**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.

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