MathDB

4

Part of 2008 Tournament Of Towns

Problems(8)

TT2008 Junior O-Level - P4

Source:

9/4/2010
Given three distinct positive integers such that one of them is the average of the two others. Can the product of these three integers be the perfect 2008th power of a positive integer?
number theory proposednumber theory
TT2008 Junior A-Level - P4

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9/4/2010
Baron Munchausen claims that he got a map of a country that consists of five cities. Each two cities are connected by a direct road. Each road intersects no more than one another road (and no more than once). On the map, the roads are colored in yellow or red, and while circling any city (along its border) one can notice that the colors of crossed roads alternate. Can Baron's claim be true?
combinatorics proposedcombinatoricsBaron Munchausen
TT2008 Senior O-Level - P4

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9/4/2010
Five distinct positive integers form an arithmetic progression. Can their product be equal to a2008a^{2008} for some positive integer aa ?
arithmetic sequencenumber theory proposednumber theory
TT2008 Senior A-Level - P4

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9/4/2010
Let ABCDABCD be a non-isosceles trapezoid. De fine a point A1A1 as intersection of circumcircle of triangle BCDBCD and line ACAC. (Choose A1A_1 distinct from CC). Points B1,C1,D1B_1, C_1, D_1 are de fined in similar way. Prove that A1B1C1D1A_1B_1C_1D_1 is a trapezoid as well.
geometrycircumcircletrapezoidpower of a pointradical axisgeometry proposed
2008 ToT Spring Junior O P4 (n + 1)! divisible by 1! + 2! + ... + n!

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3/7/2020
Find all positive integers nn such that (n+1)!(n + 1)! is divisible by 1!+2!+...+n!1! + 2! + ... + n!.
number theoryfactorialdivisible
2008 ToT Spring Junior A P4 finitely many points, 4 colours

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3/7/2020
Given are finitely many points in the plane, no three on a line. They are painted in four colours, with at least one point of each colour. Prove that there exist three triangles, distinct but not necessarily disjoint, such that the three vertices of each triangle have different colours, and none of them contains a coloured point in its interior.
combinatoricsColoring
2008 ToT Spring Senior O P4 2 copies of a convex polygon in a square

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3/7/2020
No matter how two copies of a convex polygon are placed inside a square, they always have a common point. Prove that no matter how three copies of the same polygon are placed inside this square, they also have a common point.
combinatorial geometryconvex polygonpolygonsquare tablegeometry
2008 ToT Spring Senior A P4 angles

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2/26/2020
Each of Peter and Basil draws a convex quadrilateral with no parallel sides. The angles between a diagonal and the four sides of Peter's quadrilateral are α,α,β\alpha, \alpha, \beta and γ\gamma in some order. The angles between a diagonal and the four sides of Basil's quadrilateral are also α,α,β\alpha, \alpha, \beta and γ\gamma in some order. Prove that the acute angle between the diagonals of Peter's quadrilateral is equal to the acute angle between the diagonals of Basil's quadrilateral.
geometryangles