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sqing

#1 h Mar 27, 2025, 12:35 PM

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Let $a,b,c > 0$ and $a+b+c +abc =4.$ Prove that
$a^2 + b^2 + c^2 + 3 \geq 2( ab+bc + ca )$ Let $a,b,c > 0$ and $ab+bc+ca+abc=4.$ Prove that
$a^2 + b^2 + c^2 + 2abc \geq 5$

This post has been edited 1 time. Last edited by sqing, Mar 27, 2025, 12:44 PM

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sqing

#2 h Mar 27, 2025, 12:47 PM

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Let $a,b,c > 0$ and $a^2+b^2+c^2+abc =4.$ Prove that
$a+b+c + 3 \geq 2( ab+bc + ca )$ Let $a,b,c > 0$ and $a^2+b^2+c^2+abc=4.$ Prove that
$ab+bc+ca+3 \leq 2(a+b+c )$

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#3 h Mar 27, 2025, 3:40 PM

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sqing wrote:

Let $a,b,c > 0$ and $a+b+c +abc =4.$ Prove that
$a^2 + b^2 + c^2 + 3 \geq 2( ab+bc + ca )$

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sqing

#4 h Mar 28, 2025, 2:54 AM

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Thanks.
Let $a,b,c > 0$ and $a+b+c =3abc .$ Prove that
$a^2 + b^2 + c^2 + 3 \geq 2( ab+bc + ca )$ Let $a,b,c > 0$ and $a^2+b^2+c^2+1=4a^2b^2c^2 .$ Prove that
$a^2 + b^2 + c^2 + 3 \geq 2( ab+bc + ca )$ Let $a,b,c >0$ and $abc=1$ . Prove that
$a^2+b^2+c^2 +3 \geq 2(ab+bc+ca)$ Let $x,y,z>0$ and $x^2+y^2+z^2+3=2(xy+yz+zx) .$ Show that
$\sqrt{xy}+\sqrt{yz}+\sqrt{zx}\ge 3$ Let $x,y,z>0$ and $x^2+y^2+z^2+3 \le 2(xy+yz+zx)$ . Prove that
$(x^4+y)(y^4+z)(z^4+x) \ge (x+y^2)(y+z^2)(z+x^2)$

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#5 h Mar 28, 2025, 1:03 PM

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Can you prove this inequality ?

Let a,b,c ≥ 0 real numbers with ab+bc+ca ≤4. Prove that a+b+c+3 ≥ 2(√ab + √bc + √ca )

This post has been edited 1 time. Last edited by SUN8691, Mar 28, 2025, 1:04 PM
Reason: Mistake

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#6 h Mar 30, 2025, 4:08 AM

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SUN8691 wrote:

Can you prove this inequality ?

Let a,b,c ≥ 0 real numbers with ab+bc+ca ≤4. Prove that a+b+c+3 ≥ 2(√ab + √bc + √ca )

That inequality holds on what a, b, c ? I can't find it

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#7 h Mar 30, 2025, 3:05 PM

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Let $x,y,z>0$ and $x^2+y^2+z^2+3=2(xy+yz+zx) .$ Provethat: $\text{[math]}$ $\sqrt{xy}+\sqrt{yz}+\sqrt{zx}\ge 3$

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sqing

#9 h Monday at 3:25 AM

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Thanks.

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