G
Topic
First Poster
Last Poster
Inequality involving square root cube root and 8th root
bamboozled 0
If 
and 
, then find the minimum value of ![$\sqrt{ad^3 g^4} +\sqrt[3]{be^3 h^4} + \sqrt[8]{cf^3 k^4}$](//latex.artofproblemsolving.com/b/f/a/bfa17e8bb4cc05a84ed6a757f506d6d0ed2b0081.png)
and 
, then find the minimum value of ![$\sqrt{ad^3 g^4} +\sqrt[3]{be^3 h^4} + \sqrt[8]{cf^3 k^4}$](http://latex.artofproblemsolving.com/b/f/a/bfa17e8bb4cc05a84ed6a757f506d6d0ed2b0081.png)
0 replies
and 
.
Is there any generalizations?
functional equation
henderson 4
N
by megarnie
Source: unknown
Find all functions 
, satisfying the condition
for any positive reals
and 
.
, satisfying the condition
for any positive reals
and 
.
4 replies
Parallelograms and concyclicity
Lukaluce 31
N
by Ihatecombin
Source: EGMO 2025 P4
Let 
be an acute triangle with incentre 
and 
. Let lines 
and 
intersect the circumcircle of at 
and 
, respectively. Consider points 
and 
such that 
and 
are parallelograms (with 
, and 
). Let 
be the point of intersection of lines 
and 
. Prove that points 
, and are concyclic.
be an acute triangle with incentre 
and 
. Let lines 
and 
intersect the circumcircle of at 
and 
, respectively. Consider points 
and 
such that 
and 
are parallelograms (with 
, and 
). Let 
be the point of intersection of lines 
and 
. Prove that points 
, and are concyclic.
31 replies
My Unsolved FE in R+
ZeltaQN2008 2
N
by megarnie
Source: Ho Chi Minh TST 2017 - 2018
Find all functions 
such that for all any 
such that for all any 
2 replies
1 viewing
Something nice
KhuongTrang 32
N
by arqady
Source: own
Problem. Given 
be non-negative real numbers such that 
Prove that
be non-negative real numbers such that 
Prove that
32 replies
Infimum of decreasing sequence b_n/n^2
a1267ab 35
N
by shendrew7
Source: USA Winter TST for IMO 2020, Problem 1 and TST for EGMO 2020, Problem 3, by Carl Schildkraut and Milan Haiman
Choose positive integers 
satisfying
![\[1=\frac{b_1}{1^2} > \frac{b_2}{2^2} > \frac{b_3}{3^2} > \frac{b_4}{4^2} > \dotsb\]](//latex.artofproblemsolving.com/a/e/6/ae65e3d53931fe9d60d0de76dc79540ccf44c26e.png)
and let 
denote the largest real number satisfying 
for all positive integers 
. What are the possible values of across all possible choices of the sequence 
?
Carl Schildkraut and Milan Haiman
satisfying![\[1=\frac{b_1}{1^2} > \frac{b_2}{2^2} > \frac{b_3}{3^2} > \frac{b_4}{4^2} > \dotsb\]](http://latex.artofproblemsolving.com/a/e/6/ae65e3d53931fe9d60d0de76dc79540ccf44c26e.png)
and let 
denote the largest real number satisfying 
for all positive integers 
. What are the possible values of across all possible choices of the sequence 
?Carl Schildkraut and Milan Haiman
35 replies
IMO Genre Predictions
ohiorizzler1434 52
N
by justaguy_69
Everybody, with IMO upcoming, what are you predictions for the problem genres?
Personally I predict:
Personally I predict:
52 replies
\sqrt{a^2+b^2+2}+\sqrt{b^2+c^2+2 }+\sqrt{c^2+a^2+2}\ge 6
parmenides51 19
N
by NicoN9
Source: JBMO Shortlist 2017 A1
Let 
be positive real numbers such that 
. Prove
that
.
be positive real numbers such that 
. Provethat
.
19 replies
and 
Prove that
