Let $ n$ be a natural number. A cube of edge $ n$ may be divided in 1996 cubes whose edges length are also natural numbers. Find the minimum possible value for $ n$.

For each positive integer $n$, denote by $\omega(n)$ the number of distinct prime divisors of $n$ (for example, $\omega(1)=0$ and $\omega(12)=2$). Find all polynomials $P(x)$ with integer coefficients, such that whenever $n$ is a positive integer satisfying $\omega(n)>2023^{2023}$, then $P(n)$ is also a positive integer with \[\omega(n)\ge\omega(P(n)).\] Greece (Minos Margaritis - Iasonas Prodromidis)

For integers $n \ge k \ge 0$ we define the [i]bibinomial coefficient[/i] $\left( \binom{n}{k} \right)$ by \[ \left( \binom{n}{k} \right) = \frac{n!!}{k!!(n-k)!!} .\] Determine all pairs $(n,k)$ of integers with $n \ge k \ge 0$ such that the corresponding bibinomial coefficient is an integer. [i]Remark: The double factorial $n!!$ is defined to be the product of all even positive integers up to $n$ if $n$ is even and the product of all odd positive integers up to $n$ if $n$ is odd. So e.g. $0!! = 1$, $4!! = 2 \cdot 4 = 8$, and $7!! = 1 \cdot 3 \cdot 5 \cdot 7 = 105$.[/i]

Every nonnegative periodic decimal fraction represents a rational number, also in the form $\frac{p}{q}$ can be represented ($p$ and $q$ are natural numbers and coprime, $p\ge 0$, $q > 0)$. Now let $a_1$, $a_2$, $a_3$ and $a_4$ be digits to represent numbers in the decadic system. Let $a_1 \ne a_3$ or $a_2 \ne a_4$.Prove that it for the numbers: $z_1 = 0, \overline{a_1a_2a_3a_4} = 0,a_1a_2a_3a_4a_1a_2a_3a_4...$ $z_2 = 0, \overline{a_4a_1a_2a_3}$ $z_3 = 0, \overline{a_3a_4a_1a_2}$ $z_4 = 0, \overline{a_2a_3a_4a_1}$ In the above representation $p/q$ always have the same denominator. [hide=original wording]Jeder nichtnegative periodische Dezimalbruch repr¨asentiert eine rationale Zahl, die auch in der Form p/q dargestellt werden kann (p und q nat¨urliche Zahlen und teilerfremd, p >= 0, q > 0). Nun seien a1, a2, a3 und a4 Ziffern zur Darstellung von Zahlen im dekadischen System. Dabei sei a1 $\ne$ a3 oder a2 $\ne$ a4. Beweisen Sie! Die Zahlen z1 = 0, a1a2a3a4 = 0,a1a2a3a4a1a2a3a4... z2 = 0, a4a1a2a3 z3 = 0, a3a4a1a2 z4 = 0, a2a3a4a1 haben in der obigen Darstellung p/q stets gleiche Nenner.[/hide]

Find all triples $(x,y,z)$ of positive integers such that $(x+1)^{y+1}+1=(x+2)^{z+1}$.

Let $c \ge 2$ be a fixed integer. Let $a_1 = c$ and for all $n \ge 2$ let $a_n = c \cdot \phi (a_{n-1})$. What are the numbers $c$ for which sequence $(a_n)$ will be bounded? $\phi$ denotes Euler’s Phi Function, meaning that $\phi (n)$ gives the number of integers within the set $\{1, 2, . . . , n\}$ that are relative primes to $n$. We call a sequence $(x_n)$ bounded if there exist a constant $D$ such that $|x_n| < D$ for all positive integers $n$.

Determine the number of pairs of positive integers \( (p, k) \) such that \( p \) is a prime number and \( p^2 + 2^k \) is a perfect square less than 2023. A number is called a perfect square if it is the square of an integer.

We call a positive integer $N$ [i]contagious[/i] if there are $1000$ consecutive non-negative integers such that the sum of all their digits is $N$. Find all contagious positive integers.

Let $p$ and $c$ be an prime and a composite, respectively. Prove that there exist two integers $m,n,$ such that $$0<m-n<\frac{\textup{lcm}(n+1,n+2,\cdots,m)}{\textup{lcm}(n,n+1,\cdots,m-1)}=p^c.$$

The number $1995$ is divisible by both $19$ and $95$. How many four-digit numbers are there that are divisible by two-digit numbers formed by both its first two digits and its last two digits?

Circles of radii 5, 5, 8, and $m/n$ are mutually externally tangent, where $m$ and $n$ are relatively prime positive integers. Find $m+n.$

Find the number of positive integer solutions to the equation $(x_1+x_2+x_3)^2(y_1+y_2) = 2548$.

(a) Positive integers $p,q,r,a$ satisfy $pq=ra^2$, where $r$ is prime and $p,q$ are relatively prime. Prove that one of the numbers $p,q$ is a perfect square. (b) Examine if there exists a prime $p$ such that $p(2^{p+1}-1)$ is a perfect square.

For an $n$-tuple of integers, define a transformation to be: $$(a_1,a_2,\cdots,a_{n-1},a_n)\rightarrow (a_1+a_2, a_2+a_3, \cdots, a_{n-1}+a_n, a_n+a_1)$$ Find all ordered pairs of integers $(n,k)$ with $n,k\geq 2$, such that for any $n$-tuple of integers $(a_1,a_2,\cdots,a_{n-1},a_n)$, after a finite number of transformations, every element in the of the $n$-tuple is a multiple of $k$.

Given a positive integer $k$ show that there exists a prime $p$ such that one can choose distinct integers $a_1,a_2\cdots, a_{k+3} \in \{1, 2, \cdots ,p-1\}$ such that p divides $a_ia_{i+1}a_{i+2}a_{i+3}-i$ for all $i= 1, 2, \cdots, k$. [i]South Africa [/i]

Consider a sequence denoted by $F_n$ of non-square numbers . $F_1=2$,$F_2=3$,$F_3=5$ and so on . Now , if $m^2\leq F_n<(m+1)^2$ . Then prove that $m$ is the integer closest to $\sqrt{n}$.

Let $a_1,a_2,\dots, a_n$ be positive integers with product $P,$ where $n$ is an odd positive integer. Prove that $$\gcd(a_1^n+P,a_2^n+P,\dots, a_n^n+P)\le 2\gcd(a_1,\dots, a_n)^n.$$ [i]Proposed by Daniel Liu[/i]

All members of geometrical progression $(b_n)_{n\geq1}$ are members of some arithmetical progression. It is known that $b_1$ is an integer. Prove that all members of this geometrical progression are integers. (progression is infinite)

Find the largest prime factor of $45^5-1.$

Let $a_d$ be the number of non-negative integer solutions $(a, b)$ to $a + b = d$ where $a \equiv b$ (mod $n$) for a fixed $n \in Z^+$. Consider the generating function $M(t) = a_0 + a_1t + a_2t^2 + ...$ Consider $$P(n) = \lim_{t\to 1} \left( nM(t) - \frac{1}{(1 - t)^2} \right).$$ Then $P(n)$, $n \in Z^+$ is a polynomial in $n$, so we can extend its domain to include all real numbers while having it remain a polynomial. Find $P(0)$.

For which numbers $ n$ is there a positive integer $ k$ with the following property: The sum of digits for $ k$ is $ n$ and the number $ k^2$ has sum of digits $ n^2.$

Let $f:\mathbb N\to\mathbb R$ be given by $$f(n)=n^{\frac12\tau(n)}$$for $n\in\mathbb N=\{1,2,\ldots\}$ where $\tau(n)$ is the number of divisors of $n$. Show that $f$ is an injection.

The sequence (a_n) is defined as follows: $ a_0\equal{}1, a_1\equal{}3$ For $ n\ge 2$, $ a_{n\plus{}2}\equal{}a_{n\plus{}1}\plus{}9a_n$ if n is even, $ a_{n\plus{}2}\equal{}9a_{n\plus{}1}\plus{}5a_n$ if n is odd. Prove that 1) $ (a_{1995})^2\plus{}(a_{1996})^2\plus{}...\plus{}(a_{2000})^2$ is divisible by 20 2) $ a_{2n\plus{}1}$ is not a perfect square for every natural numbers $ n$.

An infinite arithmetic progression whose terms are positive integers contains the square of an integer and the cube of an integer. Show that it contains the sixth power of an integer.

Prove that a prime $p$ is expressible in the form $x^2+3y^2;x,y\in Z$ if and only if it is expressible in the form $ m^2+mn+n^2;m,n \in Z$.Can $p$ be replaced by a natural number $n$?