MathDB

3

Part of 1988 Greece National Olympiad

Problems(3)

min max distance 2 circles 1988 Greece MO Grade X p3

Source:

9/7/2024
Two circles (O1,R1)(O_1,R_1),(O2,R2)(O_2,R_2) lie each external to the other. Find : a) the minimum length of the segment connecting points of the circles b) the max length of the segment connecting points of the circles
geometrygeometric inequalitycircles
MB/ MC+ ND/NC >1, rectangle ABCD

Source: 1988 Greece MO Grade XI p3

9/7/2024
Bisectors of ∠BAC\angle BAC, ∠CAD\angle CAD in a rectangle ABCDABCD , intersect the sides BCBC, CDCD at points MM and NN resp. Prove that (MB)(MC)+(ND)(NC)>1\frac{(MB)}{(MC)}+\frac{(ND)}{(NC)}>1
geometryrectanglegeometric inequality
matrix A^{3n} takes only two values if A^2+I=A

Source: 1988 Greece MO Grade XII p3

9/7/2024
Let AA be a n×nn \times n matrix of real numbers such that A2+I=AA^2+\mathbb{I}=A, where I\mathbb{I} is the identity n×nn \times n matrix. Prove that the matrix A3nA^{3n} , where ν∈Z\nu\in\mathbb{Z} takes only two values and find those values.
linear algebramatrix