Y by math90, NguyenVanHoa29, Zuyong, TNKT
Problem. Let 
Prove that
Prove that
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hard inequality omg
tokitaohma 3
N
by tokitaohma
1. Given 
and 
Prove that:
2. Given and 
Prove that:
and 
Prove that:
2. Given
and 
Prove that:
3 replies
Diophantine involving cube
Sadigly 1
N
by mashumaro
Source: Azerbaijan Senior NMO 2020
. Solve the equation: 
1 reply
Old hard problem
ItzsleepyXD 2
N
by ItzsleepyXD
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 
.
2 replies
The Return of Triangle Geometry
peace09 9
N
by mathfun07
Source: 2023 ISL A7
Let 
be a positive integer. Prove that there exist three permutations 
, 
, and 
of 
such that ![\[\left|\sqrt{a_k}+\sqrt{b_k}+\sqrt{c_k}-2\sqrt{N}\right|<2023\]](//latex.artofproblemsolving.com/9/8/0/98097fb721722b9103ef65d72322956877e262cb.png)
for every 
.
be a positive integer. Prove that there exist three permutations 
, 
, and 
of 
such that ![\[\left|\sqrt{a_k}+\sqrt{b_k}+\sqrt{c_k}-2\sqrt{N}\right|<2023\]](http://latex.artofproblemsolving.com/9/8/0/98097fb721722b9103ef65d72322956877e262cb.png)
for every 
.
9 replies
Set Partition
Butterfly 0
For the set of positive integers 
, no matter how its elements are partitioned into two subsets, at least one of the subsets must contain three numbers 
(
is allowed) such that 
. Find the minimal 
.
, no matter how its elements are partitioned into two subsets, at least one of the subsets must contain three numbers 
(
is allowed) such that 
. Find the minimal 
.
0 replies
Points Lying on its Cevian Triangle's Thomson Cubic
Feuerbach-Gergonne 1
N
by golue3120
Source: Own
Given 
and a point , let 
be the cevian triangle of with respect to . Let 
be the orthocenter of , and denote the isotomic conjugate of 
with respect to by 
, respectively. Let the centroid of be 
, and denote the isogonal conjugate of with respect to by 
. Prove that
or in brief
and a point , let 
be the cevian triangle of with respect to . Let 
be the orthocenter of , and denote the isotomic conjugate of 
with respect to by 
, respectively. Let the centroid of be 
, and denote the isogonal conjugate of with respect to by 
. Prove that
or in brief
1 reply
Areas of triangles AOH, BOH, COH
Arne 71
N
by EpicBird08
Source: APMO 2004, Problem 2
Let be the circumcenter and the orthocenter of an acute triangle . Prove that the area of one of the triangles 
, 
and 
is equal to the sum of the areas of the other two.
be the circumcenter and the orthocenter of an acute triangle . Prove that the area of one of the triangles 
, 
and 
is equal to the sum of the areas of the other two.
71 replies
Problem 6
termas 68
N
by HamstPan38825
Source: IMO 2016
There are 
line segments in the plane such that every two segments cross and no three segments meet at a point. Geoff has to choose an endpoint of each segment and place a frog on it facing the other endpoint. Then he will clap his hands 
times. Every time he claps,each frog will immediately jump forward to the next intersection point on its segment. Frogs never change the direction of their jumps. Geoff wishes to place the frogs in such a way that no two of them will ever occupy the same intersection point at the same time.
(a) Prove that Geoff can always fulfill his wish if is odd.
(b) Prove that Geoff can never fulfill his wish if is even.
line segments in the plane such that every two segments cross and no three segments meet at a point. Geoff has to choose an endpoint of each segment and place a frog on it facing the other endpoint. Then he will clap his hands 
times. Every time he claps,each frog will immediately jump forward to the next intersection point on its segment. Frogs never change the direction of their jumps. Geoff wishes to place the frogs in such a way that no two of them will ever occupy the same intersection point at the same time.(a) Prove that Geoff can always fulfill his wish if
is odd.(b) Prove that Geoff can never fulfill his wish if
is even.
68 replies
1 viewing
2n^2+4n-1 and 3n+4 have common powers
bin_sherlo 2
N
by Assassino9931
Source: Türkiye 2025 JBMO TST P5
Find all positive integers such that a positive integer power of 
equals to a positive integer power of 
.
such that a positive integer power of 
equals to a positive integer power of 
.
2 replies
combi/nt
blug 2
N
by aaravdodhia
Prove that every positive integer can be written in the form
where 
for some non-negative 
such that
for every 
.
can be written in the form
where 
for some non-negative 
such that
for every 
.
2 replies
1 viewing
System of equations in juniors' exam
AlperenINAN 2
N
by Assassino9931
Source: Turkey JBMO TST 2025 P3
Find all positive real solutions 
to the following system:
to the following system:
2 replies
Non-negative real variables inequality
G
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G
H
Source: own
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#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.
,
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Y
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Y by arqady
I think it is a concave function according to w^3 and the rest is easy checking.
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