query: Proof that $2(a^4+b^4+c^4)<(a^2+b^2+c^2)^2$ if and only if line segments of length $a$,$b$ and $c$ form a triangle
positive_passages.docid: 157660
positive_passages.text: proof $a$ ,$b$ and $c$ can be sides of a triangle
negative_passages.docid: 115929
negative_passages.text: In a triangle $\angle A = 2\angle B$ iff $a^2 = b(b+c)$
negative_passages.docid: 49548
negative_passages.text: If $(a + ib)^3 = 8$, prove that $a^2 + b^2 = 4$
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: 48484
negative_passages.text: Prove that $6|a+b+c$ if and only if $6|a^3+b^3+c^3$
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: 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: 95643
negative_passages.text: Proving $\sum_{cyc} \sqrt{a^2+ab+b^2}\geq\sqrt 3$ when $a+b+c=3$
negative_passages.docid: 32963
negative_passages.text: proving the inequality $\triangle\leq \frac{1}{4}\sqrt{(a+b+c)\cdot abc}$
negative_passages.docid: 46814
negative_passages.text: $ a+b+c+d=6 , a^2+b^2+c^2+d^2=12$ $\implies$ $ 36 \leq 4(a^3+b^3+c^3+d^3)-(a^4+b^4+c^4+d^4) \leq48 $
negative_passages.docid: 50417
negative_passages.text: Prove that the triangles $ABC$ and $AB^{'}C^{'}$ have the same incentre.
negative_passages.docid: 111003
negative_passages.text: Prove that if $A\triangle B = C\triangle B$, then $A = C$
negative_passages.docid: 14480
negative_passages.text: Proving $(P(AB))^2+(P(AB^c))^2+(P(A^cB))^2+(P(A^cB^c))^2\ge\frac{1}{4}$ for any events $A$ and $B$
negative_passages.docid: 43478
negative_passages.text: Prove that: $ m_a + m_b + m_c \le 4R + r $
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: 87206
negative_passages.text: Prove that a triangle can be inscribed in the hyperbola $xy=c^2$ whose sides touch the parabola $y^2=4ax$.
negative_passages.docid: 45702
negative_passages.text: For any triangle with sides $a$, $b$, $c$, show that $a^2b(a-b)+b^2c(b-c)+c^2a(c-a)\ge 0$