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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sqing

#1 h Apr 5, 2025, 1:10 PM

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Let $a,b$ be real numbers such that $a^2+b^2+a^3 +b^3=4 .$ Prove that
$a+b \leq 2$ Let $a,b$ be real numbers such that $a+b + a^2+b^2+a^3 +b^3=6 .$ Prove that
$a+b \leq 2$

This post has been edited 1 time. Last edited by sqing, Apr 5, 2025, 1:19 PM

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sqing

#2 h Apr 6, 2025, 1:41 PM

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Let $a, b, c, d\geq 0 , bc + d + a = 5, cd + a + b = 2$ and $da + b + c = 6.$ Prove that
$\frac{3}{2}(\sqrt{13}-3) \leq a b + c d+d a \leq 6$ $\frac{1}{2}(11-\sqrt{13}) \leq bc+ c d+d a \leq 6$

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DAVROS

#3 h Apr 6, 2025, 2:33 PM

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sqing wrote:

Let $a,b$ be real numbers such that $a+b + a^2+b^2+a^3 +b^3=6 .$ Prove that $a+b \leq 2$

solution

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sqing

#4 h Apr 7, 2025, 4:12 AM

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Very very nice.Thank DAVROS.

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#5 h Apr 7, 2025, 4:30 AM

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sqing wrote:

Let $a,b$ be real numbers such that $a^2+b^2+a^3 +b^3=4 .$ Prove that
$a+b \leq 2$ Let $a,b$ be real numbers such that $a+b + a^2+b^2+a^3 +b^3=6 .$ Prove that
$a+b \leq 2$

This inequality is readily derived from the following inequality:
For real numbers $a$ and $b$ satisfying $a+b>2$ and any positive integer $k$ , it holds that $a^k+b^k>2$ .

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sqing

#6 h Apr 7, 2025, 4:38 AM

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Good.Thank lbh_qys.

This post has been edited 1 time. Last edited by sqing, Apr 7, 2025, 4:39 AM

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sqing

#7 h Apr 7, 2025, 4:43 AM

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Let $a,b$ be real numbers such that $a^3 +b^3+ab=3 .$ Prove that
$a+b \leq 2$ Let $a,b$ be real numbers such that $a^2+b^2+a^3 +b^3+ab=5 .$ Prove that
$a+b \leq 2$ Let $a,b$ be real numbers such that $a+b + a^2+b^2+a^3 +b^3+ab=7 .$ Prove that
$a+b \leq 2$

This post has been edited 1 time. Last edited by sqing, Apr 7, 2025, 4:46 AM

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sqing

#8 h Apr 8, 2025, 2:51 AM

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Let $a,b$ be real numbers such that $a+ b\leq -4$ . Prove that
$a^2 + b^2 +a^3 + b^3 + \frac{9}{2} ab \leq \frac{2197}{216}$ $a^2 + b^2 +a^3 + b^3 +5ab \leq \frac{343}{27}$ $a^2 + b^2 +a^3 + b^3 +6ab \leq \frac{512}{27}$

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#9 h Apr 8, 2025, 4:33 AM

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sqing wrote:

Let $a,b$ be real numbers such that $a+ b\leq -4$ . Prove that
$a^2 + b^2 +a^3 + b^3 +5ab \leq \frac{343}{27}$

fakesolve
i think there is probably a way to make this rigorous

This post has been edited 2 times. Last edited by pieMax2713, Apr 8, 2025, 4:40 AM

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sqing

#10 h Apr 8, 2025, 8:35 AM

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Thanks.

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sqing

#11 h Apr 9, 2025, 2:26 AM

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Let $a,b,c>1$ and $ab+bc+ca+a+b+c \geq 36$ . Prove that
$\dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c}\leq a+b+c-8$ Let $a,b,c>1$ and $ab+bc+ca \geq 27$ . Prove that
$\dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c}\leq a+b+c-8$

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