A two-variable & non-homogenous inequality that seems hard to me

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0 replies

jlacosta

Mar 2, 2025

0 replies

Coloring

demmy 6

N 4 minutes ago by Kaimiaku

Source: Thailand TST 2015

What is the maximum number of squares in an $8 \times 8$ board that can be colored so that for each square in the board, at most one square adjacent to it is colored.

6 replies

demmy

Dec 2, 2023

Kaimiaku

4 minutes ago

Nice and easy FE on R+

sttsmet 21

N 9 minutes ago by bo18

Source: EMC 2024 Problem 4, Seniors

Find all functions $f: \mathbb{R}^{+} \to \mathbb{R}^{+}$ such that $f(x+yf(x)) = xf(1+y)$
for all x, y positive reals.

21 replies

sttsmet

Dec 23, 2024

bo18

9 minutes ago

max power of 2 that divides \lceil(1+\sqrt{3})^{2n}\rceil for pos. integer n

parmenides51 2

N 18 minutes ago by Inspector_Maygray

Source: Gulf Mathematical Olympiad GMO 2017 p4

1 - Prove that $55 < (1+\sqrt{3})^4 < 56$ .

2 - Find the largest power of $2$ that divides $\lceil(1+\sqrt{3})^{2n}\rceil$ for the positive integer $n$

2 replies

parmenides51

Aug 23, 2019

Inspector_Maygray

18 minutes ago

Points in general position

AshAuktober 1

N 20 minutes ago by Rdgm

Source: 2025 Nepal ptst p1 of 4

Shining tells Prajit a positive integer $n \ge 2025$ . Prajit then tries to place n points such that no four points are concyclic and no $3$ points are collinear in Euclidean plane, such that Shining cannot find a group of three points such that their circumcircle contains none of the other remaining points. Is he always able to do so?

(Prajit Adhikari, Nepal and Shining Sun, USA)

1 reply

AshAuktober

Yesterday at 2:15 PM

Rdgm

20 minutes ago

No more topics!

A two-variable & non-homogenous inequality that seems hard to me

MyLifeMyChoice 3

N Yesterday at 7:45 PM by Radin_

Source: Developing from a larger, three-variable one

For $a,b>0$ , prove/disprove the following claim: :maybe:

$a^3b^3+\frac{1}{a^3}+\frac{1}{b^3}+3\stackrel{?}{\ge}a^2b+b^2a+\frac{1}{a^2b}+\frac{1}{b^2a}+\frac{a}{b}+\frac{b}{a}$

3 replies

MyLifeMyChoice

Mar 13, 2025

Radin_

Yesterday at 7:45 PM

A two-variable & non-homogenous inequality that seems hard to me

G H J

Reveal topic tags

inequalities

hint

advice

G H BBookmark kLocked kLocked NReply

Source: Developing from a larger, three-variable one

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MyLifeMyChoice

50 posts

MyLifeMyChoice

#1 h Mar 13, 2025, 5:27 AM

Y by

For $a,b>0$ , prove/disprove the following claim: :maybe:

$a^3b^3+\frac{1}{a^3}+\frac{1}{b^3}+3\stackrel{?}{\ge}a^2b+b^2a+\frac{1}{a^2b}+\frac{1}{b^2a}+\frac{a}{b}+\frac{b}{a}$

This post has been edited 1 time. Last edited by MyLifeMyChoice, Yesterday at 6:54 PM

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DKI

27 posts

DKI

#2 h Mar 13, 2025, 8:57 AM • 1 Y

Y by MyLifeMyChoice

Equivalent to Schur's inequality.

Z K Y

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MyLifeMyChoice

50 posts

MyLifeMyChoice

#3 h Yesterday at 6:56 PM

Y by

DKI wrote:

Equivalent to Schur's inequality.

I haven't get it yet dear friend. Could you please explain your observation just a little bit :huh:

This post has been edited 2 times. Last edited by MyLifeMyChoice, Yesterday at 7:00 PM

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Radin_

1 post

Radin_

#4 h Yesterday at 7:45 PM • 1 Y

Y by ehuseyinyigit

by Schur's inequality we have that: $a^3+b^3+c^3+3abc \ge a^2 b+a^2 c+b^2 a+b^2 c+c^2 a+c^2 b$ now simply replace a with $\frac{1}{a}$ , b with $\frac{1}{b}$ and c with ab to get the desired result

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