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Imtersecting two regular pentagons
Miquel-point 2
N
by ohiorizzler1434
Source: KoMaL B. 5093
The intersection of two congruent regular pentagons is a decagon with sides of 
in this order. Prove that
![\[a_1a_3+a_3a_5+a_5a_7+a_7a_9+a_9a_1=a_2a_4+a_4a_6+a_6a_8+a_8a_{10}+a_{10}a_2.\]](//latex.artofproblemsolving.com/9/e/e/9ee73bbdc4b4f2cad2eb3fcfb3dbdf76b6200b4d.png)
in this order. Prove that![\[a_1a_3+a_3a_5+a_5a_7+a_7a_9+a_9a_1=a_2a_4+a_4a_6+a_6a_8+a_8a_{10}+a_{10}a_2.\]](http://latex.artofproblemsolving.com/9/e/e/9ee73bbdc4b4f2cad2eb3fcfb3dbdf76b6200b4d.png)
2 replies
monving balls in 2018 boxes
parmenides51 1
N
by venhancefan777
Source: 1st Mathematics Regional Olympiad of Mexico Northwest 2018 P1
There are 
boxes 
, 
, 
,..,
. The 
-th box 
contains balls.
A move consists of the following steps:
a) Choose an integer
greater than 
and choose 
a multiple of .
b) Take a ball from each of the consecutive boxes
, 
, 
and move the 
balls to the box 
.
With these movements, what is the largest number of balls we can get in the box?
boxes 
, 
, 
,..,
. The 
-th box 
contains balls.A move consists of the following steps:
a) Choose an integer
greater than 
and choose 
a multiple of .b) Take a ball from each of the consecutive boxes
, 
, 
and move the 
balls to the box 
.With these movements, what is the largest number of balls we can get in the box
?
1 reply
inequality
danilorj 0
Let 
be nonnegative real numbers such that 
. Prove that
![\[
\frac{a}{4 - b} + \frac{b}{4 - c} + \frac{c}{4 - a} + \frac{1}{16}(1 - a)^2(1 - b)^2(1 - c)^2 \leq 1,
\]](//latex.artofproblemsolving.com/5/b/d/5bd3349071e075519bd986c845c500125b7d46f8.png)
and determine all such triples 
where the equality holds.
be nonnegative real numbers such that 
. Prove that![\[
\frac{a}{4 - b} + \frac{b}{4 - c} + \frac{c}{4 - a} + \frac{1}{16}(1 - a)^2(1 - b)^2(1 - c)^2 \leq 1,
\]](http://latex.artofproblemsolving.com/5/b/d/5bd3349071e075519bd986c845c500125b7d46f8.png)
and determine all such triples 
where the equality holds.
0 replies
P,Q,B are collinear
MNJ2357 28
N
by Ilikeminecraft
Source: 2020 Korea National Olympiad P2
is the orthocenter of an acute triangle 
, and let 
be the midpoint of 
. Suppose 
meets 
and 
at 
respectively. 
meets 
at 
, and the line through perpendicular to meets 
at 
. Prove that 
are collinear.
28 replies
Chinese Girls Mathematical Olympiad 2017, Problem 7
Hermitianism 45
N
by Ilikeminecraft
Source: Chinese Girls Mathematical Olympiad 2017, Problem 7
This is a very classical problem.
Let the
be a cyclic quadrilateral with circumcircle 
.Lines 
and 
intersect at point 
,and lines 
,
intersect at point 
.Circle 
is tangent to segments 
at points 
respectively,and intersects with circle at points 
.Lines 
intersect line 
at 
respectively.Show that 
are concyclic.
Let the
be a cyclic quadrilateral with circumcircle 
.Lines and 
intersect at point 
,and lines 
, intersect at point 
.Circle 
is tangent to segments 
at points 
respectively,and intersects with circle at points 
.Lines 
intersect line 
at 
respectively.Show that 
are concyclic.
45 replies
D1031 : A general result on polynomial 1
Dattier 1
N
by Dattier
Source: les dattes à Dattier
Let 
with 
.
Is it true that![$P(x,y) \in \mathbb Q[x,y]$](//latex.artofproblemsolving.com/b/3/7/b37942b05292a493c8e350322adc122205bc778d.png)
?
with 
.Is it true that
![$P(x,y) \in \mathbb Q[x,y]$](http://latex.artofproblemsolving.com/b/3/7/b37942b05292a493c8e350322adc122205bc778d.png)
?
1 reply
Asymmetric FE
sman96 18
N
by jasperE3
Source: BdMO 2025 Higher Secondary P8
Find all functions 
such that
for all 
.
such that
for all 
.
18 replies
Easy Geometry
pokmui9909 6
N
by reni_wee
Source: FKMO 2025 P4
Triangle 
satisfies 
. Let the incenter of triangle be 
, which touches 
at 
, respectively. Let 
be the midpoint of . Let the circle centered at passing through 
intersect 
at 
, respecively. Let line 
meet at 
, line 
meet 
at 
. Prove that the three lines 
are concurrent.
satisfies 
. Let the incenter of triangle be 
, which touches 
at 
, respectively. Let be the midpoint of . Let the circle centered at passing through 
intersect 
at 
, respecively. Let line 
meet at 
, line 
meet 
at 
. Prove that the three lines 
are concurrent.
6 replies
Old hard problem
ItzsleepyXD 3
N
by Funcshun840
Source: IDK
Let be a triangle and let 
be its circumcenter and 
its incenter.
Let
be the radical center of its three mixtilinears and let 
be the isogonal conjugate of .
Let
be the Gergonne point of the triangle .
Prove that line
is parallel with line 
.
be a triangle and let 
be its circumcenter and 
its incenter.Let
be the radical center of its three mixtilinears and let be the isogonal conjugate of .Let
be the Gergonne point of the triangle .Prove that line
is parallel with line 
.
3 replies
