query: You are given the sides of a triangle $a,b,c$ Prove that: $a^2(b+c-a)+b^2(c+a-b)+c^2(a+b-c)\le 3abc$ (Sweden 1950)
positive_passages.docid: 23080
positive_passages.text: Prove that $a^2(b+c-a)+b^2(a+c-b)+c^2(a+b-c) \leq 3abc.$
negative_passages.docid: 73799
negative_passages.text: In a convex Quadrilateral ABCD, $\measuredangle ABC = \measuredangle BCD = 120^{\circ}$.Prove that: $ AC + BD \ge AB + BC + CD$
negative_passages.docid: 59014
negative_passages.text: Prove that $\sin(2A)+\sin(2B)+\sin(2C)=4\sin(A)\sin(B)\sin(C)$ when $A,B,C$ are angles of a triangle
negative_passages.docid: 157660
negative_passages.text: proof $a$ ,$b$ and $c$ can be sides of a triangle
negative_passages.docid: 62126
negative_passages.text: Constructing a triangle given 2 sides and the inscribed circle's radius
negative_passages.docid: 37536
negative_passages.text: Inequality: $ab^2+bc^2+ca^2 \le 4$, when $a+b+c=3$.
negative_passages.docid: 19940
negative_passages.text: How to show that $A^3+B^3+C^3 - 3ABC = (A+B+C)(A+B\omega+C\omega^2)(A+B\omega^2+C\omega)$ indirectly?
negative_passages.docid: 54170
negative_passages.text: Prove that $\triangle ABC$ is equilateral, If $(a+b-c)^{a}\cdot (b+c-a)^{b}\cdot (c+a-b)^{c}\geq a^a\cdot b^b\cdot c^c$
negative_passages.docid: 5729
negative_passages.text: If $a^2+b^2+c^2=1$ then prove the following.
negative_passages.docid: 22237
negative_passages.text: Proof of triangle inequality
negative_passages.docid: 123136
negative_passages.text: Proof of the identity: $c\sin \frac{A-B}{2} \equiv (a-b) \cos \frac{C}{2}$
negative_passages.docid: 42545
negative_passages.text: If $ a,b,c\in \left(0,\frac{\pi}{2}\right)\;,$ Then prove that $\frac{\sin (a+b+c)}{\sin a+\sin b+\sin c}<1$
negative_passages.docid: 32197
negative_passages.text: Prove that $\tan A + \tan B + \tan C = \tan A\tan B\tan C,$ $A+B+C = 180^\circ$
negative_passages.docid: 30044
negative_passages.text: Inequality with triangle sides
negative_passages.docid: 9986
negative_passages.text: Prove $ \sin(A+B)\sin(A-B)=\sin^2A-\sin^2B $
negative_passages.docid: 12646
negative_passages.text: In acute angle triangle, prove that $\cos (A-B) \cos (B-C) \cos (C-A)\ge 8\cos A \cos B \cos C$
negative_passages.docid: 50417
negative_passages.text: Prove that the triangles $ABC$ and $AB^{'}C^{'}$ have the same incentre.