Questions on proving and manipulating inequalities.

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4
votes
0answers
132 views
+50

How to prove there exists $n_{1}a_{n_{0}}+n_{2}a_{n_{1}}+\cdots+n_{k}a_{n_{k-1}}<3(a_{1}+a_{2}+\cdots+a_{N})$

Let $a_{1},a_{2},\cdots,a_{N}$ be nonnegative reals, not all $0$. Prove that there exists a sequence $$1=n_{0}<n_{1}<\cdots<n_{k}=N+1$$ of integers such that ...
6
votes
2answers
105 views
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How prove this inequality $\prod_{1\le i<j\le 5}|z_{i}-z_{j}|^2\le 5^5$

let $z_{1},z_{2},z_{3},z_{4},z_{5}$ are complex numbers,and such $$|z_{1}|^2+|z_{2}|^2+|z_{3}|^2+|z_{4}|^2+|z_{5}|^2=5$$ ...
6
votes
2answers
165 views
+100

Find Minimum value of $P=\frac{1}{1+2x}+\frac{1}{1+2y}+\frac{3-2xy}{5-x^2-y^2}$

Given: $x,y\in (-\sqrt2;\sqrt2)$ and $x^4+y^4+4=\dfrac{6}{xy}$ Find Minimum value Of $$P=\frac{1}{1+2x}+\frac{1}{1+2y}+\frac{3-2xy}{5-x^2-y^2}$$ Could someone help me ?
12
votes
2answers
203 views
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Bernoulli's inequality and an unexpected limit

This question is inspired by What would happen to Bernoulli's inequality if $x<-1$?. Let $x_n=\min\{x\in{\bf R}:(1+x)^n\geq 1+nx\}$, where $n$ is natural and odd (my mistake in the first ...
4
votes
1answer
120 views
+100

A Cauchy-Schwartz type inequality

Given positive integers $k<n$ and positive real numbers $x_1$, $x_2, \dots, x_n$. Denote $$ A={x_1\over x_2+x_3+\dots+x_{k+1}}+{x_2\over x_3+x_4+\dots+x_{k+2}}+\ldots+{x_n\over x_1+x_2+\dots+x_k}$$ ...