MathDB

4

Part of 2004 IMC

Problems(2)

Problem 4 IMC 2004 Macedonia

Source:

7/25/2004
Suppose n≥4n\geq 4 and let SS be a finite set of points in the space (R3\mathbb{R}^3), no four of which lie in a plane. Assume that the points in SS can be colored with red and blue such that any sphere which intersects SS in at least 4 points has the property that exactly half of the points in the intersection of SS and the sphere are blue. Prove that all the points of SS lie on a sphere.
geometry3D geometrysphereinequalitiesIMCcollege contests
Problem 10 IMC 2004 Macedonia

Source:

7/26/2004
For n≥1n\geq 1 let MM be an n×nn\times n complex array with distinct eigenvalues λ1,λ2,…,λk\lambda_1,\lambda_2,\ldots,\lambda_k, with multiplicities m1,m2,…,mkm_1,m_2,\ldots,m_k respectively. Consider the linear operator LML_M defined by LMX=MX+XMTL_MX=MX+XM^T, for any complex n×nn\times n array XX. Find its eigenvalues and their multiplicities. (MTM^T denotes the transpose matrix of MM).
linear algebramatrixalgebrapolynomialvectorIMCcollege contests