# Questions tagged [karamata-inequality]

Inequalities, which we can prove by the Karamata's inequal;ity

36 questions
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### Find the number of natural solutions of $5^x+7^x+11^x=6^x+8^x+9^x$

Find the number of natural solutions of $5^x+7^x+11^x=6^x+8^x+9^x$ It's easy to see that $x=0$ and $x=1$ are solutions but are these the only one? How do I demonstrate that? I've tried to write them ...
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### If a,b,c are sides of a triangle, prove: $\sqrt{a+b-c} + \sqrt{b+c-a} + \sqrt{c+a-b} \le \sqrt{a} + \sqrt{b} + \sqrt{c}$

I did substitute $a=x+y, b=x+z, c=y+z$ and I arrived at $\sqrt{2x} + \sqrt{2y} + \sqrt{2z} \le \sqrt{x+y} + \sqrt{x+z} + \sqrt{y+z}$. However, after this, I tried various methods like AM-GM and Cauchy-...
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### Somebody help me please. I have a difficult inequality.

Let $ab+bc+ca=1$. Prove that $2 \ge \sqrt{1+a^2} + \sqrt{1+b^2}+\sqrt{1+c^2}-a-b-c \geq \sqrt3$.
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### Find maximize and minimize of $P=x+y$

For $\{x,y\}\subset\mathbb R$ such that $\sqrt{x+1}+\sqrt{y+1}=\sqrt{2}\left(x+y\right)$ find maximize and minimize of $P=x+y$ I found the maximize but minimize I have no idea. Help me.
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