MathDB

5

Part of 2023 Tuymaada Olympiad

Problems(2)

Graph numbering

Source: 2023 Tuymaada Junior P5

7/12/2023
A graph contains pp vertices numbered from 11 to pp, and qq edges numbered from p+1p + 1 to p+qp + q. It turned out that for each edge the sum of the numbers of its ends and of the edge itself equals the same number ss. It is also known that the numbers of edges starting in all vertices are equal. Prove that s=12(4p+q+3).s = \dfrac{1}{2} (4p+q+3).
graphcombinatorics
small ship sails on a polynomial determined coordinate

Source: Tuymaada 2023 Senior P5

7/8/2023
A small ship sails on an infinite coordinate sea. At the moment tt the ship is at the point with coordinates (f(t),g(t))(f(t), g(t)), where ff and gg are two polynomials of third degree. Yesterday at 14:0014:00 the ship was at the same point as at 13:0013:00, and at 20:0020:00, it was at the same point as at 19:0019:00. Prove that the ship sails along a straight line.
analytic geometrypolynomialalgebraTuymaada