Inequalities

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High School Math Grades 9-12, Ages 13-18

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High School Math Grades 9-12, Ages 13-18

Grades 9-12, Ages 13-18

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42229 posts

sqing

#1 h Dec 22, 2023, 4:06 AM

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Let $a,b,c\geq 0,ab+bc+ca>0$ and $a+b+c=3$ . Prove that
$\frac{8}{3}\leq\frac{(a+b)(b+c)(c+a)}{ab+bc+ca}\leq 3$ $3\leq\frac{(a+b)(2b+c)(c+a)}{ab+bc+ca}\leq 6$ $\frac{3}{2}\leq\frac{(a+b)(2b+c)(c+ a)}{ab+bc+2ca}\leq 6$ $1\leq\frac{(a+b)(2b+c)(c+ a)}{ab+bc+ 3ca}\leq 6$

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DAVROS

#2 h Dec 22, 2023, 6:34 AM

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sqing wrote:

Let $a,b,c\geq 0,ab+bc+ca>0$ and $a+b+c=3$ . Prove that $\frac{8}{3}\leq\frac{(a+b)(b+c)(c+a)}{ab+bc+ca}\leq 3$

solution

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DAVROS

#3 h Dec 22, 2023, 7:23 AM

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sqing wrote:

Let $a,b,c\geq 0,ab+bc+ca>0$ and $a+b+c=3$ . Prove that $3\leq\frac{(a+b)(2b+c)(c+a)}{ab+bc+ca}\leq 6$

solution

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sqing

#4 h Dec 22, 2023, 8:04 AM

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Very very nice. Thank DAVROS .

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42229 posts

sqing

#5 h Dec 22, 2023, 8:05 AM

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Let $a,b,c\geq 0,ab+bc+ca>0$ and $a+b+c=3$ . Prove that
$\frac{(a+b)(2b+c)(c+ a)}{2ab+bc+ ca}\geq\frac{9(3-\sqrt 3)}{4}$ $\frac{72}{49}\leq\frac{(a+b)(b+c)(c+a)}{ab+2bc+2ca}\leq 3$

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42229 posts

sqing

#6 h Dec 23, 2023, 8:02 AM

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Let $a,b,c \geq 0$ and $a^2+b^2+c^2+abc=4.$ Prove that
$2\sqrt 2\geq a+b+c-\sqrt{abc}\geq 2$ $4\sqrt 2\geq 2a+b+2c-\sqrt{abc}\geq 2$ Let $a,b,c$ be reals such that $a^3+b^3+c^3+2abc=5.$ Prove that
$3\geq a+b+c\geq \sqrt[3] 5$ $2\sqrt[3] {20}\geq 2a+b+2c\geq \sqrt[3] 5$

This post has been edited 1 time. Last edited by sqing, Dec 23, 2023, 8:10 AM

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sqing

#7 h Apr 29, 2025, 8:58 AM

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sqing wrote:

Let $a,b,c \geq 0$ and $a^2+b^2+c^2+abc=4.$ Prove that
$2\sqrt 2\geq a+b+c-\sqrt{abc}\geq 2$ $4\sqrt 2\geq 2a+b+2c-\sqrt{abc}\geq 2$

Let $a,b,c>0$ and $a^2+b^2+c^2+1=4abc$ . Prove that $a+b+c \geq \sqrt{abc}+2$ $a+b+c \geq2\sqrt{abc}+1$ https://artofproblemsolving.com/community/c6h3353507p31120910
https://artofproblemsolving.com/community/c6h3513571p34100500
https://artofproblemsolving.com/community/c4h3452846p33333465
https://artofproblemsolving.com/community/c6h3532244p34366370

This post has been edited 3 times. Last edited by sqing, Apr 29, 2025, 2:03 PM

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