MathDB

4

Part of 2005 Romania Team Selection Test

Problems(2)

sequence of digits interlaced with another sequence of digit

Source: Romanian IMO TST 2005 - day 4, problem 4

4/23/2005
a) Prove that there exists a sequence of digits {cn}n≥1\{c_n\}_{n\geq 1} such that or each n≥1n\geq 1 no matter how we interlace knk_n digits, 1≤kn≤91\leq k_n\leq 9, between cnc_n and cn+1c_{n+1}, the infinite sequence thus obtained does not represent the fractional part of a rational number. b) Prove that for 1≤kn≤101\leq k_n\leq 10 there is no such sequence {cn}n≥1\{c_n\}_{n\geq 1}. Dan Schwartz
modular arithmeticnumber theory proposednumber theory
again a polyhedra problem

Source: Romanian IMO TST 2005 - day 5, problem 4

4/24/2005
We consider a polyhedra which has exactly two vertices adjacent with an odd number of edges, and these two vertices are lying on the same edge. Prove that for all integers n≥3n\geq 3 there exists a face of the polyhedra with a number of sides not divisible by nn.
combinatorics proposedcombinatorics