query: Let $p$ and $q$ be positive numbers having a sum of $1.$ What is the minimum value of $(p+(1/p))^2 + (q+(1/q))^2?$
positive_passages.docid: 40700
positive_passages.text: Proving inequality $(a+\frac{1}{a})^2 + (b+\frac{1}{b})^2 \geq \frac{25}{2}$ for $a+b=1$
negative_passages.docid: 45090
negative_passages.text: Given any positive real numbers $a,b,c$, we have $(a^{2}+2)(b^{2}+2)(c^{2}+2)\geq 9(ab+bc+ca)$
negative_passages.docid: 54473
negative_passages.text: Value of $\sum 1/p^p$
negative_passages.docid: 92300
negative_passages.text: For positive $a$, $b$, $c$ with $abc=1$, show that $\sum_{cyc}\sqrt{a^2-a+1}\geq a+b+c$
negative_passages.docid: 708
negative_passages.text: Find the minimum of $(u-v)^2+ (\sqrt{2-u^2}-\frac{9}{v})^2$ for $00$.
negative_passages.docid: 54078
negative_passages.text: The minimum value of $\frac{(x+\frac{1}{x})^6-(x^6+\frac{1}{x^6})-2}{(x+\frac{1}{x})^3+x^3+\frac{1}{x^3}}$
negative_passages.docid: 67710
negative_passages.text: Determine the smallest positive value of x(in degrees) for which: $\tan(x+100^{\circ}) = \tan(x+50^{\circ})\tan (x)\tan(x-50^{\circ})$
negative_passages.docid: 115850
negative_passages.text: Minimum of $|ax-by+c|$
negative_passages.docid: 130499
negative_passages.text: How can $p^{q+1}+q^{p+1}$ be a perfect square?
negative_passages.docid: 62462
negative_passages.text: Problem with inequality: $ \left| \sqrt{2}-\frac{p}{q} \right| > \frac{1}{3q^2}$
negative_passages.docid: 141529
negative_passages.text: Proving existence of a minimum
negative_passages.docid: 140599
negative_passages.text: Inequality Proof in Real Numbers: If $p+q=1$ then $pq\le\frac14$
negative_passages.docid: 137105
negative_passages.text: The minimum value of $x^8 – 8x^6 + 19x^4 – 12x^3 + 14x^2 – 8x + 9$ is
negative_passages.docid: 127901
negative_passages.text: If $x^2+y^2 \equiv 0\pmod{p}$, then $p \equiv 1 \pmod{4}$
negative_passages.docid: 45454
negative_passages.text: Pairs of $(p,q)$ such that $p^2 + 1 = 2 q^2$?
negative_passages.docid: 51649
negative_passages.text: prove there is no smallest positive rational number
negative_passages.docid: 62501
negative_passages.text: Minimum value of $\frac{4}{4-x^2}+\frac{9}{9-y^2}$