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2

Part of 2014 CHMMC (Fall)

Problems(2)

2014 Fall Team #2

Source:

3/26/2022
Consider two overlapping regular tetrahedrons of side length 22 in space. They are centered at the same point, and the second one is oriented so that the lines from its center to its vertices are perpendicular to the faces of the first tetrahedron. What is the volume encompassed by the combined solid?
combinatoricsgeometry
2014 CHMMC Tiebreaker 2 - square routs od 2x2 matrix

Source:

3/1/2024
A matrix [xyzw]\begin{bmatrix} x & y \\ z & w \end{bmatrix} has square root [abcd]\begin{bmatrix} a & b \\ c & d \end{bmatrix} if [abcd]2=[a2+bcab+bdac+cdbc+d2]=[xyzw]\begin{bmatrix} a & b \\ c & d \end{bmatrix}^2 = \begin{bmatrix} a^2 + bc &ab + bd \\ ac + cd & bc + d^2 \end{bmatrix} = \begin{bmatrix} x & y \\ z & w \end{bmatrix} Determine how many square roots the matrix [2234]\begin{bmatrix} 2 & 2 \\ 3 & 4 \end{bmatrix} has (complex coefficients are allowed).




matrixMatriceslinear algebraCHMMC