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Topic
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Concurrency in Parallelogram
amuthup 89
N
by happypi31415
Source: 2021 ISL G1
Let 
be a parallelogram with 
A point 
is chosen on the extension of ray 
past 
The circumcircle of 
meets the segment 
again at 
The circumcircle of triangle 
meets the segment 
at 
Prove that lines 
are concurrent.
be a parallelogram with 
A point 
is chosen on the extension of ray 
past 
The circumcircle of 
meets the segment 
again at 
The circumcircle of triangle 
meets the segment 
at 
Prove that lines 
are concurrent.
89 replies
deleting multiple or divisor in pairs from 2-50 on a blackboard
parmenides51 1
N
by TheBaiano
Source: 2023 May Olympiad L2 p3
The 
numbers 
are written on the blackboard . An allowed operation consists of choosing two different numbers 
and 
of the blackboard such that is a multiple of and delete exactly one of the two. María performs a sequence of permitted operations until she observes that it is no longer possible to perform any more. Determine the minimum number of numbers that can remain on the board at that moment.
numbers 
are written on the blackboard . An allowed operation consists of choosing two different numbers 
and 
of the blackboard such that is a multiple of and delete exactly one of the two. María performs a sequence of permitted operations until she observes that it is no longer possible to perform any more. Determine the minimum number of numbers that can remain on the board at that moment.
1 reply
at everystep a, b, c are replaced by a+\gcd(b,c), b+\gcd(a,c), c+\gcd(a,b)
NJAX 9
N
by atdaotlohbh
Source: 2nd Al-Khwarizmi International Junior Mathematical Olympiad 2024, Day2, Problem 8
Three positive integers are written on the board. In every minute, instead of the numbers 
, Elbek writes 
. Prove that there will be two numbers on the board after some minutes, such that one is divisible by the other.
Note.
- Greatest common divisor of numbers 
and 
Proposed by Sergey Berlov, Russia
, Elbek writes 
. Prove that there will be two numbers on the board after some minutes, such that one is divisible by the other.Note.
- Greatest common divisor of numbers 
and 
Proposed by Sergey Berlov, Russia
9 replies
Easy complete system of residues problem in Taiwan TST
Fysty 6
N
by Primeniyazidayi
Source: 2025 Taiwan TST Round 1 Independent Study 1-N
Find all positive integers 
such that there exist two permutations 
and 
of the set 
, satisfying the condition
for all 
.
Proposed by Fysty
such that there exist two permutations 
and 
of the set 
, satisfying the condition
for all 
.Proposed by Fysty
6 replies
JBMO Shortlist 2022 A2
Lukaluce 13
N
by Rayvhs
Source: JBMO Shortlist 2022
Let 
and 
be positive real numbers such that 
. Prove that
Proposed by Petar Filipovski, Macedonia
and 
be positive real numbers such that 
. Prove that
Proposed by Petar Filipovski, Macedonia
13 replies
A very beautiful geo problem
TheMathBob 4
N
by ravengsd
Source: Polish MO Finals P2 2023
Given an acute triangle 
with their incenter 
. Point 
lies on 
on the same side as 
wrt 
. Point 
lies on the shorter arc of the circumcircle . It is given that 
Prove that 
is the angle bisector of 
.
with their incenter 
. Point 
lies on 
on the same side as 
wrt 
. Point 
lies on the shorter arc of the circumcircle . It is given that 
Prove that 
is the angle bisector of 
.
4 replies
Prove that
A Duality Operation on Decreasing Integer Sequences
Ritangshu 0
Let 
be the set of all sequences 
of non-negative integers such that
(i)
; and
(ii) there exists a positive integer
such that 
for all 
.
Define the dual of the sequence
to be the sequence 
, where, for 
,
is the number of 
's which are greater than or equal to 
.
(i) Show that the dual of a sequence in belongs to .
(ii) Show that the dual of the dual of a sequence in is the original sequence itself.
(iii) Show that the duals of distinct sequences in are distinct.
be the set of all sequences 
of non-negative integers such that(i)
; and(ii) there exists a positive integer
such that 
for all 
.Define the dual of the sequence
to be the sequence 
, where, for 
,
is the number of 
's which are greater than or equal to 
.(i) Show that the dual of a sequence in
belongs to .(ii) Show that the dual of the dual of a sequence in
is the original sequence itself.(iii) Show that the duals of distinct sequences in
are distinct.0 replies
Property of a function
Ritangshu 0
Let 
, where 
and 
.
Prove that the function
satisfies the following property:
![\[
f\left( \lambda x + (1 - \lambda)x',\; \lambda y + (1 - \lambda)y' \right) > \min\{f(x, y),\; f(x', y')\}
\]](//latex.artofproblemsolving.com/3/b/d/3bd9fbf82b0b1d98a3c68745bb3f159cd9e2529a.png)
for all
and for all 
.
, where 
and 
.Prove that the function
satisfies the following property:![\[
f\left( \lambda x + (1 - \lambda)x',\; \lambda y + (1 - \lambda)y' \right) > \min\{f(x, y),\; f(x', y')\}
\]](http://latex.artofproblemsolving.com/3/b/d/3bd9fbf82b0b1d98a3c68745bb3f159cd9e2529a.png)
for all
and for all 
.0 replies
Subset of digits to express as a sum
anantmudgal09 46
N
by anudeep
Source: INMO 2020 P3
Let 
be a subset of 
. Suppose there is a positive integer 
such that for any integer 
, one can find positive integers 
so that 
and all the digits in the decimal representations of (expressed without leading zeros) are in . Find the smallest possible value of 
.
Proposed by Sutanay Bhattacharya
be a subset of 
. Suppose there is a positive integer 
such that for any integer 
, one can find positive integers 
so that 
and all the digits in the decimal representations of (expressed without leading zeros) are in . Find the smallest possible value of 
.Proposed by Sutanay Bhattacharya
Let 
be such that any positive integer 
where the non-negative integers 
have all their digits in 
Find the smallest possible number of elements in 46 replies
