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Dear Sqing: So Many Inequalities...
hashtagmath 33
N
by GeoMorocco
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 :)
33 replies
3 knightlike moves is enough
sarjinius 1
N
by markam
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.
1 reply
Weird Geo
Anto0110 0
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 
.
0 replies
Is the geometric function injective?
Project_Donkey_into_M4 1
N
by Funcshun840
Source: Mock RMO TDP and Kayak 2018, P3
A non-degenerate triangle 
is given in the plane, let 
be the set of points which lie strictly inside it. Also let 
be the set of circles in the plane. For a point 
, let 
be the reflection of 
in sides 
respectively. Define a function 
such that 
is the circumcircle of 
. Is 
injective?
Note: The function is called injective if for any 
, 
is given in the plane, let 
be the set of points which lie strictly inside it. Also let 
be the set of circles in the plane. For a point 
, let 
be the reflection of 
in sides 
respectively. Define a function 
such that 
is the circumcircle of 
. Is 
injective?Note: The function
is called injective if for any 
, 
1 reply
numbers at vertices of triangle / tetrahedron, consecutive and gcd related
parmenides51 1
N
by TheBaiano
Source: 2022 May Olympiad L2 p4
a) A positive integer is written at each vertex of a triangle. Then on each side of the triangle the greatest common divisor of its ends is written. It is possible that the numbers written on the sides be three consecutive integers, in some order?
b) A positive integer is written at each vertex of a tetrahedron. Then, on each edge of the tetrahedron is written the greatest common divisor of its ends . It is possible that the numbers written in the edges are six consecutive integers, in some order?
b) A positive integer is written at each vertex of a tetrahedron. Then, on each edge of the tetrahedron is written the greatest common divisor of its ends . It is possible that the numbers written in the edges are six consecutive integers, in some order?
1 reply
red squares in a 7x7 board
parmenides51 2
N
by TheBaiano
Source: 2022 May Olympiad L2 p1
In a 
board, some squares are painted red. Let be the number of rows that have an odd number of red squares and let be the number of columns that have an odd number of red squares. Find all possible values of 
. For each value found, give a example of how the board can be painted.
board, some squares are painted red. Let be the number of rows that have an odd number of red squares and let be the number of columns that have an odd number of red squares. Find all possible values of 
. For each value found, give a example of how the board can be painted.
2 replies
winning strategy, vertices of regular n-gon
parmenides51 1
N
by TheBaiano
Source: 2022 May Olympiad L2 p5
The vertices of a regular polygon with 
sides are marked on the blackboard. Ana and Beto play alternately, Ana begins. Each player, in turn, must do the following:
join two vertices with a segment, without cutting another already marked segment; or
delete a vertex that does not belong to any marked segment.
The player who cannot take any action on his turn loses the game. Determine which of the two players can guarantee victory:
a) if
b) if
sides are marked on the blackboard. Ana and Beto play alternately, Ana begins. Each player, in turn, must do the following:
join two vertices with a segment, without cutting another already marked segment; or delete a vertex that does not belong to any marked segment.The player who cannot take any action on his turn loses the game. Determine which of the two players can guarantee victory:
a) if
b) if
1 reply
another functional inequality?
Scilyse 31
N
by Andyexists
Source: 2023 ISL A4
Let 
be the set of positive real numbers. Determine all functions 
such that ![\[x \big(f(x) + f(y)\big) \geqslant \big(f(f(x)) + y\big) f(y)\]](//latex.artofproblemsolving.com/e/7/1/e71b6e0d0b858eb46b81149f1e6be8c41e13d301.png)
for every 
.
be the set of positive real numbers. Determine all functions 
such that ![\[x \big(f(x) + f(y)\big) \geqslant \big(f(f(x)) + y\big) f(y)\]](http://latex.artofproblemsolving.com/e/7/1/e71b6e0d0b858eb46b81149f1e6be8c41e13d301.png)
for every 
.
31 replies
Apple sharing in Iran
mojyla222 2
N
by sami1618
Source: Iran 2025 second round p6
Ali is hosting a large party. Together with his 
friends, 
people are seated around a circular table in a fixed order. Ali places apples for serving directly in front of himself and wants to distribute them among everyone. Since Ali and his friends dislike eating alone and won't start unless everyone receives an apple at the same time, in each step, each person who has at least one apple passes one apple to the first person to their right who doesn't have an apple (in the clockwise direction).
Find all values of such that after some number of steps, the situation reaches a point where each person has exactly one apple.
friends, 
people are seated around a circular table in a fixed order. Ali places apples for serving directly in front of himself and wants to distribute them among everyone. Since Ali and his friends dislike eating alone and won't start unless everyone receives an apple at the same time, in each step, each person who has at least one apple passes one apple to the first person to their right who doesn't have an apple (in the clockwise direction).Find all values of
such that after some number of steps, the situation reaches a point where each person has exactly one apple.
2 replies
such that: 
.
