MATH PROBLEM : International Mathematical Talent Search

Problem 1/1. For every positive integer n, form the number n/s(n), where s(n) is the sum of the digits of n in base 10. Determine the minimum value of  n/s(n) in each of the following cases:

(i)                10 ≤ n ≤ 99

(ii)              (ii) 100 ≤ n ≤ 999

(iii)            1000 ≤ n ≤ 9999

(iv)            10000 ≤ n ≤ 99999

Problem 2/1. Find all pairs of integers, n and k, 2 < k < n, such that the binomial coefficients

form an increasing arithmetic series.

Problem 3/1. On an 8 x 8 board we place n dominoes, each covering two adjacent squares, so that no more dominoes can be placed on the remaining squares. What is the smallest value of n for which the above statement is true?

Problem 4/1. Show that an arbitrary acute triangle can be dissected by straight line segments into three parts in three different ways so that each part has a line of symmetry.

Problem 5/1. Show that it is possible to dissect an arbitrary tetrahedron into six parts by planes or portions thereof so that each of the parts has a plane of symmetry.