5
Part of 2013 Tournament of Towns
Problems(8)
2013 ToT Spring Junior O p5, angle chasing, equal segments, 150^o
Source:
3/4/2020
In a quadrilateral , angle is equal to , angle is right, and sides and are equal. Determine the angle between and the line connecting the midpoints of sides and .
geometryanglesequal segmentsright angle
2013 ToT Spring Junior A p5, lattice triangle with 2 lattice points interior
Source:
3/4/2020
A point in the plane is called a node if both its coordinates are integers. Consider a triangle with vertices at nodes containing exactly two nodes inside. Prove that the straight line connecting these nodes either passes through a vertex or is parallel to a side of the triangle.
geometrylattice pointsparallelcombinatorial geometry
2013 ToT Spring Senior O p5 admissible integer trinomial, coefficients<=2013
Source:
3/5/2020
A quadratic trinomial with integer coefficients is called admissible if its leading coeffi cient is , its roots are integers and the absolute values of coefficients do not exceed . Basil has summed up all admissible quadratic trinomials. Prove that the resulting trinomial has no real roots.
algebratrinomialinteger roots
2013 ToT Spring Senior A p5 coloring points in red, blue and yellow
Source:
3/5/2020
On an initially colourless plane three points are chosen and marked in red, blue and yellow.
At each step two points marked in different colours are chosen. Then one more point is painted in the third colour so that these three points form a regular triangle with the vertices coloured clockwise in ''red, blue, yellow". A point already marked may be marked again so that it may have several colours. Prove that for any number of moves all the points containing the same colour lie on the same line.
combinatorial geometrycombinatoricsColoring
2013 ToT Fall Junior O p5 8 rooks are placed on a chessboard
Source:
3/22/2020
Eight rooks are placed on a chessboard so that no two rooks attack each other. Prove that one can always move all rooks, each by a move of a knight so that in the final position no two rooks attack each other as well. (In intermediate positions several rooks can share the same square).
Chessboardcombinatorics
2013 ToT Fall Junior A p5 101-gon inscribed in circle, perpendicular each vertex
Source:
3/22/2020
A -gon is inscribed in a circle. From each vertex of this polygon a perpendicular is dropped to the opposite side or its extension. Prove that at least one perpendicular drops to the side.
perpendicularpoygoncombinatorial geometrygeometry
2013 ToT Fall Senior O p5 asteroid is a ball or a cube?
Source:
3/22/2020
A spacecraft landed on an asteroid. It is known that the asteroid is either a ball or a cube. The rover started its route at the landing site and finished it at the point symmetric to the landing site with respect to the center of the asteroid. On its way, the rover transmitted its spatial coordinates to the spacecraft on the landing site so that the trajectory of the rover movement was known. Can it happen that this information is not suffcient to determine whether the asteroid is a ball or a cube?
combinatorics3D geometrycombinatorial geometryspherecube
2013 ToT Fall Senior O p5 f(f(x))=x, g(g(x))=x, f(g(x))>x, g(f(x))>x
Source:
3/22/2020
Do there exist two integer-valued functions and such that for every integer we have
(a) ?
(b) ?
functionalgebrainequalities