Inequalities

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

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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41157 posts

sqing

#1 h Mar 8, 2025, 12:43 PM

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Let $a,b$ be real numbers such that $a + b \geq |ab + 1|.$ Prove that $a^3 + b^3 \geq |a^3 b^3 + 1|$ Let $a,b$ be real numbers such that $2(a + b ) \geq |ab + 1|.$ Prove that $26( a^3 + b^3) \geq |a^3 b^3 + 1|$ Let $a,b$ be real numbers such that $4(a + b) \geq 3|ab + 1|.$ Prove that $148(a^3 + b^3) \geq27 |a^3 b^3 + 1|$

This post has been edited 1 time. Last edited by sqing, Mar 8, 2025, 12:54 PM

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pennypc123456789

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pennypc123456789

#2 h Mar 8, 2025, 1:02 PM

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sqing wrote:

Let $a,b$ be real numbers such that $a + b \geq |ab + 1|.$ Prove that $a^3 + b^3 \geq |a^3 b^3 + 1|$
Let $a,b$ be real numbers such that $4(a + b) \geq 3|ab + 1|.$ Prove that $148(a^3 + b^3) \geq27 |a^3 b^3 + 1|$

since $a^2b^2 - ab + 1 > 0$ , we have :
$|a^3b^3 + 1| = |ab + 1|( a^2b^2 - ab + 1) = |ab + 1| [(ab+1)^2-3ab] \le (a+b)[(a+b)^2-3ab] = (a + b )(a^2-ab+b^2) = a^3 +b^3 .$

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sqing

#3 h Mar 8, 2025, 1:23 PM • 1 Y

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Good.Thanks.

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DAVROS

#4 h Mar 9, 2025, 10:20 AM

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sqing wrote:

Let $a,b$ be real numbers such that $4(a + b) \geq 3|ab + 1|.$ Prove that $148(a^3 + b^3) \geq27 |a^3 b^3 + 1|$

solution

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sqing

#5 h Mar 9, 2025, 2:37 PM

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Very very nice.Thank DAVROS.

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sqing

#6 h Mar 11, 2025, 9:10 AM

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Let $a,b,c \in [0,1].$ Prove that $-\frac{3}{4}\leq (a-b)(b-2c)(c-3a) \leq 4$ $-6\leq (a-2b)(b-3c)(c-4a) \leq 9$ $-4\leq (a- 2b)(b+c)(c-2a) \leq \frac{9}{4}$

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sqing

#7 h Mar 19, 2025, 4:16 AM

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Let $a,b\ge 0$ and $2a + 2b \ge 4 + ab.$ Prove that
$a^2 + b^2 +ab \ge 4$ $a^2 + b^2 +ab +a + b \ge 6$ $a^2 + b^2 +ab + 2a + 2b \ge 8$ Let $a,b\ge 1$ and $2a + 2b\ge 4 + ab.$ Prove that
$a^2 + b^2 +ab \ge 7$ $a^2 + b^2 +ab +a + b \ge 10$ $a^2 + b^2 +ab + 2ab + 2b \ge 13$

This post has been edited 1 time. Last edited by sqing, Mar 19, 2025, 4:21 AM

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DAVROS

#8 h Today at 6:57 AM

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sqing wrote:

Let $a,b\ge 0$ and $2a + 2b \ge 4 + ab.$ Prove that $a^2 + b^2 +ab \ge 4$

solution

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DAVROS

#9 h Today at 7:31 AM

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sqing wrote:

Let $a,b\ge 1$ and $2a + 2b\ge 4 + ab.$ Prove that $a^2 + b^2 +ab \ge 7$

solution

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sqing

#10 h Today at 7:57 AM

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Very very nice.Thank DAVROS.

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sqing

#11 h Today at 8:49 AM

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Let $a,b,c\geq 1.$ Prove that
$(a+1)(b+2)(c +1)-6 abc\leq 6$ $(a+1)(b+3)(c +1)-8abc\leq 8$ $(a+2)(b+1)(c +2)-9a b c \leq 9$ $(a+2)(b+3)(c +2)-12a b c \leq 24$ $(a+3)(b+1)(c +3)-16a b c \leq 16$ $(a+3)(b+2)(c +3)-16a b c \leq 32$

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DAVROS

#12 h Today at 9:34 AM

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sqing wrote:

Let $a,b,c\geq 1.$ Prove that $(a+1)(b+2)(c +1)-6 abc\leq 6$

solution

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41157 posts

sqing

#13 h 4 hours ago

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Very very nice.Thank DAVROS.

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