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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
Strange Geometry
Itoz 1
N
by hukilau17
Source: Own
Given a fixed circle 
with its center 
. There are two fixed points 
and one moving point 
on . The midpoint of the line segment 
is 
. 
is a fixed point on . Line 
intersects
at 
, and line 
intersects 
at 
.
Find all the fixed points such that is always tangent to 
when varies.
with its center 
. There are two fixed points 
and one moving point 
on . The midpoint of the line segment 
is 
. 
is a fixed point on . Line 
intersects
at 
, and line 
intersects 
at 
.Find all the fixed points
such that is always tangent to 
when varies.1 reply
find all pairs of positive integers
Khalifakhalifa 2
N
by Haris1
Find all pairs of positive integers
such that:![\[
a^2 + b^2 \mid a^3 + b^3
\]](http://latex.artofproblemsolving.com/a/1/d/a1ddbd20c854e291da5670de449229d1664fa56c.png)
2 replies
D860 : Flower domino and unconnected
Dattier 4
N
by Haris1
Source: les dattes à Dattier
Let G be a grid of size m*n.
We have 2 dominoes in flowers and not connected like here
IMAGE
Determine a necessary and sufficient condition on m and n, so that G can be covered with these 2 kinds of dominoes.
We have 2 dominoes in flowers and not connected like here
IMAGE
Determine a necessary and sufficient condition on m and n, so that G can be covered with these 2 kinds of dominoes.
4 replies
Equal Distances in an Isosceles Setting
mojyla222 3
N
by sami1618
Source: IDMC 2025 P4
Let 
be an isosceles triangle with 
. The circle 
, passing through 
and 
, intersects segment 
at 
. The circle 
is tangent to at and passes through 
. Let and 
be the midpoints of segments and 
, respectively. The line 
intersects and at points 
and 
, respectively, where and are the intersections closer to . Prove that 
.
Proposed by Hooman Fattahi
be an isosceles triangle with 
. The circle 
, passing through 
and 
, intersects segment 
at 
. The circle 
is tangent to at and passes through 
. Let and 
be the midpoints of segments and 
, respectively. The line 
intersects and at points 
and 
, respectively, where and are the intersections closer to . Prove that 
.Proposed by Hooman Fattahi
3 replies
standard Q FE
jasperE3 1
N
by ErTeeEs06
Source: gghx, p19004309
Find all functions 
such that for any 
:
such that for any 
:
1 reply
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
