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Romania District Olympiad 2004 - Grade XI

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April 10, 2011
vectorlinear algebramatrixlinear algebra unsolved

Problem Statement

a) Let x1,x2,x3,y1,y2,y3∈Rx_1,x_2,x_3,y_1,y_2,y_3\in \mathbb{R} and aij=sin⁡(xi−yj), i,j=1,3‾a_{ij}=\sin(x_i-y_j),\ i,j=\overline{1,3} and A=(aij)∈M3A=(a_{ij})\in \mathcal{M}_3 Prove that det⁡A=0\det A=0.
b) Let z1,z2,…,z2n∈C∗, n≥3z_1,z_2,\ldots,z_{2n}\in \mathbb{C}^*,\ n\ge 3 such that ∣z1∣=∣z2∣=…=∣zn+3∣|z_1|=|z_2|=\ldots=|z_{n+3}| and arg⁡z1≥arg⁡z2≥…≥arg⁡(zn+3)\arg z_1\ge \arg z_2\ge \ldots\ge \arg(z_{n+3}). If bij=∣zi−zj+n∣, i,j=1,n‾b_{ij}=|z_i-z_{j+n}|,\ i,j=\overline{1,n} and B=(bij)∈MnB=(b_{ij})\in \mathcal{M}_n, prove that det⁡B=0\det B=0.
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