and 
.
Y by
For all 
, find the maximum 
which satisfies
, find the maximum 
which satisfies
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Bounding number of solutions for floor function equation
Ciobi_ 1
N
by sarjinius
Source: Romania NMO 2025 9.3
Let 
be a positive integer. Consider the following equation: ![\[ \{x\}+\{2x\}+ \dots + \{nx\} = \lfloor x \rfloor + \lfloor 2x \rfloor + \dots + \lfloor 2nx \rfloor\]](//latex.artofproblemsolving.com/8/2/9/8299b3ceb1650418c0a373536cb1d2931900efd9.png)
a) For 
, solve the given equation in 
.
b) Prove that, for any, the equation has at most 
real solutions.
be a positive integer. Consider the following equation: ![\[ \{x\}+\{2x\}+ \dots + \{nx\} = \lfloor x \rfloor + \lfloor 2x \rfloor + \dots + \lfloor 2nx \rfloor\]](http://latex.artofproblemsolving.com/8/2/9/8299b3ceb1650418c0a373536cb1d2931900efd9.png)
a) For 
, solve the given equation in 
.b) Prove that, for any
, the equation has at most 
real solutions.
1 reply
nice system of equations
outback 4
N
by Raj_singh1432
Solve in positive numbers the system
4 replies
Two circles, a tangent line and a parallel
Valentin Vornicu 103
N
by zuat.e
Source: IMO 2000, Problem 1, IMO Shortlist 2000, G2
Two circles 
and 
intersect at two points 
and 
. Let 
be the line tangent to these circles at 
and 
, respectively, so that lies closer to than . Let 
be the line parallel to and passing through the point , with 
on and 
on . Lines 
and 
meet at 
; lines 
and meet at 
; lines 
and meet at 
. Show that 
.
and 
intersect at two points 
and 
. Let 
be the line tangent to these circles at 
and 
, respectively, so that lies closer to than . Let 
be the line parallel to and passing through the point , with 
on and 
on . Lines 
and 
meet at 
; lines 
and meet at 
; lines 
and meet at 
. Show that 
.
103 replies
Inequalities
idomybest 3
N
by damyan
Source: The Interesting Around Technical Analysis Three Variable Inequalities
The problem is in the attachment below.
3 replies
Function on positive integers with two inputs
Assassino9931 2
N
by Assassino9931
Source: Bulgaria Winter Competition 2025 Problem 10.4
The function 
is such that 
for any positive integers 
. Assume there exists a positive integer 
such that 
for all positive integers 
. Determine all possible values of 
.
is such that 
for any positive integers 
. Assume there exists a positive integer 
such that 
for all positive integers 
. Determine all possible values of 
.
2 replies
Normal but good inequality
giangtruong13 4
N
by IceyCold
Source: From a province
Let 
satisfy that 
. Prove that: 
satisfy that 
. Prove that: 
4 replies
Tiling rectangle with smaller rectangles.
MarkBcc168 61
N
by YaoAOPS
Source: IMO Shortlist 2017 C1
A rectangle 
with odd integer side lengths is divided into small rectangles with integer side lengths. Prove that there is at least one among the small rectangles whose distances from the four sides of are either all odd or all even.
Proposed by Jeck Lim, Singapore
with odd integer side lengths is divided into small rectangles with integer side lengths. Prove that there is at least one among the small rectangles whose distances from the four sides of are either all odd or all even.Proposed by Jeck Lim, Singapore
61 replies
A magician has one hundred cards numbered 1 to 100
Valentin Vornicu 49
N
by YaoAOPS
Source: IMO 2000, Problem 4, IMO Shortlist 2000, C1
A magician has one hundred cards numbered 1 to 100. He puts them into three boxes, a red one, a white one and a blue one, so that each box contains at least one card. A member of the audience draws two cards from two different boxes and announces the sum of numbers on those cards. Given this information, the magician locates the box from which no card has been drawn.
How many ways are there to put the cards in the three boxes so that the trick works?
How many ways are there to put the cards in the three boxes so that the trick works?
49 replies
Nice inequality
sqing 2
N
by Seungjun_Lee
Source: WYX
Let 
be real numbers . Prove that : There exist positive integer 
such that 
Where 
be real numbers . Prove that : There exist positive integer 
such that 
Where 
2 replies
Concurrency
Dadgarnia 27
N
by zuat.e
Source: Iranian TST 2020, second exam day 2, problem 4
Let 
be an isosceles triangle (
) with incenter 
. Circle 
passes through 
and and is tangent to 
. intersects 
and circumcircle of at 
and 
, respectively. Let 
be the midpoint of 
and 
be the midpoint of 
. Prove that 
, 
and 
are concurrent.
Proposed by Alireza Dadgarnia
be an isosceles triangle (
) with incenter 
. Circle 
passes through 
and and is tangent to 
. intersects 
and circumcircle of at 
and 
, respectively. Let 
be the midpoint of 
and 
be the midpoint of 
. Prove that 
, 
and 
are concurrent.Proposed by Alireza Dadgarnia
27 replies
A weird inequality
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