query: If $a$ and $b$ are positive real numbers such that $a+b=1$, prove that $(a+1/a)^2+(b+1/b)^2\ge 25/2$
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: 136868
negative_passages.text: Prove that $\sqrt{a^2+3b^2}+\sqrt{b^2+3c^2}+\sqrt{c^2+3a^2}\geq6$ if $(a+b+c)^2(a^2+b^2+c^2)=27$
negative_passages.docid: 133382
negative_passages.text: Proof that $a+b+\frac{1}{1+\sqrt{ab}}\geq\frac{5}{2}$
negative_passages.docid: 137626
negative_passages.text: If $a,b,c$ are positive integers,and $\cfrac{a}{b}+\cfrac{b}{c}+\cfrac{c}{a}$ and $\cfrac{b}{a}+\cfrac{c}{b}+\cfrac{a}{c}$ are integers then $a=b=c$
negative_passages.docid: 101362
negative_passages.text: Let a,b,c be positive real number, proof.
negative_passages.docid: 56311
negative_passages.text: Proving $(1+a_{2})^{2}(1+a_{3})^{3}\cdots(1+a_{n})^{n}\ge n^n$ for positive reals $a_2,\ldots, a_n$ whose product is $1$
negative_passages.docid: 19909
negative_passages.text: if $abc=1$, then $a^2+b^2+c^2\ge a+b+c$
negative_passages.docid: 142992
negative_passages.text: Prove that for every positive integer $n$, $1/1^2+1/2^2+1/3^2+\cdots+1/n^2\le2-1/n$
negative_passages.docid: 68705
negative_passages.text: Prove that $A\ge0, B\ge0$ and $A\ge B$ implies $B^{-1}\ge A^{-1}$
negative_passages.docid: 17100
negative_passages.text: Prove that $\frac{b^2+c^2}{b+c}+\frac{c^2+a^2}{c+a}+\frac{a^2+b^2}{a+b} \ge a+b+c$
negative_passages.docid: 5729
negative_passages.text: If $a^2+b^2+c^2=1$ then prove the following.
negative_passages.docid: 141102
negative_passages.text: If $a,b,c$ are positive integers, with $a^2+b^2-ab=c^2$ prove that $(a-b)(b-c)\le0$.
negative_passages.docid: 23080
negative_passages.text: Prove that $a^2(b+c-a)+b^2(a+c-b)+c^2(a+b-c) \leq 3abc.$
negative_passages.docid: 153313
negative_passages.text: Prove that for all integers $ a $ and $ b $, if $ a $ divides $ b $, then $ a^{2} $ divides $ b^{2} $.
negative_passages.docid: 97750
negative_passages.text: Let $a, b, c$ be positive real numbers such that $abc = 1$. Prove that $a^2 + b^2 + c^2 \ge a + b + c$.
negative_passages.docid: 130327
negative_passages.text: Prove that $a^2 > b^2$ ; if $a > b \geq 0$.
negative_passages.docid: 70006
negative_passages.text: $a,b,m,n$ be positive integers such that g.c.d.$(a,b)=1$ and $a^m+b^m|a^n+b^n$ , then $m|n$?