query: A simple proof for $\prod_{i=1}^d a_i+\prod_{i=1}^d b_i \le \prod_{i=1}^d (a_i^d+b_i^d)^{1/d}$?
positive_passages.docid: 25648
positive_passages.text: Prove $(a_1+b_1)^{1/n}\cdots(a_n+b_n)^{1/n}\ge \left(a_1\cdots a_n\right)^{1/n}+\left(b_1\cdots b_n\right)^{1/n}$
negative_passages.docid: 136652
negative_passages.text: Closed form expression for the product $\prod\limits_{k=1}^{n}\left(1 - \frac{1}{ak}\right)$
negative_passages.docid: 133228
negative_passages.text: Prove $(|x| + |y|)^p \le |x|^p + |y|^p$ for $x,y \in \mathbb R$ and $p \in (0,1]$.
negative_passages.docid: 7201
negative_passages.text: Prove that $x_1^2+x_2^2+x_3^2=1$ yields $ \sum_{i=1}^{3}\frac{x_i}{1+x_i^2} \le \frac{3\sqrt{3}}{4} $
negative_passages.docid: 26741
negative_passages.text: How to prove for $s<1,|a+b|^s\le|a|^s+|b|^s$
negative_passages.docid: 126574
negative_passages.text: Proving $\prod_k \sin \pi k / n = n / 2^{n-1}$
negative_passages.docid: 128108
negative_passages.text: $Y=\prod_{n\in\mathbb{N}}X$.
negative_passages.docid: 41558
negative_passages.text: Prove that $\frac4{abcd} \geq \frac a b + \frac bc + \frac cd +\frac d a$
negative_passages.docid: 39082
negative_passages.text: Proving $x^\alpha-\alpha x \le 1- \alpha $
negative_passages.docid: 93427
negative_passages.text: Show that $\sum_{i=1}^{n} \frac {a_i}{1+a-a_i} + \prod_{i=1}^{n} {(1-a_i)} \leq 1$
negative_passages.docid: 39953
negative_passages.text: $a\mid b,\ c\mid d\,\Rightarrow\ ac\mid bd $ $\ \, \bf\small [Divisibility\ Product\ Rule]$
negative_passages.docid: 27449
negative_passages.text: Product of Divisors of some $n$ proof
negative_passages.docid: 82591
negative_passages.text: Prove $(x+r_1) \cdots (x+r_n) \geq (x+(r_1 \cdots r_n)^{1/n})^{n}$.
negative_passages.docid: 281
negative_passages.text: How one can prove $\prod_{k=1}^{n-1} (\cos\theta-\cos(k\pi/n)) = 2^{1-n}\sin(n\theta)\csc\theta$?
negative_passages.docid: 91081
negative_passages.text: Prove that, for every natural n, $\sum_{d|n} \frac{\mu(d)}{d}=\prod_{p|n}(1-\frac{1}{p})$
negative_passages.docid: 71299
negative_passages.text: Prove that $\frac{a}{b} < \frac{a+c}{b+d} <\frac{ c}{d}$
negative_passages.docid: 80789
negative_passages.text: Prove that $\forall a,b,c,d\in \mathbb{R}\,$, $(a^2-b^2)(c^2-d^2)\leq(ac-bd)^2$ and equalities hold if and only if $ad=bc$.