# 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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