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 p and every natural number n, the numbers p^n,p^{n+1} and p^{n+2} 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 P(x) be a polynomial with real coefficients. Prove that there exist positive integers n and k such that k has n digits and more than P(n) positive divisors.

Bài 48. Prove that for all integers a > 1 and b > 1 there exists a function f from the positive integers to the positive integers such that f(a\cdot f(n)) = b\cdot n for all n positive integer.

Bài 49. Let \, P(z) \, be a polynomial with complex coefficients which is of degree \, 1992 \, and has distinct zeros. Prove that there exist complex numbers \, a_1, a_2, \ldots, a_{1992} \, such that \, P(z) \, divides the polynomial

\left( \cdots \left( (z-a_1)^2 - a_2 \right)^2 \cdots - a_{1991} \right)^2 - a_{1992}.

Bài 50. Suppose \, q_{0}, \, q_{1}, \, q_{2}, \ldots \; \, is an infinite sequence of integers satisfying the following two conditions:

(i)  \, m-n \, divides \, q_{m}-q_{n}\, for \, m > n \geq 0,

(ii) there is a polynomial \, P \, such that \, |q_{n}| < P(n) \, for all \, n.

Prove that there is a polynomial \, Q \, such that \, q_{n}= Q(n) \, for all \, n.

Bài 51. For any nonempty set S of real numbers, let \sigma(S) denote the sum of the elements of S. Given a set A of n positive integers, consider the collection of all distinct sums \sigma(S) as S ranges over the nonempty subsets of A.  Prove that this collection of sums can be partitioned into n classes so that in each class, the ratio of the largest sum to the smallest sum does not exceed 2.

Bài 52. Determine (with proof) whether there is a subset X of the integers with the following property: for any integer n there is exactly one solution of a + 2b = n with a,b \in X.

Bài 53. Let S be a set with 2002 elements, and let N be an integer with 0 \leq N \leq 2^{2002}. Prove that it is possible to color every subset of S 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 N white subsets.

Bài 54. Prove that for every positive integer n there exists an n-digit number divisible by 5^n all of whose digits are odd.

Bài 55. At the vertices of a regular hexagon are written six nonnegative integers whose sum is 2003. 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 0 appears at all six vertices.

Bài 56. Let S be a set containing n^{2}+n-1 elements, for some positive integer n. Suppose that the n- element subsets of S are partitioned into two classes. Prove that there are at least n pairwise disjoint sets in the same class.

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