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Coins in a circle
JuanDelPan 15
N
by Ilikeminecraft
Source: Pan-American Girls’ Mathematical Olympiad 2021, P1
There are 
coins numbered from 
to 
. These coins are placed around a circle, not necesarily in order.
In each turn, if we are on the coin numbered
, we will jump to the one places from it, always in a clockwise order, beginning with coin number 1. For an example, see the figure below.
Find all values of for which there exists an arrangement of the coins in which every coin will be visited.
coins numbered from 
to 
. These coins are placed around a circle, not necesarily in order.In each turn, if we are on the coin numbered
, we will jump to the one places from it, always in a clockwise order, beginning with coin number 1. For an example, see the figure below.Find all values of
for which there exists an arrangement of the coins in which every coin will be visited.
15 replies
Exponential + factorial diophantine
62861 34
N
by ali123456
Source: USA TSTST 2017, Problem 4, proposed by Mark Sellke
Find all nonnegative integer solutions to 
.
Proposed by Mark Sellke
.Proposed by Mark Sellke
34 replies
Everybody has 66 balls
YaoAOPS 3
N
by Blast_S1
Source: 2025 CTST P5
There are 
people and 
colors, where each person has one ball of each color. For each person, their balls have positive mass summing to one. Find the smallest constant 
such that regardless of the mass distribution, each person can choose one ball such that the sum of the chosen balls of each color does not exceed .
people and 
colors, where each person has one ball of each color. For each person, their balls have positive mass summing to one. Find the smallest constant 
such that regardless of the mass distribution, each person can choose one ball such that the sum of the chosen balls of each color does not exceed .
3 replies
Inspired by IMO 1984
sqing 4
N
by SunnyEvan
Source: Own
Let 
and 
. Prove that
Equality holds when 
Equality holds when 
and 
. Prove that
Equality holds when 
Equality holds when 
4 replies
Partition set with equal sum and differnt cardinality
psi241 73
N
by mananaban
Source: IMO Shortlist 2018 C1
Let 
be an integer. Prove that there exists a set 
of 
positive integers satisfying the following property: For every 
the set can be partitioned into two subsets with equal sums of elements, with one of subsets of cardinality 
.
be an integer. Prove that there exists a set 
of 
positive integers satisfying the following property: For every 
the set can be partitioned into two subsets with equal sums of elements, with one of subsets of cardinality 
.
73 replies
IMO 2018 Problem 5
orthocentre 75
N
by VideoCake
Source: IMO 2018
Let 
, 
, 
be an infinite sequence of positive integers. Suppose that there is an integer 
such that, for each 
, the number
is an integer. Prove that there is a positive integer 
such that 
for all 
.
Proposed by Bayarmagnai Gombodorj, Mongolia
, 
, 
be an infinite sequence of positive integers. Suppose that there is an integer 
such that, for each 
, the number
is an integer. Prove that there is a positive integer 
such that 
for all 
.Proposed by Bayarmagnai Gombodorj, Mongolia
75 replies
Ornaments and Christmas trees
Morskow 29
N
by gladIasked
Source: Slovenia IMO TST 2018, Day 1, Problem 1
Let be a positive integer. On the table, we have 
ornaments in different colours, not necessarily of each colour. Prove that we can hang the ornaments on Christmas trees in such a way that there are exactly ornaments on each tree and the ornaments on every tree are of at most 
different colours.
be a positive integer. On the table, we have 
ornaments in different colours, not necessarily of each colour. Prove that we can hang the ornaments on Christmas trees in such a way that there are exactly ornaments on each tree and the ornaments on every tree are of at most 
different colours.
29 replies
Another square grid :D
MathLuis 42
N
by gladIasked
Source: USEMO 2021 P1
Let be a fixed positive integer and consider an 
grid of real numbers. Determine the greatest possible number of cells 
in the grid such that the entry in is both strictly greater than the average of 's column and strictly less than the average of 's row.
Proposed by Holden Mui
be a fixed positive integer and consider an 
grid of real numbers. Determine the greatest possible number of cells 
in the grid such that the entry in is both strictly greater than the average of 's column and strictly less than the average of 's row.Proposed by Holden Mui
42 replies
,
Maximum of Incenter-triangle
mpcnotnpc 2
N
by mpcnotnpc
Triangle 
has side lengths 
, 
, and . Select a point 
inside , and construct the incenters of 
, 
, and 
and denote them as 
, 
, 
. What is the maximum area of the triangle 
?
has side lengths 
, 
, and . Select a point 
inside , and construct the incenters of 
, 
, and 
and denote them as 
, 
, 
. What is the maximum area of the triangle 
?
2 replies
