Some number theory problems: tutorial set II: RMO and INMO

1. A simplified form of Fermat’s theorem: If x, y, z, n are natural numbers, and n \geq z, prove that the relation x^{n} + y^{n} = z^{n} does not hold.

2. Distribution of numbers: Find ten numbers x_{1}, x_{2}, \ldots, x_{10} such that (a) the number x_{1} is contained in the closed interval [0,1] (b) the numbers x_{1} and x_{2} lie in different halves of the closed interval [0,1] (c) the numbers x_{1}, x_{2}, x_{3} lie in different thirds of the closed interval [0,1] (d) the numbers x_{1}, x_{2}, x_{3} and x_{4} lie in different quarters of the closed interval [0,1],  etc., and finally, (e) the numbers x_{1}, x_{2}, x_{3}, \ldots, x_{10} lie in different tenths of the closed interval [0,1]

3. Is generalization of the above possible?

4. Proportions: Consider numbers A, B, C, p, q, r such that: A:B =p, B:C=q, C:A=r, write the proportion A:B:C = \Box : \Box : \Box in such a way that in the empty squares, there will appear expressions containing p, q, r only; these expressions being obtained by cyclic permutation of one another expressions.

5. Give an elementary proof of the fact that the positive root of x^{5} + x = 10 is irrational.

I will give you sufficient time to try these. Later, I will post the solutions.

Cheers,

Nalin Pithwa.

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