MathDB

3

Part of 2016 VJIMC

Problems(2)

parallel lines in a simplex

Source: 26th annual VJIMC (2016), Category I, Problem 3

4/10/2016
Let d≥3d \geq 3 and let A1…Ad+1A_1 \dots A_{d + 1} be a simplex in Rd\mathbb{R}^d. (A simplex is the convex hull of d+1d + 1 points not lying in a common hyperplane.) For every i=1,…,d+1i = 1, \dots , d + 1 let OiO_i be the circumcentre of the face A1…Ai−1Ai+1…Ad+1A_1 \dots A_{i - 1}A_{i+1}\dots A_{d+1}, i.e. OiO_i lies in the hyperplane A1…Ai−1Ai+1…Ad+1A_1 \dots A_{i - 1}A_{i+1}\dots A_{d+1} and it has the same distance from all points A1,…,Ai−1,Ai+1,…,Ad+1A_1, \dots , A_{i-1}, A_{i+1}, \dots , A_{d+1}. For each ii draw a line through AiA_i perpendicular to the hyperplane O1…Oi−1Oi+1…Od+1O_1 \dots O_{i-1}O_{i+1} \dots O_{d+1}. Prove that either these lines are parallel or they have a common point.
geometrycollege contests
eigenvalues of a nxn matrix

Source: 26th annual VJIMC (2016), Category II, Problem 3

4/10/2016
For n≥3n \geq 3 find the eigenvalues (with their multiplicities) of the n×nn \times n matrix
[101000……00020100……00102010……00010201……00001020……00000102……00⋮⋮⋮⋮⋮⋮⋱⋮⋮⋮⋮⋮⋮⋮⋮⋱⋮⋮000000……20000000……01]\begin{bmatrix} 1 & 0 & 1 & 0 & 0 & 0 & \dots & \dots & 0 & 0\\ 0 & 2 & 0 & 1 & 0 & 0 & \dots & \dots & 0 & 0\\ 1 & 0 & 2 & 0 & 1 & 0 & \dots & \dots & 0 & 0\\ 0 & 1 & 0 & 2 & 0 & 1 & \dots & \dots & 0 & 0\\ 0 & 0 & 1 & 0 & 2 & 0 & \dots & \dots & 0 & 0\\ 0 & 0 & 0 & 1 & 0 & 2 & \dots & \dots & 0 & 0\\ \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \ddots & & \vdots & \vdots\\ \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & & \ddots & \vdots & \vdots\\ 0 & 0 & 0 & 0 & 0 & 0 & \dots & \dots & 2 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & \dots & \dots & 0 & 1 \end{bmatrix}
linear algebramatrixcollege contestseigenvalue