Some Number Theory Questions for RMO and INMO

1) Let $n \geq 2$ and k be any positive integers. Prove that $(n-1)^{2}\mid (n^{k}-1)$ if and only if $(n-1) \mid k$.

2) Prove that there are no positive integers a, b, $n >1$ such that $(a^{n}-b^{n}) \mid (a^{n}+b^{n})$.

3) If a and $b>2$ are any positive integers, prove that $2^{a}+1$ is not divisible by $2^{b}-1$.

4) The integers 1,3,6,10, $\ldots$, $n(n+1)/2$, …are called the triangular numbers because they are the numbers of dots needed to make successive triangular arrays of dots. For example, the number 10 can be perceived as the number of acrobats in a human triangle, 4 in a row at the bottom, 3 at the next level, then 2, then 1 at the top. The square numbers are $1, 4, 9, \ldots, n^{2}, \ldots$ The pentagonal numbers 1, 5, 12, 22, $\ldots$, $(3n^{2}-n)/2$, $\ldots$, can be seen in a geometric array in the following way: Start with n equally spaced dots $P_{1}, P_{2}, \ldots, P_{n}$ on a straight line in a plane, with distance 1 between consecutive dots. Using $P_{1}P_{2}$ as a base side, draw a regular pentagon in the plane. Similarly, draw $n-2$ additional regular pentagons on base sides $P_{1}P_{3}$, $P_{1}P_{4}$, $\ldots$, $P_{1}P_{n}$, all pentagons lying on the same side of the line $P_{1}P_{n}$. Mark dots at each vertex and at unit intervals along the sides of these pentagons. Prove that the total number of dots in the array is $(3n^{2}-n)/2$. In general, if regular k-gons are constructed on the sides $P_{1}P_{2}$, $P_{1}P_{3}$, …, $P_{1}P_{n}$, with dots marked again at unit intervals, prove that the total number of dots is $1+kn(n-1)/2 -(n-1)^{2}$. This is the nth k-gonal number.

5) Prove that if $m>n$, then $a^{2^{n}}+1$ is a divisor of $a^{2^{m}}-1$. Show that if a, m, n are positive with $m \neq n$, then

$( a^{2^{m}}+1, a^{2^{n}}+1) = 1$, if a is even; and is 2, if a is odd.

6) Show that if $(a,b)=1$ then $(a+b, a^{2}-ab+b^{2})=1$ or 3.

7) Show that if $(a,b)=1$ and p is an odd prime, then $( a+b, \frac{a^{p}+b^{p}}{a+b})=p$ or 1.

8) Suppose that $2^{n}+1=xy$, where x and y are integers greater than 1 and $n>0$. Show that $2^{a}\mid (x-1)$ if and only if $2^{a}\mid (y-1)$.

9) Prove that $(n!+1, (n+1)!+1)=1$.

10) Let a and b be positive integers such that $(1+ab) \mid (a^{2}+b^{2})$. Show that the integer $(a^{2}+b^{2})/(1+ab)$ must be a perfect square.

Note that in the above questions, in general, (a,b) means the gcd of a and b.

More later,
Nalin Pithwa.

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