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Advanced topics in Inequalities
va2010 18
N
by sqing
So a while ago, I compiled some tricks on inequalities. You are welcome to post solutions below!
18 replies
two subsets with no fewer than four common elements.
micliva 39
N
by de-Kirschbaum
Source: All-Russian Olympiad 1996, Grade 9, First Day, Problem 4
In the Duma there are 1600 delegates, who have formed 16000 committees of 80 persons each. Prove that one can find two committees having no fewer than four common members.
A. Skopenkov
A. Skopenkov
39 replies
3 knightlike moves is enough
sarjinius 2
N
by cooljoseph
Source: Philippine Mathematical Olympiad 2025 P6
An ant is on the Cartesian plane. In a single move, the ant selects a positive integer 
, then either travels [list]
[*] units vertically (up or down) and 
units horizontally (left or right); or
[*] units horizontally (left or right) and units vertically (up or down).
[/list]
Thus, for any, the ant can choose to go to one of eight possible points.
Prove that, for any integers
and 
, the ant can travel from 
to 
using at most 
moves.
, then either travels [list][*]
units vertically (up or down) and 
units horizontally (left or right); or[*]
units horizontally (left or right) and units vertically (up or down).[/list]
Thus, for any
, the ant can choose to go to one of eight possible points.Prove that, for any integers
and 
, the ant can travel from 
to 
using at most 
moves.
2 replies
16th ibmo - uruguay 2001/q3.
carlosbr 21
N
by de-Kirschbaum
Source: Spanish Communities
Let 
be a set of 
elements and 
are subsets of (
), such that every one of them has at least 
elements.
Show that there exists
and 
, with 
, such that the number of common elements of 
and 
is greater or equal to: 
be a set of 
elements and 
are subsets of (
), such that every one of them has at least 
elements.Show that there exists
and 
, with 
, such that the number of common elements of 
and 
is greater or equal to: 
21 replies
Weird Geo
Anto0110 1
N
by cooljoseph
In a trapezium 
, the sides 
and 
are parallel and the angles 
and 
are acute. Show that it is possible to divide the triangle 
into 4 disjoint triangle 
and the triangle 
into 4 disjoint triangles 
such that the triangles 
and 
are congruent for all .
, the sides 
and 
are parallel and the angles 
and 
are acute. Show that it is possible to divide the triangle 
into 4 disjoint triangle 
and the triangle 
into 4 disjoint triangles 
such that the triangles 
and 
are congruent for all .
1 reply
such that
Maximum of Incenter-triangle
mpcnotnpc 4
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 
?
4 replies
Something nice
KhuongTrang 26
N
by KhuongTrang
Source: own
Problem. Given 
be non-negative real numbers such that 
Prove that
be non-negative real numbers such that 
Prove that
26 replies
1 viewing
Tiling rectangle with smaller rectangles.
MarkBcc168 59
N
by Bonime
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
59 replies
Existence of AP of interesting integers
DVDthe1st 34
N
by DeathIsAwe
Source: 2018 China TST Day 1 Q2
A number is interesting if 2018 divides 
(the number of positive divisors of ). Determine all positive integers such that there exists an infinite arithmetic progression with common difference whose terms are all interesting.
is interesting if 2018 divides 
(the number of positive divisors of ). Determine all positive integers such that there exists an infinite arithmetic progression with common difference whose terms are all interesting.
34 replies
