query: Show that if $a+b+c=0$, $2(a^4 + b^4+ c^4)$ is a perfect square
positive_passages.docid: 27387
positive_passages.text: Show that $a+b+c=0$ implies that $32(a^4+b^4+c^4)$ is a perfect square.
negative_passages.docid: 74222
negative_passages.text: If $a+b=1$ then $a^{4b^2}+b^{4a^2}\leq1$
negative_passages.docid: 48787
negative_passages.text: A number is a perfect square if and only if it has odd number of positive divisors
negative_passages.docid: 111847
negative_passages.text: For $a,b,c>0$, prove that $\frac{a^2}{b}+\frac{b^2}{c}+\frac{c^2}{a}\ge a+b+c+\frac{4(a-b)^2}{a+b+c}$
negative_passages.docid: 84220
negative_passages.text: Questions on perfect squares
negative_passages.docid: 41641
negative_passages.text: $(n-7)(n+7)=$ some perfect square
negative_passages.docid: 87522
negative_passages.text: If $ a + b + c \mid a^2 + b^2 + c^2$ then $ a + b + c \mid a^n + b^n + c^n$ for infinitely many $n$
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: 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: 128387
negative_passages.text: Given that $a+b\sqrt[3]{2} +c\sqrt[3]{4} =0$, where $a,b,c$ are integers. Show $a=b=c=0$
negative_passages.docid: 96327
negative_passages.text: Show that $a^x+b^x+c^x>(a+b+c)^x$ for all $a,b,c>0$ and $0<x<1$
negative_passages.docid: 134121
negative_passages.text: Determining all the positive integers $n$ such that $n^4+n^3+n^2+n+1$ is a perfect square.
negative_passages.docid: 125164
negative_passages.text: How to prove the existence of $b$ in $Q$ such that $a<b^2<c$ in $Q$?
negative_passages.docid: 142513
negative_passages.text: Prove that $2xy\mid x^2+y^2-x$ implies $x$ is a perfect square.
negative_passages.docid: 7491
negative_passages.text: How to show that $7\mid a^2+b^2$ implies $7\mid a$ and $7\mid b$?
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: 31104
negative_passages.text: $a+b+c =0$; $a^2+b^2+c^2=1$. Prove that: $a^2 b^2 c^2 \le \frac{1}{54}$