Find all the pairs of positive integers $ (a,b)$ such that $ a^2 \plus{} b \minus{} 1$ is a power of prime number $ ; a^2 \plus{} b \plus{} 1$ can divide $ b^2 \minus{} a^3 \minus{} 1,$ but it can't divide $ (a \plus{} b \minus{} 1)^2.$

For any positive integer $b\ge2$, we write the base-$b$ numbers as follows: \[(d_kd_{k-1}\dots d_0)_b=d_kb^k+d_{k-1}b^{k-1}+\dots+d_1b^1+d_0b^0,\]where each digit $d_i$ is a member of the set $S=\{0,1,2,\dots,b-1\}$ and either $d_k\not=0$ or $k=0$. There is a unique way to write any nonnegative integer in the above form. If we select the digits from a different set $S$ instead, we may obtain new representations of all positive integers or, in some cases, all integers. For example, if $b=3$ and the digits are selected from $S=\{-1,0,1\}$, we obtain a way to uniquely represent all integers, known as a $\emph{balanced ternary}$ representation. As further examples, the balanced ternary representation of numbers $5$, $-3$, and $25$ are: \[5=(1\ {-1}\ {-1})_3,\qquad{-3}=({-1}\ 0)_3,\qquad25=(1\ 0\ {-1}\ 1)_3.\]However, not all digit sets can represent all integers. If $b=3$ and $S=\{-2,0,2\}$, then no odd number can be represented. Also, if $b=3$ and $S=\{0,1,2\}$ as in the usual base-$3$ representation, then no negative number can be represented. Given a set $S$ of four integers, one of which is $0$, call $S$ a $\emph{4-basis}$ if every integer $n$ has at least one representation in the form \[n=(d_kd_{k-1}\dots d_0)_4=d_k4^k+d_{k-1}4^{k-1}+\dots+d_14^1+d_04^0,\]where $d_k,d_{k-1},\dots,d_0$ are all elements of $S$ and either $d_k\not=0$ or $k=0$. [list=a] [*]Show that there are infinitely many integers $a$ such that $\{-1,0,1,4a+2\}$ is not a $4$-basis. [*]Show that there are infinitely many integers $a$ such that $\{-1,0,1,4a+2\}$ is a $4$-basis.[/list]

1. Around a circumference are written $2012$ number, each of with is equal to $1$ or $-1$. If there are not $10$ consecutive numbers that sum $0$, find all possible values of the sum of the $2012$ numbers.

Find all triples $(x, y, z)$ such that $x, y, z, x - y, y - z, x - z$ are all prime positive integers.

Prove that for each positive integer $ n$ there exist $ n$ consecutive positive integers none of which is an integral power of a prime number.

Let $p$ be a prime number. Prove that there is no number divisible by $p$ in the $n-th$ row of Pascal's triangle if and only if $n$ can be represented in the form $n = p^sq - 1$, where $s$ and $q$ are integers with $s \geq 0, 0 < q < p$.

A function $f : \mathbb{Z}_+ \to \mathbb{Z}_+$, where $\mathbb{Z}_+$ is the set of positive integers, is non-decreasing and satisfies $f(mn) = f(m)f(n)$ for all relatively prime positive integers $m$ and $n$. Prove that $f(8)f(13) \ge (f(10))^2$.

In a competition, there were $2n+1$ teams. Every team plays exatly once against every other team. Every match finishes with the victory of one of the teams. We call cyclical a 3-subset of team ${ A,B,C }$ if $A$ won against $B$, $B$ won against $C$ , $C$ won against $A$. (a) Find the minimum of cyclical 3-subset (depending on $n$); (b) Find the maximum of cyclical 3-subset (depending on $n$).

Prove that there exist infinitely many positive integers $n$ such that $3^n+2$ and $5^n+2$ are all composite numbers.

$(CZS 2)$ Let $p$ be a prime odd number. Is it possible to find $p-1$ natural numbers $n + 1, n + 2, . . . , n + p -1$ such that the sum of the squares of these numbers is divisible by the sum of these numbers?

In 1772 Euler discovered the curious fact that $n^2 +n+41$ is prime when $n$ is any of $0,1,2, \cdots, 39$. Show that there exist $40$ consecutive integer values of $n$ for which this polynomial is not prime.

Suppose that for some $m,n\in\mathbb{N}$ we have $\varphi (5^m-1)=5^n-1$, where $\varphi$ denotes the Euler function. Show that $(m,n)>1$.

Suppose $\, q_{0}, \, q_{1}, \, q_{2}, \ldots \; \,$ is an infinite sequence of integers satisfying the following two conditions: (i) $\, m-n \,$ divides $\, q_{m}-q_{n}\,$ for $\, m > n \geq 0,$ (ii) there is a polynomial $\, P \,$ such that $\, |q_{n}| < P(n) \,$ for all $\, n$ Prove that there is a polynomial $\, Q \,$ such that $\, q_{n}= Q(n) \,$ for all $\, n$.

If $ n$ is any whole number, $ n^2(n^2 \minus{} 1)$ is always divisible by $ \textbf{(A)}\ 12 \qquad\textbf{(B)}\ 24 \qquad\textbf{(C)}\ \text{any multiple of }12 \qquad\textbf{(D)}\ 12 \minus{} n \qquad\textbf{(E)}\ 12\text{ and }24$

Prove that for every prime number $p$, there are infinitely many positive integers $n$ such that $p$ divides $2^n - n$.

Let $P$ be a polynomial with integer coefficients such that $P(0)=0$ and \[\gcd(P(0), P(1), P(2), \ldots ) = 1.\] Show there are infinitely many $n$ such that \[\gcd(P(n)- P(0), P(n+1)-P(1), P(n+2)-P(2), \ldots) = n.\]

Already the complete problem: Determine all pairs $(a,b)$ of integers different from $0$ for which it is possible to find a positive integer $x$ and an integer $y$ such that $x$ is relatively prime to $b$ and in the following list there is an infinity of integers: $\rightarrow\qquad\frac{a + xy}{b}$, $\frac{a + xy^2}{b^2}$, $\frac{a + xy^3}{b^3}$, $\ldots$, $\frac{a + xy^n}{b^n}$, $\ldots$ One idea? :arrow: [b][url=http://www.mathlinks.ro/Forum/viewtopic.php?t=61319]View all the problems from XIX Mexican Mathematical Olympiad[/url][/b]

Each term of a sequence of natural numbers is obtained from the previous term by adding to it its largest digit. What is the maximal number of successive odd terms in such a sequence?

Let $n>3$ be an integer. Integers $a_1, \dots, a_n$ are given so that $a_k\in \{k, -k\}$ for all $1\leq k\leq n$. Prove that there is a sequence of indices $1\leq k_1, k_2, \dots, k_n\leq n$, not necessarily distinct, for which the sums \[a_{k_1}\] \[a_{k_1}+a_{k_2}\] \[a_{k_1}+a_{k_2}+a_{k_3}\] \[\vdots\] \[a_{k_1}+a_{k_2}+\cdots+a_{k_n}\] have distinct residues modulo $2n+1$, and so that the last one is divisible by $2n+1$.

Determine all integers $a$ such that $a^k + 1$ is divisible by $12321$ for some $k$

Let $ n, k$ be positive integers and suppose that the polynomial $ x^{2k}\minus{}x^k\plus{}1$ divides $ x^{2n}\plus{}x^n\plus{}1$. Prove that $ x^{2k}\plus{}x^k\plus{}1$ divides $ x^{2n}\plus{}x^n\plus{}1$.

The cells of a square $2011 \times 2011$ array are labelled with the integers $1,2,\ldots, 2011^2$, in such a way that every label is used exactly once. We then identify the left-hand and right-hand edges, and then the top and bottom, in the normal way to form a torus (the surface of a doughnut). Determine the largest positive integer $M$ such that, no matter which labelling we choose, there exist two neighbouring cells with the difference of their labels at least $M$. (Cells with coordinates $(x,y)$ and $(x',y')$ are considered to be neighbours if $x=x'$ and $y-y'\equiv\pm1\pmod{2011}$, or if $y=y'$ and $x-x'\equiv\pm1\pmod{2011}$.) [i](Romania) Dan Schwarz[/i]

Let $a_{1}, \ldots, a_{n}$ be an infinite sequence of strictly positive integers, so that $a_{k} < a_{k+1}$ for any $k.$ Prove that there exists an infinity of terms $ a_{m},$ which can be written like $a_m = x \cdot a_p + y \cdot a_q$ with $x,y$ strictly positive integers and $p \neq q.$

Let $S$ be a subset of $\{1,2,3,\dots,30\}$ with the property that no pair of distinct elements in $S$ has a sum divisible by $5$. What is the largest possible size of $S$? $ \textbf{(A)}\ 10\qquad\textbf{(B)}\ 13\qquad\textbf{(C)}\ 15\qquad\textbf{(D)}\ 16\qquad\textbf{(E)}\ 18 $

Find the number of integers $n$ that satisfy $\textit{both}$ of the following conditions: [list=i] [*]$208<n<2008$, [*]$n$ has the same remainder when divided by $24$ or by $30$.[/list]