Y by math90, NguyenVanHoa29, Zuyong, TNKT
Problem. Let 
Prove that
Prove that
G
Topic
First Poster
Last Poster
Grid combo with tilings
a_507_bc 7
N
by john0512
Source: All-Russian MO 2023 Final stage 10.6
A square grid 
is tiled in two ways - only with dominoes and only with squares 
. What is the least number of dominoes that are entirely inside some square ?
is tiled in two ways - only with dominoes and only with squares 
. What is the least number of dominoes that are entirely inside some square ?
7 replies
sqrt(2)<=|1+z|+|1+z^2|<=4
SuiePaprude 3
N
by alpha31415
let z be a complex number with |z|=1 show that sqrt2 <=|1+z|+|1+z^2|<=4
3 replies
Simple but hard
Lukariman 5
N
by Giant_PT
Given triangle ABC. Outside the triangle, construct rectangles ACDE and BCFG with equal areas. Let M be the midpoint of DF. Prove that CM passes through the center of the circle circumscribing triangle ABC.
5 replies
inequalities
Ducksohappi 0
let a,b,c be non-negative numbers such that ab+bc+ca>0. Prove:
P/s: I have analysed:
, similarly to 
, by SOS
P/s: I have analysed:
, similarly to 
, by SOS
0 replies
bulgarian concurrency, parallelograms and midpoints related
parmenides51 7
N
by Ilikeminecraft
Source: Bulgaria NMO 2015 p5
In a triangle 
points 
and 
lie on the segments 
and 
, respectively, and are such that 
is a parallelogram. The circle with center the midpoint 
of the segment 
and radius 
and the circle of diameter 
intersect for the second time at the point 
. Prove that the lines 
and 
intersect in a point.
points 
and 
lie on the segments 
and 
, respectively, and are such that 
is a parallelogram. The circle with center the midpoint 
of the segment 
and radius 
and the circle of diameter 
intersect for the second time at the point 
. Prove that the lines 
and 
intersect in a point.
7 replies
Interesting inequalities
sqing 3
N
by sqing
Source: Own
Let 
and 
Show that
Where 
Let and 
Show that
Where
and 
Show that
Where 
Let
and 
Show that
Where
3 replies
Fast-growing sequence is always an integer
MarkBcc168 18
N
by Thapakazi
Source: Fake USAMO 2020 P1, Fake USAJMO 2020 P3
Suppose that 
is a positive integer and 
is an infinite sequence of positive integers satisfying
for all positive integers 
. Prove that this sequence must be eventually constant, i.e. there exists a positive integer 
such that 
.
Proposed by YRNG-BCC168.
is a positive integer and 
is an infinite sequence of positive integers satisfying
for all positive integers 
. Prove that this sequence must be eventually constant, i.e. there exists a positive integer 
such that 
.Proposed by YRNG-BCC168.
18 replies
AC bisects BE, BC = DE, CD//BE, <BAC = <DAE, AB/BD=AE/ED
parmenides51 3
N
by pku
Source: China Northern MO 2012 p7 CNMO
As shown in figure , in the pentagon 
, 
, 
, 
. If 
and 
. Prove that 
bisects the line segment 
.
IMAGE
, 
, 
, 
. If 
and 
. Prove that 
bisects the line segment 
.IMAGE
3 replies
Angle Relationships in Triangles
steven_zhang123 3
N
by Bergo1305
In , 
. The internal angle bisector of 
and the external angle bisector of intersect the ray at points 
and 
, respectively. Given that 
, prove that 
.
, 
. The internal angle bisector of 
and the external angle bisector of intersect the ray at points 
and 
, respectively. Given that 
, prove that 
.
3 replies
D1032 : A general result on polynomial 2
Dattier 4
N
by alpha31415
Source: les dattes à Dattier
Let ![$P \in \mathbb Q[x,y]$](//latex.artofproblemsolving.com/a/3/8/a384cbf38fd3a7f812b22baaeb0860e7f6eb26c0.png)
with 
and ![$\forall (a,b) \in \mathbb Z^2 \cap [0,d]^2, P(a,b) \in \mathbb Z$](//latex.artofproblemsolving.com/a/6/5/a65ecb4bb2d9b5e6096f322c2de8bd11b945ed6a.png)
.
Is it true that
?
![$P \in \mathbb Q[x,y]$](http://latex.artofproblemsolving.com/a/3/8/a384cbf38fd3a7f812b22baaeb0860e7f6eb26c0.png)
with 
and ![$\forall (a,b) \in \mathbb Z^2 \cap [0,d]^2, P(a,b) \in \mathbb Z$](http://latex.artofproblemsolving.com/a/6/5/a65ecb4bb2d9b5e6096f322c2de8bd11b945ed6a.png)
.Is it true that
?
4 replies
Non-negative real variables inequality
G
H
G
H
Source: own
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
#2
Apr 25, 2025, 3:31 AM
Y by
Are there any easier methods?
After multiplying both sides by
, we need to show that
![\[\frac13\sum_{\rm cyc}a^2b^2(a+3b)(3a+b)(a-b)^2+abc\cdot f(a,b,c)\ge0,\]](//latex.artofproblemsolving.com/2/4/3/243941650f929368784625dffef0acd7bc2c2e2a.png)
where 
is a fifth degree polynomial:
![\[\sum_{\rm cyc}\left(2a^5+\frac83a^4b+\frac83ab^4+\frac{22}3a^2b^2c-\frac{14}3a^3b^2-\frac{14}3a^2b^3-\frac{16}3a^2b^2c\right).\]](//latex.artofproblemsolving.com/a/b/f/abf1e0242ef9a8fa9a5338cf5ab57065c22e79b0.png)
By the 
method, it is not hard to show that 
is true.
After multiplying both sides by
, we need to show that![\[\frac13\sum_{\rm cyc}a^2b^2(a+3b)(3a+b)(a-b)^2+abc\cdot f(a,b,c)\ge0,\]](http://latex.artofproblemsolving.com/2/4/3/243941650f929368784625dffef0acd7bc2c2e2a.png)
where 
is a fifth degree polynomial:![\[\sum_{\rm cyc}\left(2a^5+\frac83a^4b+\frac83ab^4+\frac{22}3a^2b^2c-\frac{14}3a^3b^2-\frac{14}3a^2b^3-\frac{16}3a^2b^2c\right).\]](http://latex.artofproblemsolving.com/a/b/f/abf1e0242ef9a8fa9a5338cf5ab57065c22e79b0.png)
By the 
method, it is not hard to show that 
is true.
,
Z
K
Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
Y by arqady
I think it is a concave function according to w^3 and the rest is easy checking.
Z
K
Y
N Quick Reply
G
H
