query: Given $a,b,x>0$, $x (a^y+b^y)^{1/y}$
positive_passages.docid: 25419
positive_passages.text: If $n\geq m$ then $(x^m+y^m)^{1/m} \ge (x^n+y^n)^{1/n}$
negative_passages.docid: 91877
negative_passages.text: Let $|x|\leq1, |y|\leq 1$. Show that$0 \leq x^{2}+y^{2} -2x^{2} y^{2}+2xy \sqrt{1-x^{2}} \sqrt{1-y^{2}} \leq 1$
negative_passages.docid: 12954
negative_passages.text: Prove if $ac>bc$ and $c>0$, then $a>b$
negative_passages.docid: 15753
negative_passages.text: Solve the following equation in positive integers $x$ and $y$
negative_passages.docid: 72470
negative_passages.text: Prove that $2|ab| \leq a^2 + b^2$ and $|a|+|b| \leq \sqrt {2}(a^2+b^2)^{1/2}$
negative_passages.docid: 130327
negative_passages.text: Prove that $a^2 > b^2$ ; if $a > b \geq 0$.
negative_passages.docid: 15232
negative_passages.text: Proving an inequality involving real powers
negative_passages.docid: 147785
negative_passages.text: Proving inequality $x^xy^y \geq (\frac{x+y}{2})^{x+y}$
negative_passages.docid: 84946
negative_passages.text: If $0<a<b$, prove that $a<\sqrt{ab}<\frac{a+b}{2}<b$
negative_passages.docid: 125707
negative_passages.text: A simple proof (hopefully) for $2^x \neq 0$
negative_passages.docid: 45918
negative_passages.text: Inequality: $x^2+y^2+xy\ge 0$
negative_passages.docid: 3206
negative_passages.text: Given $ax^2+by^2+cxy > 0$, what can I deduce about $a$, $b$, and $c$?
negative_passages.docid: 25258
negative_passages.text: Prove that if $ a,b,c > 0 $, then $ [(1 + a) (1 + b) (1 + c)]^{7} > 7^{7} (a^{4} b^{4} c^{4}) $.
negative_passages.docid: 25078
negative_passages.text: What is the solution set for the equation$ x ^ y = y ^ x$ in positive real numbers, x not equal to y?
negative_passages.docid: 7775
negative_passages.text: Show that $|x-y|\leq |x|+|y|$.
negative_passages.docid: 91617
negative_passages.text: Prove this inequality $a \sqrt{1-b^2}+b\sqrt{1-a^2}\le1 $
negative_passages.docid: 44403
negative_passages.text: Prove that $|a+b|^p \leq 2^p \{ |a|^p +|b|^p \}$