Bài 46. Prove that the natural numbers can be divided into two groups in a way that both conditions are fulfilled:
1) For every prime number and every natural number , the numbers and do not have the same colour.
2) There does not exist an infinite geometric sequence of natural numbers of the same colour.
Bài 47. Let be a polynomial with real coefficients. Prove that there exist positive integers and such that has digits and more than positive divisors.
Bài 48. Prove that for all integers and there exists a function from the positive integers to the positive integers such that for all positive integer.
Bài 49. Let be a polynomial with complex coefficients which is of degree and has distinct zeros. Prove that there exist complex numbers such that divides the polynomial
Bài 50. Suppose is an infinite sequence of integers satisfying the following two conditions:
(i) divides for
(ii) there is a polynomial such that for all .
Prove that there is a polynomial such that for all .
Bài 51. For any nonempty set of real numbers, let denote the sum of the elements of . Given a set of positive integers, consider the collection of all distinct sums as ranges over the nonempty subsets of . Prove that this collection of sums can be partitioned into classes so that in each class, the ratio of the largest sum to the smallest sum does not exceed .
Bài 52. Determine (with proof) whether there is a subset of the integers with the following property: for any integer there is exactly one solution of with .
Bài 53. Let be a set with elements, and let be an integer with . Prove that it is possible to color every subset of either black or white so that the following conditions hold:
(a) the union of any two white subsets is white;
(b) the union of any two black subsets is black;
(c) there are exactly white subsets.
Bài 54. Prove that for every positive integer there exists an -digit number divisible by all of whose digits are odd.
Bài 55. At the vertices of a regular hexagon are written six nonnegative integers whose sum is . Bert is allowed to make moves of the following form: he may pick a vertex and replace the number written there by the absolute value of the difference between the numbers written at the two neighboring vertices. Prove that Bert can make a sequence of moves, after which the number appears at all six vertices.
Bài 56. Let be a set containing elements, for some positive integer . Suppose that the element subsets of are partitioned into two classes. Prove that there are at least pairwise disjoint sets in the same class.