MathDB

2

Part of 2009 Germany Team Selection Test

Problems(4)

Show that there is an infinite number of primes p

Source: German TST 3, P2, 2009, Exam set by Gunther Vogel

7/18/2009
Let (an)n∈N \left(a_n \right)_{n \in \mathbb{N}} defined by a_1 \equal{} 1, and a_{n \plus{} 1} \equal{} a^4_n \minus{} a^3_n \plus{} 2a^2_n \plus{} 1 for n≥1. n \geq 1. Show that there is an infinite number of primes p p such that none of the an a_n is divisible by p. p.
inductionnumber theoryprime numbersnumber theory unsolved
Tracy has been baking a rectangular cake

Source: German TST 4, P2, 2009, Exam set by Christian Reiher

7/18/2009
Tracy has been baking a rectangular cake whose surface is dissected by grid lines in square fields. The number of rows is 2n 2^n and the number of columns is 2^{n \plus{} 1} where n≥1,n∈N. n \geq 1, n \in \mathbb{N}. Now she covers the fields with strawberries such that each row has at least 2n \plus{} 2 of them. Show that there four pairwise distinct strawberries A,B,C A,B,C and D D which satisfy those three conditions: (a) Strawberries A A and B B lie in the same row and A A further left than B. B. Similarly D D lies in the same row as C C but further left. (b) Strawberries B B and C C lie in the same column. (c) Strawberries A A lies further up and further left than D. D.
combinatorics unsolvedcombinatorics
In Skinien there 2009 towns where each of them is connected

Source: German TST 7, P2, 2009, Exam set by Christian Reiher

7/18/2009
In Skinien there 2009 towns where each of them is connected with exactly 1004 other town by a highway. Prove that starting in an arbitrary town one can make a round trip along the highways such that each town is passed exactly once and finally one returns to its starting point.
combinatorics unsolvedcombinatorics
CP bisects angle C at triangle ABC

Source: AIMO 2, German Pre-TST 2009

7/16/2011
Let triangle ABCABC be perpendicular at A.A. Let MM be the midpoint of segment BC‾.\overline{BC}. Point DD lies on side AC‾\overline{AC} and satisfies ∣AD∣=∣AM∣.|AD|=|AM|. Let P≠CP \neq C be the intersection of the circumcircle of triangles AMCAMC and BDC.BDC. Prove that CPCP bisects the angle at CC of triangle ABC.ABC.
geometrycircumcircleincentergeometry unsolved