GULF 2017

1- Find a pair $(m,n)$ of positive integers such that $K = |2^m-3^n|$ in all of this cases : $a) K=5$ $b) K=11$ $c) K=19$ 2-Is there a pair $(m,n)$ of positive integers such that : $$|2^m-3^n| = 2017$$ 3-Every prime number less than $41$ can be represented in the form $|2^m-3^n|$ by taking an Appropriate pair $(m,n)$ of positive integers. Prove that the number $41$ cannot be represented in the form $|2^m-3^n|$ where $m$ and $n$ are positive integers

4-Note that $2^5+3^2=41$ . The number $53$ is the least prime number that cannot be represented as a sum or an difference of a power of $2$ and a power of $3$ . Prove that the number $53$ cannot be represented in any of the forms $2^m-3^n$ , $3^n-2^m$ , $2^m-3^n$ where $m$ and $n$ are positive integers

One country consists of islands $A_1,A_2,\cdots,A_N$ ,The ministry of transport decided to build some bridges such that anyone will can travel by car from any of the islands $A_1,A_2,\cdots,A_N$ to any another island by one or more of these bridges. For technical reasons the only bridges that can be built is between $A_i$ and $A_{i+1}$ where $i = 1,2,\cdots,N-1$ , and between $A_i$ and $A_N$ where $i<N$ .

We say that a plan to build some bridges is good if it is satisfies the above conditions , but when we remove any bridge it will not satisfy this conditions. We assume that there is $a_N$ of good plans. Observe that $a_1 = 1$ (The only good plan is to not build any bridge) , and $a_2 = 1$ (We build one bridge).

1-Prove that $a_3 = 3$ 2-Draw at least $5$ different good plans in the case that $N=4$ and the islands are the vertices of a square 3-Compute $a_4$ 4-Compute $a_6$ 5-Prove that there is a positive integer $i$ such that $1438$ divides $a_i$

1 - Prove that $55 < (1+\sqrt{3})^4 < 56$ .

2 - Find the largest power of $2$ that divides $\lceil(1+\sqrt{3})^{2n}\rceil$ for the positive integer $n$