from the teacher. Then, the teacher chooses a student , who has never been chosen before, such that the number of marbles that he owns in a multiple of 
. That chosen student then equally distribute half of his marbles to 
other students. The same goes on until the teacher is not able to choose anymore student.Find all integer
, such that for some initial numbers of marbles that the students receive, the teacher can choose all the student(according to the rule above), so that each student receiving equal amount of marbles at the end.
0 replies
geometry
EeEeRUT 1
N
by ItzsleepyXD
Source: TMO 2025
Let 
and 
be touch points of the incenter of 
at 
and 
, respectively. Let 
and 
be the circumcenter of triangles 
and 
, respectively. Show that 
and 
concurrent.
and 
be touch points of the incenter of 
at 
and 
, respectively. Let 
and 
be the circumcenter of triangles 
and 
, respectively. Show that 
and 
concurrent.
1 reply
+1 wSpanish Mathematical Olympiad 2002, Problem 1
OmicronGamma 3
N
by NicoN9
Source: Spanish Mathematical Olympiad 2002
Find all the polynomials 
of one variable that fullfill the following for all real numbers 
and 
:
.
of one variable that fullfill the following for all real numbers 
and 
:
.
3 replies
Additive set with special property
the_universe6626 1
N
by jasperE3
Source: Janson MO 1 P2
Let be a nonempty set of positive integers such that:
if 
then 
.
for any prime 
, there exists 
such that 
.
Prove that the set of all positive integers not in is finite.
(Proposed by cknori)
be a nonempty set of positive integers such that:
if 
then 
. for any prime 
, there exists 
such that 
.Prove that the set of all positive integers not in
is finite.(Proposed by cknori)
1 reply
ISI UGB 2025 P4
SomeonecoolLovesMaths 8
N
by chakrabortyahan
Source: ISI UGB 2025 P4
Let 
be the unit circle in the complex plane. Let 
be the map given by 
. We define 
and 
for 
. The smallest positive integer 
such that 
is called the period of 
. Determine the total number of points in 
of period 
.
(Hint :
)
be the unit circle in the complex plane. Let 
be the map given by 
. We define 
and 
for 
. The smallest positive integer 
such that 
is called the period of 
. Determine the total number of points in 
of period 
.(Hint :
)
8 replies
So Many Terms
oVlad 7
N
by NuMBeRaToRiC
Source: KöMaL A. 765
Find all functions 
which satisfy the following equality for all 
![\[f(x)f(y)-f(x-1)-f(y+1)=f(xy)+2x-2y-4.\]](//latex.artofproblemsolving.com/2/0/a/20ae7491750bac4c94deaed32d5b9aa66b26491f.png)
Proposed by Dániel Dobák, Budapest
which satisfy the following equality for all 
![\[f(x)f(y)-f(x-1)-f(y+1)=f(xy)+2x-2y-4.\]](http://latex.artofproblemsolving.com/2/0/a/20ae7491750bac4c94deaed32d5b9aa66b26491f.png)
Proposed by Dániel Dobák, Budapest
7 replies
Cauchy like Functional Equation
ZETA_in_olympiad 3
N
by jasperE3
Find all functions 
such that 
for all 
and 
such that 
for all 
and 
3 replies
special polynomials and probability
harazi 12
N
by MathLuis
Source: USA TST 2005, Problem 3, created by Harazi and Titu
We choose random a unitary polynomial of degree and coefficients in the set 
. Prove that the probability for this polynomial to be special is between 
and 
, where a polynomial 
is called special if for every 
in the sequence 
there are infinitely many numbers relatively prime with 
.
and coefficients in the set 
. Prove that the probability for this polynomial to be special is between 
and 
, where a polynomial 
is called special if for every 
in the sequence 
there are infinitely many numbers relatively prime with 
.
12 replies
