# Tagged Questions

Questions on proving, manipulating and applying inequalities.

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### The number of positive integer solutions to the equation $x_1+x_2+…+x_n=n^2.$

I'm working on this problem. To solve it I need this lemma: Let $n\ge2, n\in \mathbb N$. Let $X$ be the number of solutions in positive integers to the equation $x_1+x_2+...+x_n=n^2$. Let $Y$ be ...
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### Solve an easy inequality

How can I find the solution of the following inequality analytically in terms of $x$? $$i_1x^3+i_2x^2+i_3x \ge 0$$ where $i_k$ is a constant value.
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### Convex subset and linear equalities

Let S denote the set of $(a,b,c)$ $\in$ ${\mathbb{R^3}}$ which satisfies the following equalities: $-2a+b+c \leq 4$ $a-2b+c \leq 1$ $2a+2b-c \leq 5$ $a \geq 1$ $b \geq 2$ $c \geq 3$ ...
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### Why is $|\psi_n-f|^p \leq2^p |f|^p$ when $|\psi_n|\leq |f|$?

Why is $|\psi_n-f|^p\leq 2^p |f|^p$ when $|\psi_n|\leq|f|$? Is it true that $|a+b|^p\leq 2^p (|a|^p+|b|^p)$?
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### Inequality involving a convex function

I am stuck, showing the following inequality in an easy way (using only inequalities or something): Let $x\in [-a,a]$ for some $a>0$ and $p\in (1,2)$. I want to show that there then exists a ...
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### Show the roots of the quadratic equation $z^2 +bz+ c = 0$ lie in or on the unit circle

So I need a little help with the following: Considering separately the cases of real and complex roots show that the roots of the quadratic equation $z^2 +bz+ c = 0$ lie in or on the unit circle (i.e....
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### Prove that $\frac{1}{4\cdot 1976^3}-\frac{1}{16\cdot 1976^7}>10^{-19.76}$

Prove that $\frac{1}{4\cdot 1976^3}-\frac{1}{16\cdot 1976^7}>10^{-19.76}$ without using a calculator. I rearraged to get $4 \cdot 1976^4-1 > 10^{-19.76} \cdot 16 \cdot 1976^7$ and so we have ...
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### Trace class norm and rank inequality

I am quite new to operators in Hilbert spaces and I have been trying to show that for any linear and bounded operator $T : \mathcal{H} \rightarrow \mathcal{H}$ \vert \vert T \vert \...
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### Prove that $\sqrt{n+1}>\sqrt{n}+\frac{1}{2\sqrt{n}}-\frac{1}{8n\sqrt{n}}$

Prove that $\sqrt{n+1}>\sqrt{n}+\frac{1}{2\sqrt{n}}-\frac{1}{8n\sqrt{n}}$ if $n>0$. I didn't see an easy way of proving this without doing a lot of algebra and rearranging. Is there an easier ...
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### Inequality involving ArcTan

How to prove that for $x\in[0, +\infty]$ the following inequality is true: $$\arctan x\geq\frac{3 x}{1+2\sqrt{1+x^2}}?$$ I don't have idea from where to start, so any hint is welcome. Thanks in ...
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