query: Prove that solution of the equation $8x^3-4x^2-4x+1= 0$ has roots $\cos\frac{\pi}{7},\cos\frac{3 \pi}{7},\cos\frac{5 \pi}{7}$
positive_passages.docid: 22058
positive_passages.text: Prove that $\cos(\pi/7)$ is root of equation $8x^3-4x^2-4x+1=0$
negative_passages.docid: 90261
negative_passages.text: Find all solutions of the equation $n^m=x^2+py^2$ which satisfy the following properties
negative_passages.docid: 51804
negative_passages.text: How to show that a root of the equation $x (x+1)(x+2) ....... (x+2009) = c $ can have multiplicity at most 2?
negative_passages.docid: 35529
negative_passages.text: Prove that $\sin x+2x\ge\frac{3x(x+1)}{\pi}$ for all $x\in [0,\frac{\pi}{2}]$
negative_passages.docid: 56877
negative_passages.text: Show that $\sqrt{4 + 2\sqrt{3}} - \sqrt{3}$ is rational.
negative_passages.docid: 144851
negative_passages.text: For a prime $p \equiv 1$ or $3$ (mod $8$), show that the equation $x^{2} + 2y^{2} = p$ has a solution.
negative_passages.docid: 114872
negative_passages.text: If $ax^2-bx+c=0$ has two distinct real roots lying in the interval $(0,1)$ $a,b,c$ belongs to natural prove that $\log_5 {abc}\geq2$
negative_passages.docid: 73277
negative_passages.text: Prove $[\sqrt{x}+\sqrt{x+1}]=[\sqrt{4x+2}], x \in \Bbb N$
negative_passages.docid: 153170
negative_passages.text: Find the number of solutions of the equation $e^{4x}+e^{3x}-4e^{2x}+e^x+1=0$
negative_passages.docid: 146294
negative_passages.text: Find all positive integers $a,b$ such that each of the equations $x^2-ax+b=0$ and $x^2-bx+a=0$ has distinct positive integral roots.
negative_passages.docid: 61148
negative_passages.text: Finding real roots of $ P(x)=x^8 - x^7 +x^2 -x +15$
negative_passages.docid: 68164
negative_passages.text: $(2x^2+3y^2-7)xdx-(3x^2+2y^2-8)ydy=0$
negative_passages.docid: 50777
negative_passages.text: Prove that $4\sin\frac{2\pi}{7}- \tan\frac{\pi}{7}= \sqrt{7}$ .
negative_passages.docid: 140763
negative_passages.text: How to prove that $\sum_{n=1}^{2012} \frac{1}{x_n} = 1 $ has finitely many solutions for positive integer $x_i$?
negative_passages.docid: 41073
negative_passages.text: Prove $ax^2+bx+c=0$ has no rational roots if $a,b,c$ are odd
negative_passages.docid: 135858
negative_passages.text: Prove $x^5 + x^4 + x^3 + x^2 + 1 = 0$ has no rational solution
negative_passages.docid: 126445
negative_passages.text: Proving the second root of a quadratic equation