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k a April Highlights and 2025 AoPS Online Class Information
jlacosta   0
Apr 2, 2025
Spring is in full swing and summer is right around the corner, what are your plans? At AoPS Online our schedule has new classes starting now through July, so be sure to keep your skills sharp and be prepared for the Fall school year! Check out the schedule of upcoming classes below.

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0 replies
jlacosta
Apr 2, 2025
0 replies
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Connecting chaos in a grid
Assassino9931   3
N 7 minutes ago by dgrozev
Source: Bulgaria National Olympiad 2025, Day 1, Problem 2
Exactly \( n \) cells of an \( n \times n \) square grid are colored black, and the remaining cells are white. The cost of such a coloring is the minimum number of white cells that need to be recolored black so that from any black cell \( c_0 \), one can reach any other black cell \( c_k \) through a sequence \( c_0, c_1, \ldots, c_k \) of black cells where each consecutive pair \( c_i, c_{i+1} \) are adjacent (sharing a common side) for every \( i = 0, 1, \ldots, k-1 \). Let \( f(n) \) denote the maximum possible cost over all initial colorings with exactly \( n \) black cells. Determine a constant $\alpha$ such that
\[ \frac{1}{3}n^{\alpha} \leq f(n) \leq 3n^{\alpha} \]for any $n\geq 100$.
3 replies
Assassino9931
Apr 8, 2025
dgrozev
7 minutes ago
J
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Balanced Tournaments
anantmudgal09   7
N 8 minutes ago by Mathgloggers
Source: The 1st India-Iran Friendly Competition Problem 1
A league consists of $2024$ players. A round involves splitting the players into two different teams and having every member of one team play with every member of the other team. A round is called balanced if both teams have an equal number of players. A tournament consists of several rounds at the end of which any two players have played each other. The committee organised a tournament last year which consisted of $N$ rounds. Prove that the committee can organise a tournament this year with $N$ balanced rounds.

Proposed by Anant Mudgal and Navilarekallu Tejaswi
7 replies
anantmudgal09
Jun 12, 2024
Mathgloggers
8 minutes ago
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2 var inequalities
sqing   1
N 8 minutes ago by sqing
Source: Own
Let $ a,b \in [0 ,1] . $ Prove that
$$ \frac{a}{ 1+a+b^2 }+\frac{b }{ 1+b+a^2 }\leq \frac{2}{3}$$$$ \frac{a}{ 1+a^2+b }+\frac{b }{ 1+b^2+a }\leq \frac{2}{3}$$$$ \frac{a}{ 1+a^2+b }+\frac{b }{ 1+b^2+a }+\frac{ab }{1+ab }\leq \frac{7}{6}$$$$ \frac{a}{ 1+a^2+b }+\frac{b }{ 1+b^2+a }+\frac{ab }{2+ab }\leq1$$$$ \frac{a}{ 1+a^2+b }+\frac{b }{ 1+b^2+a }+\frac{ab }{1+2ab }\leq1$$
1 reply
sqing
16 minutes ago
sqing
8 minutes ago
J
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nice [symmedians in a triangle, < ABM = < BAN]
grodij   10
N 12 minutes ago by Lemmas
Source: IMO Shortlist 2000, G5
The tangents at $B$ and $A$ to the circumcircle of an acute angled triangle $ABC$ meet the tangent at $C$ at $T$ and $U$ respectively. $AT$ meets $BC$ at $P$, and $Q$ is the midpoint of $AP$; $BU$ meets $CA$ at $R$, and $S$ is the midpoint of $BR$. Prove that $\angle ABQ=\angle BAS$. Determine, in terms of ratios of side lengths, the triangles for which this angle is a maximum.
10 replies
grodij
Nov 14, 2004
Lemmas
12 minutes ago
J
No more topics!
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k An old inequality of Iran 96
Hamath   5
N Jun 15, 2011 by Amir Hossein
For $a,b,c > 0$
$ \frac{1}{(a+b)^2}+ \frac{1}{(b+c)^2}+ \frac{1}{(c+a)^2} \geq \frac{9}{4(ab+bc+ca)}$
5 replies
Hamath
Jun 15, 2011
Amir Hossein
Jun 15, 2011
J
An old inequality of Iran 96
G H J
Reveal topic tags
inequalities
inequalities unsolved
L
G H BBookmark kLocked kLocked NReply
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Hamath
64 posts
Hamath
#1 h  Jun 15, 2011, 3:48 AM • 2 Y
Y by Adventure10, Mango247
For $a,b,c > 0$
$ \frac{1}{(a+b)^2}+ \frac{1}{(b+c)^2}+ \frac{1}{(c+a)^2} \geq \frac{9}{4(ab+bc+ca)}$
This post has been edited 1 time. Last edited by Hamath, Jun 15, 2011, 4:07 AM
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arqady
30213 posts
arqady
#2 h  Jun 15, 2011, 4:06 AM • 1 Y
Y by Adventure10
Hamath wrote:
For $a,b,c > 0$
$ \frac{1}{(a+b)^2}+ \frac{1}{(b+c)^2}+ \frac{1}{(c+a)} \geq \frac{9}{4(ab+bc+ca)}$
Are you mean that $ \frac{1}{(a+b)^2}+ \frac{1}{(b+c)^2}+ \frac{1}{(c+a)^2} \geq \frac{9}{4(ab+bc+ca)}$?
If so it's obviously true by $uvw$.
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Hamath
64 posts
Hamath
#3 h  Jun 15, 2011, 4:09 AM • 1 Y
Y by Adventure10
Sorry,
what uvw?
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shinichiman
3212 posts
shinichiman
#4 h  Jun 15, 2011, 4:38 AM • 1 Y
Y by Adventure10
There are many solution for this old Problem.
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Hamath
64 posts
Hamath
#5 h  Jun 15, 2011, 9:02 AM • 1 Y
Y by Adventure10
shinichiman wrote:
There are many solution for this old Problem.

can you show they?
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Amir Hossein
5452 posts
Amir Hossein
#6 h  Jun 15, 2011, 9:07 AM • 1 Y
Y by Adventure10
I found this problem for the 34th time. See here, and locked.
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