×

# classic probs..

Inequalities 1 Classical Theorems 1. Show that for positive reals a, b, c

a 2 b + b 2 c + c 2 a  ab2 + bc2 + ca2  ≥ 9a 2 b 2 c 2 . 2. Let a, b, c be positive reals such that abc = 1. Prove that a + b + c ≤ a 2 + b 2 + c 2 . 3. Let P(x) be a polynomial with positive coefficients. Prove that if P  1 x  ≥ 1 P(x) holds for x = 1, then it holds for all x > 0. 4. Show that for all positive reals a, b, c, d, 1 a + 1 b + 4 c + 16 d ≥ 64 a + b + c + d . 5. (USAMO 1980/5) Show that for all non-negative reals a, b, c ≤ 1, a b + c + 1 + b c + a + 1 + c a + b + 1 + (1 − a)(1 − b)(1 − c) ≤ 1. 6. (USAMO 1977/5) If a, b, c, d, e are positive reals bounded by p and q with 0 < p ≤ q, prove that (a + b + c + d + e)  1 a + 1 b + 1 c + 1 d + 1 e  ≤ 25 + 6 rp q − rq p 2 and determine when equality holds. 7. Let a, b, c, be non-negative reals such that a + b + c = 1. Prove that a 3 + b 3 + c 3 + 6abc ≥ 1 4 . 8. (IMO 1995/2) a, b, c are positive reals with abc = 1. Prove that 1 a 3(b + c) + 1 b 3(c + a) + 1 c 3(a + b) ≥ 3 2 . 9. Let a, b, c be positive reals such that abc = 1. Show that 2 (a + 1)2 + b 2 + 1 + 2 (b + 1)2 + c 2 + 1 + 2 (c +

Note by Aayush Srivastava
3 years, 1 month ago

MarkdownAppears as
*italics* or _italics_ italics
**bold** or __bold__ bold
- bulleted- list
• bulleted
• list
1. numbered2. 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 1paragraph 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} $$\boxed{123}$$

Sort by:

What did you want to talk about? Check out the inequalities wikis for more examples!

Staff - 3 years, 1 month ago