MathDB

1

Part of 2000 Romania Team Selection Test

Problems(3)

f mapping {1,2...,n} to {1,2,3,4,5} with condition - Rom TST

Source: Romanian TST 2000

2/14/2011
Let n≥2n\ge 2 be a positive integer. Find the number of functions f:{1,2,…,n}→{1,2,3,4,5}f:\{1,2,\ldots ,n\}\rightarrow\{1,2,3,4,5 \} which have the following property: ∣f(k+1)−f(k)∣≥3|f(k+1)-f(k)|\ge 3, for any k=1,2,…n−1k=1,2,\ldots n-1.
Vasile Pop
functionalgebra proposedalgebra
Least n such that 2^2000 divides a^n-1

Source: Romanian TST 2000

2/18/2011
Let a>1a>1 be an odd positive integer. Find the least positive integer nn such that 220002^{2000} is a divisor of an−1a^n-1.
Mircea Becheanu
modular arithmeticnumber theory proposednumber theory
There exists a monochromatic isosceles trapezium ABCD

Source:

2/18/2011
Let P1P_1 be a regular nn-gon, where n∈Nn\in\mathbb{N}. We construct P2P_2 as the regular nn-gon whose vertices are the midpoints of the edges of P1P_1. Continuing analogously, we obtain regular nn-gons P3,P4,…,PmP_3,P_4,\ldots ,P_m. For m≥n2−n+1m\ge n^2-n+1, find the maximum number kk such that for any colouring of vertices of P1,…,PmP_1,\ldots ,P_m in kk colours there exists an isosceles trapezium ABCDABCD whose vertices A,B,C,DA,B,C,D have the same colour.
Radu Ignat
geometrytrapezoidceiling functioncombinatorics proposedcombinatorics