MathDB

3

Part of 1997 Vietnam Team Selection Test

Problems(2)

Find the greatest real number...

Source: Vietnam TST 1997, Problem 3

7/28/2008
Find the greatest real number α \alpha for which there exists a sequence of infinitive integers (an) (a_n), ( n \equal{} 1, 2, 3, \ldots) satisfying the following conditions: 1) an>1997n a_n > 1997n for every n∈N∗ n \in\mathbb{N}^{*}; 2) For every n≥2 n\ge 2, Un≥anα U_n\ge a^{\alpha}_n, where U_n \equal{} \gcd\{a_i \plus{} a_k | i \plus{} k \equal{} n\}.
inductioninequalitiesalgebrapolynomialnumber theory unsolvednumber theory
Colors of points on a circle

Source: Vietnam TST 1997, Problem 6

7/28/2008
Let n n, k k, p p be positive integers with 2 \le k \le \frac {n}{p \plus{} 1}. Let n n distinct points on a circle be given. These points are colored blue and red so that exactly k k points are blue and, on each arc determined by two consecutive blue points in clockwise direction, there are at least p p red points. How many such colorings are there?
combinatorics unsolvedcombinatorics