query: Proving an inequality without using $ AM\geq GM$
positive_passages.docid: 21986
positive_passages.text: What is the minimum value of $(1 + a_1)(1 + a_2). . .(1 + a_n)$?
negative_passages.docid: 142785
negative_passages.text: How can I prove Holder inequality for $0<p<1$?
negative_passages.docid: 126032
negative_passages.text: Proving a certain inequality
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: 110790
negative_passages.text: Prove the following summation inequality: $\sum_{i=1}^n(a_i)\sum_{i=1}^n(\frac{1}{a_i})\ge n^2$
negative_passages.docid: 81618
negative_passages.text: Prove $||a|-|b| | \leq |a-b|$
negative_passages.docid: 1305
negative_passages.text: Proving the AM-GM inequality for 2 numbers $\sqrt{xy}\le\frac{x+y}2$
negative_passages.docid: 33451
negative_passages.text: Sylvester rank inequality: $\operatorname{rank} A + \operatorname{rank}B \leq \operatorname{rank} AB + n$
negative_passages.docid: 60055
negative_passages.text: Solving a fractional inequality without trial and error
negative_passages.docid: 68986
negative_passages.text: Use the triangle inequality to show that $|a|+|b| \leq |a+b|+|a-b|$
negative_passages.docid: 13228
negative_passages.text: Proofs of AM-GM inequality
negative_passages.docid: 29679
negative_passages.text: Prove: $|x-y|\leq |x|+|y|$
negative_passages.docid: 25419
negative_passages.text: If $n\geq m$ then $(x^m+y^m)^{1/m} \ge (x^n+y^n)^{1/n}$
negative_passages.docid: 91871
negative_passages.text: Proving quadratic inequalities?
negative_passages.docid: 76863
negative_passages.text: How to show $ \sup \inf g(x,y) \leq \inf \sup g(x,y)$?
negative_passages.docid: 66742
negative_passages.text: Proving the AM-GM Inequality with Lagrange Multipliers
negative_passages.docid: 5369
negative_passages.text: Proving the inequality $|a-b| \leq |a-c| + |c-b|$ for real $a,b,c$