Waste less time on Facebook — follow Brilliant.
×

Absolute 2

Let \(a, b\) and \(c\) be positive real numbers such that \(a + b + c \leq 4\) and \(ab + bc + ca \geq 4\).

Prove that at least two of the inequalities

\(|a - b| \leq 2, |b - c| \leq 2, |c - a| \leq 2\)

are true.

Note by Sharky Kesa
3 years, 9 months ago

No vote yet
1 vote

  Easy Math Editor

MarkdownAppears as
*italics* or _italics_ italics
**bold** or __bold__ bold

- bulleted
- list

  • bulleted
  • list

1. numbered
2. list

  1. numbered
  2. list
Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1

paragraph 2

paragraph 1

paragraph 2

[example link](https://brilliant.org)example link
> This is a quote
This is a quote
    # I indented these lines
    # 4 spaces, and now they show
    # up as a code block.

    print "hello world"
# I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
MathAppears as
Remember to wrap math in \( ... \) or \[ ... \] to ensure proper formatting.
2 \times 3 \( 2 \times 3 \)
2^{34} \( 2^{34} \)
a_{i-1} \( a_{i-1} \)
\frac{2}{3} \( \frac{2}{3} \)
\sqrt{2} \( \sqrt{2} \)
\sum_{i=1}^3 \( \sum_{i=1}^3 \)
\sin \theta \( \sin \theta \)
\boxed{123} \( \boxed{123} \)

Comments

Sort by:

Top Newest

For sake of contradiction let us suppose \(|a-b| > 2\), \(|b-c| > 2\),\(|c-a| > 2\). which implies \[(a-b)^{2} + (b-c)^{2} + (c-a)^{2} > 8\] ---(1)

Given \(a + b + c \le 4\) \[a^{2} + b^{2} + c^{2} + 2ab + 2bc + 2ca \le 16\] \[a^{2} + b^{2} + c^{2} - ab - bc - ca \le 16 - 3ab -3bc - 3ca\] \[a^{2} + b^{2} + c^{2} - ab - bc - ca \le 4\] \[2a^{2} + 2b^{2} + 2c^{2} - 2ab - 2bc - 2ca \le 8\] \[(a-b)^{2} + (b-c)^{2} + (c-a)^{2} \le 8\].

This contradicts (1) and hence our desired proof follows.

Eddie The Head - 3 years, 9 months ago

Log in to reply

its not true

Mayank Singh - 3 years, 9 months ago

Log in to reply

It actually is true.

Sharky Kesa - 3 years, 9 months ago

Log in to reply

×

Problem Loading...

Note Loading...

Set Loading...