MathDB

6

Part of 2012 Tournament of Towns

Problems(4)

2012 ToT Spring Junior A p6 find PIN number of Inspector Gadget

Source:

3/4/2020
A bank has one million clients, one of whom is Inspector Gadget. Each client has a unique PIN number consisting of six digits. Dr. Claw has a list of all the clients. He is able to break into the account of any client, choose any nn digits of the PIN number and copy them. The n digits he copies from different clients need not be in the same nn positions. He can break into the account of each client, but only once. What is the smallest value of nn which allows Dr.Claw to determine the complete PIN number of Inspector Gadget?
combinatorics
2012 ToT Spring Senior A p6 cover plane with rectangles

Source:

3/5/2020
We attempt to cover the plane with an infi nite sequence of rectangles, overlapping allowed. (a) Is the task always possible if the area of the nnth rectangle is n2n^2 for each nn? (b) Is the task always possible if each rectangle is a square, and for any number NN, there exist squares with total area greater than NN?
combinatoricscombinatorial geometrySquaresRectanglesgeometryrectangle
2012 ToT Fall Junior A p6 centre of mass of 2n-gon

Source:

3/22/2020
(a) A point AA is marked inside a circle. Two perpendicular lines drawn through AA intersect the circle at four points. Prove that the centre of mass of these four points does not depend on the choice of the lines. (b) A regular 2n2n-gon (n≥2n \ge 2) with centre AA is drawn inside a circle (A does not necessarily coincide with the centre of the circle). The rays going from AA to the vertices of the 2n2n-gon mark 2n2n points on the circle. Then the 2n2n-gon is rotated about AA. The rays going from AA to the new locations of vertices mark new 2n2n points on the circle. Let OO and NN be the centres of gravity of old and new points respectively. Prove that O=NO = N.
geometrycombinatorial geometryperpendicularCentroidregular polygon
2012 ToT Fall Senior A p6 centroid of points of icosahedron

Source:

3/22/2020
(a) A point AA is marked inside a sphere. Three perpendicular lines drawn through AA intersect the sphere at six points. Prove that the centre of gravity of these six points does not depend on the choice of such three lines. (b) An icosahedron with the centre AA is placed inside a sphere (its centre does not necessarily coincide with the centre of the sphere). The rays going from AA to the vertices of the icosahedron mark 1212 points on the sphere. Then the icosahedron is rotated about its centre. New rays mark new 1212 points on the sphere. Let OO and NN be the centres of mass of old and new points respectively. Prove that O=NO = N.
icosahedron3D geometryCentroidgeometryspherecombinatorial geometry