Interesting Inequality Problem

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0 replies

jlacosta

May 1, 2025

0 replies

Inspired by a9opsow_

sqing 0

33 minutes ago

Source: Own

Let $a,b > 0 .$ Prove that
$\frac{(a^2 - ab-b)^2 + (b^2 - ab-a)^2 + ( ab-a-b)^2}{a^2 +b^2+(a - b)^2 } \geq 1$ $\frac{(2a^2 - 2ab-b)^2 + (2b^2 - 2ab-a)^2 + (ab-2a-2b )^2}{ (2a+b)^2 + (2b+a)^2+(a - b)^2 }\geq \frac 19$

0 replies

+3 w

sqing

33 minutes ago

0 replies

Cute NT Problem

M11100111001Y1R 5

N 39 minutes ago by compoly2010

Source: Iran TST 2025 Test 4 Problem 1

A number $n$ is called lucky if it has at least two distinct prime divisors and can be written in the form:
$n = p_1^{\alpha_1} + \cdots + p_k^{\alpha_k}$ where $p_1, \dots, p_k$ are distinct prime numbers that divide $n$ . (Note: it is possible that $n$ has other prime divisors not among $p_1, \dots, p_k$ .) Prove that for every prime number $p$ , there exists a lucky number $n$ such that $p \mid n$ .

5 replies

+2 w

M11100111001Y1R

Tuesday at 7:20 AM

compoly2010

39 minutes ago

3 var inequality

SunnyEvan 13

N an hour ago by Nguyenhuyen_AG

Let $a,b,c \in R$ ,such that $a^2+b^2+c^2=4(ab+bc+ca)$ Prove that : $\frac{7-2\sqrt{14}}{48} \leq \frac{a^3b+b^3c+c^3a}{(a^2+b^2+c^2)^2} \leq \frac{7+2\sqrt{14}}{48}$

13 replies

SunnyEvan

May 17, 2025

Nguyenhuyen_AG

an hour ago

trigonometric inequality

MATH1945 13

N 2 hours ago by sqing

Source: ?

In triangle $ABC$ , prove that $sin^2(A)+sin^2(B)+sin^2(C) \leq \frac{9}{4}$

13 replies

MATH1945

May 26, 2016

sqing

2 hours ago

No more topics!

Interesting Inequality Problem

Omerking 1

N Apr 10, 2025 by KhuongTrang

Let $a,b,c$ be three non-negative real numbers satisfying $a+b+c+abc=4.$
Prove that
$\frac{a}{a^{2}+1}+\frac{b}{b^{2}+1}+\frac{c}{c^{2}+1} \leq\frac{6}{13-3ab-3bc-3ca}$

1 reply

Omerking

Apr 10, 2025

KhuongTrang

Apr 10, 2025

Interesting Inequality Problem

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Omerking

11 posts

Omerking

#1 h Apr 10, 2025, 6:54 PM • 3 Y

Y by PikaPika999, TunarHasanzade, TDVOLIMPTEAM

Let $a,b,c$ be three non-negative real numbers satisfying $a+b+c+abc=4.$
Prove that
$\frac{a}{a^{2}+1}+\frac{b}{b^{2}+1}+\frac{c}{c^{2}+1} \leq\frac{6}{13-3ab-3bc-3ca}$

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KhuongTrang

731 posts

KhuongTrang

#2 h Apr 10, 2025, 11:12 PM • 4 Y

Y by PikaPika999, NguyenVanHoa29, Zuyong, TNKT

What is the source of this nice inequality?
https://math.stackexchange.com/questions/5054871/fracaa21-fracbb21-fraccc21-le-frac613-3ab-3bc-3c

This post has been edited 1 time. Last edited by KhuongTrang, Apr 11, 2025, 3:33 AM

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