Found problems: 6
2017 Gulf Math Olympiad, 2
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 $a_i$
2016 Gulf Math Olympiad, 2
Let $x$ be a real number that satisfies $x^1 + x^{-1} = 3$
Prove that $x^n + x^{-n}$ is an positive integer , then prove that the positive integer $x^{3^{1437}}+x^{3^{-1437}}$ is divisible by at least $1439 \times 2^{1437}$ positive integers
2014 Contests, 1
A sequence $a_0,a_1,a_2,\cdots$ satisfies the conditions $a_0 = 0$ , $a_{n-1}^2 - a_{n-1} = a_n^2 + a_n$
1) determine the two possible values of $a_1$ . then determine all possible values of $a_2$ .
2)for each $n$, prove that $a_{n+1}=a_n+1$ or $a_{n+1} = -a_n$
3)Describe the possible values of $a_{1435}$
4)Prove that the values that you got in (3) are correct
2017 Gulf Math Olympiad, 1
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
2014 Gulf Math Olympiad, 1
A sequence $a_0,a_1,a_2,\cdots$ satisfies the conditions $a_0 = 0$ , $a_{n-1}^2 - a_{n-1} = a_n^2 + a_n$
1) determine the two possible values of $a_1$ . then determine all possible values of $a_2$ .
2)for each $n$, prove that $a_{n+1}=a_n+1$ or $a_{n+1} = -a_n$
3)Describe the possible values of $a_{1435}$
4)Prove that the values that you got in (3) are correct
2016 Gulf Math Olympiad, 1
Consider sequences $a_0$,$a_1$,$a_2$,$\cdots$ of non-negative integers defined by selecting any $a_0$,$a_1$,$a_2$ (not all 0) and for each $n$ $\geq$ 3 letting
$a_n$ = |$a_n-1$ - $a_n-3$|
1-In the particular case that $a_0$ = 1,$a_1$ = 3 and $a_2$ = 2, calculate the beginning of the sequence, listing
$a_0$,$a_1$,$\cdots$,$a_{19}$,$a_{20}$.
2-Prove that for each sequence, there is a constant $c$ such that $a_i$ $\leq$ $c$ for all $i$ $\geq$ 0. Note that the constant $c$ my depend on the numbers $a_0$,$a_1$ and $a_2$
3-Prove that, for each choice of $a_0$,$a_1$ and $a_2$, the resulting sequence is eventually periodic.
4-Prove that, the minimum length p of the period described in (3) is the same for all permitted starting values
$a_0$,$a_1$,$a_2$ of the sequence