Inspired by IMO 1984

Art of Problem Solving

AoPS Online

High School Olympiads Regional, national, and international math olympiads

Regional, national, and international math olympiads

3 M G

BBookmark VNew Topic kLocked

High School Olympiads Regional, national, and international math olympiads

Regional, national, and international math olympiads

3 M G

BBookmark VNew Topic kLocked

No tags match your search

M

Inspired by IMO 1984

G H J

Reveal topic tags

inequalities proposed

L

G H BBookmark kLocked kLocked NReply

Source: Own

The post below has been deleted. Click to close.

This post has been deleted. Click here to see post.

41310 posts

sqing

#1 h Mar 26, 2025, 7:42 AM

Y by

Let $a,b,c\geq 0$ and $a+b+c=1$ . Prove that
$a^2+b^2+c+ ab +9abc\leq 1$ $a^2+b^2+c +ab+10 abc\leq\frac{28}{27}$ $a^2+b^2+c+ ab +\frac{19}{2}abc\leq\frac{55}{54}$

This post has been edited 1 time. Last edited by sqing, Mar 26, 2025, 7:43 AM

Z K Y

The post below has been deleted. Click to close.

This post has been deleted. Click here to see post.

41310 posts

sqing

#2 h Mar 26, 2025, 8:30 AM

Y by

Let $a,b,c\geq 0$ and $a^3+b^2+c+ ab +\frac{19}{2}abc\geq\frac{55}{54}$ . Prove that
$a+b+c \geq \frac{1}{3}\sqrt{\frac{55}{2}}$ $a+b+\frac{9}{10}c \geq \frac{11}{12}$ Let $a,b,c\geq 0$ and $a^3+b^2+c +ab+10 abc\geq\frac{28}{27}$ . Prove that
$a+b+c \geq \frac{\sqrt[3]{28}}{3}$ $a+b+\frac{9}{10}c \geq \frac{14}{15}$

This post has been edited 1 time. Last edited by sqing, Mar 26, 2025, 8:35 AM

Z K Y

The post below has been deleted. Click to close.

This post has been deleted. Click here to see post.

450 posts

#3 h Mar 26, 2025, 8:41 AM

Y by

sqing wrote:

Let $a,b,c\geq 0$ and $a+b+c=1$ . Prove that
$a^2+b^2+c+ ab +9abc\leq 1$ $a^2+b^2+c +ab+10 abc\leq\frac{28}{27}$ $a^2+b^2+c+ ab +\frac{19}{2}abc\leq\frac{55}{54}$

(1).
$a^2 + b^2 + c + ab + 9abc \leq a^2 + b^2 + c + ab + (ab+bc+ca) = 1$ (2).(3).
This is merely based on (1) by employing $abc \leq \frac{1}{27}$ .

Z K Y

The post below has been deleted. Click to close.

This post has been deleted. Click here to see post.

41310 posts

sqing

#4 h Mar 26, 2025, 9:30 AM

Y by

Very nice.Thank lbh_qys.

Z K Y

N Quick Reply

G

H

=

© 2025 AoPS Incorporated

a