Show that for any positive real numbers $a,b,c$ the following inequality is true: $$4(a^3+b^3+c^3+3)\geq 3(a+1)(b+1)(c+1)$$When does equality hold?
Problem
Source: Kosovo MO 2019 Grade 10, Problem 2
Tags: inequalities, High School Olympiads
02.03.2019 22:29
One can show $4a^3+4 \geq (a+1)^3$, similarly for $b$ and $c$. The rest is AM-GM, equality occurs at $a=b=c=1$.
03.03.2019 04:04
Show that for any positive real numbers $a,b$ the following inequality is true: $$2(a^2+b^2+2)\geq 2(a+1)(b+1)$$ dangerousliri wrote: Show that for any positive real numbers $a,b,c$ the following inequality is true: $$4(a^3+b^3+c^3+3)\geq 3(a+1)(b+1)(c+1)$$When does equality hold? Proof of Zhangyunhua:
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03.03.2019 08:16
nice problem,very nice solutions!
03.03.2019 09:35
teomihai wrote: nice problem,very nice solutions! Yeah.
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25.05.2019 18:01
If 4a^3+4>=(a+1)^3 we are done.We simplify this and we get a^3+1>=a^2+a which is equivalent to a^2-2a+1>=0 which is absolutely true
06.06.2019 22:19
$4 \sum (a^3 + 1) \ge 4 \sum \frac{(a + 1)^3}{4} = \sum (a + 1)^3 \ge 3 \prod (a + 1)$, using Holder (or Chebyshev) and $AM-GM$, respectively