Prove that 
Y by Adventure10, Mango247
Given a rhombus 
, find the locus of the points 
lying inside the rhombus and satisfying 
.
, find the locus of the points 
lying inside the rhombus and satisfying 
.G
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First Poster
Last Poster
Checking a summand property for integers sufficiently large.
DinDean 2
N
by DinDean
For any fixed integer 
, prove that there exists a positive integer 
, such that for any integer 
, 
can be expressed by a sum of positive integers 
's as
![\[n=a_1+a_2+\dots+a_m,\]](//latex.artofproblemsolving.com/6/c/2/6c2a979a5ff6c40bf2d7152c8c32088fc36f848b.png)
where 
, 
, 
, 
and 
.
, prove that there exists a positive integer 
, such that for any integer 
, 
can be expressed by a sum of positive integers 
's as![\[n=a_1+a_2+\dots+a_m,\]](http://latex.artofproblemsolving.com/6/c/2/6c2a979a5ff6c40bf2d7152c8c32088fc36f848b.png)
where 
, 
, 
, 
and 
.
2 replies
1 viewing
Bunnies hopping around in circles
popcorn1 22
N
by awesomeming327.
Source: USA December TST for IMO 2023, Problem 1 and USA TST for EGMO 2023, Problem 1
There are 
equally spaced points on a circular track 
of circumference . The points are labeled 
in some order, each label used once. Initially, Bunbun the Bunny begins at 
. She hops along from to 
, then from to 
, until she reaches 
, after which she hops back to . When hopping from to 
, she always hops along the shorter of the two arcs 
of ; if 
is a diameter of , she moves along either semicircle.
Determine the maximal possible sum of the lengths of the arcs which Bunbun traveled, over all possible labellings of the points.
Kevin Cong
equally spaced points on a circular track 
of circumference . The points are labeled 
in some order, each label used once. Initially, Bunbun the Bunny begins at 
. She hops along from to 
, then from to 
, until she reaches 
, after which she hops back to . When hopping from to 
, she always hops along the shorter of the two arcs 
of ; if 
is a diameter of , she moves along either semicircle.Determine the maximal possible sum of the lengths of the
arcs which Bunbun traveled, over all possible labellings of the points.Kevin Cong
22 replies
Iran second round 2025-q1
mohsen 4
N
by MathLuis
Find all positive integers n>2 such that sum of n and any of its prime divisors is a perfect square.
4 replies
Dear Sqing: So Many Inequalities...
hashtagmath 37
N
by hashtagmath
I have noticed thousands upon thousands of inequalities that you have posted to HSO and was wondering where you get the inspiration, imagination, and even the validation that such inequalities are true? Also, what do you find particularly appealing and important about specifically inequalities rather than other branches of mathematics? Thank you :)
37 replies
integer functional equation
ABCDE 148
N
by Jakjjdm
Source: 2015 IMO Shortlist A2
Determine all functions 
with the property that ![\[f(x-f(y))=f(f(x))-f(y)-1\]](//latex.artofproblemsolving.com/f/2/5/f25cc1e8ae0be1fd02b347fd94be4fab88af1d46.png)
holds for all 
.
with the property that ![\[f(x-f(y))=f(f(x))-f(y)-1\]](http://latex.artofproblemsolving.com/f/2/5/f25cc1e8ae0be1fd02b347fd94be4fab88af1d46.png)
holds for all 
.
148 replies
IMO Shortlist 2013, Number Theory #1
lyukhson 152
N
by Jakjjdm
Source: IMO Shortlist 2013, Number Theory #1
Let 
be the set of positive integers. Find all functions 
such that
![\[ m^2 + f(n) \mid mf(m) +n \]](//latex.artofproblemsolving.com/2/f/4/2f409d1de993f1af8fd839bb8e9f87a57e1b8608.png)
for all positive integers
and .
be the set of positive integers. Find all functions 
such that![\[ m^2 + f(n) \mid mf(m) +n \]](http://latex.artofproblemsolving.com/2/f/4/2f409d1de993f1af8fd839bb8e9f87a57e1b8608.png)
for all positive integers
and .
152 replies
9x9 Board
mathlover314 8
N
by sweetbird108
There is a 
board with a number written in each cell. Every two neighbour rows sum up to at least 
, and every two neighbour columns sum up to at most 
. Find the sum of all numbers on the board.
board with a number written in each cell. Every two neighbour rows sum up to at least 
, and every two neighbour columns sum up to at most 
. Find the sum of all numbers on the board.
8 replies
Estonian Math Competitions 2005/2006
STARS 3
N
by Darghy
Source: Juniors Problem 4
A 
square is divided into unit squares. Is it possible to fill each unit square with a number 
in such a way that, whenever one places the tile so that it fully covers nine unit squares, the tile will cover nine different numbers?
square is divided into unit squares. Is it possible to fill each unit square with a number 
in such a way that, whenever one places the tile so that it fully covers nine unit squares, the tile will cover nine different numbers?
3 replies
Woaah a lot of external tangents
egxa 1
N
by HormigaCebolla
Source: All Russian 2025 11.7
A quadrilateral 
with no parallel sides is inscribed in a circle 
. Circles 
are inscribed in triangles 
, respectively. Common external tangents are drawn between 
and 
, and 
, and 
, and and , not containing any sides of quadrilateral . A quadrilateral whose consecutive sides lie on these four lines is inscribed in a circle 
. Prove that the lines joining the centers of and , and , and the centers of and all intersect at one point.
with no parallel sides is inscribed in a circle 
. Circles 
are inscribed in triangles 
, respectively. Common external tangents are drawn between 
and 
, and 
, and 
, and and , not containing any sides of quadrilateral . A quadrilateral whose consecutive sides lie on these four lines is inscribed in a circle 
. Prove that the lines joining the centers of and , and , and the centers of and all intersect at one point.1 reply
The locus of P with supplementary angles condition
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H
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Source: Baltic Way 2001
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Y by Adventure10, Mango247
Let be just a parallelogram and 
circumcircles 
of 
are congruent. Translating 
by 
into a 
creates a cyclic quadrilateral 
and parallelogram 
circumcircles 
of 
are also congruent. It follows that 
which means that isogonal conjugate 
of WRT 
is on perpendicular bisector of 
As a result, the locus of is a rectangular circum-hyperbola of the parallelogram 
centered at its diagonal intersection 
If is a rhombus, this hyperbola degenerates to its diagonals 
be just a parallelogram and 
circumcircles 
of 
are congruent. Translating 
by 
into a 
creates a cyclic quadrilateral 
and parallelogram 
circumcircles 
of 
are also congruent. It follows that 
which means that isogonal conjugate 
of WRT 
is on perpendicular bisector of 
As a result, the locus of is a rectangular circum-hyperbola of the parallelogram 
centered at its diagonal intersection 
If is a rhombus, this hyperbola degenerates to its diagonals 
Z
K
Y
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or 
.N Quick Reply
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