let p be a prime and a and n be natural numbers such that (p^a -1 )/ (p-1) = 2 ^n find the number of natural divisors of na. :)

(a) Prove that there exist infinitely many natural numbers $n$ such that the sum of the digits of $11n$ is twice the sum of the digits of $n$. (b) Prove that there exist infinitely many natural numbers $n$ such that the sum of the digits of $4n + 3$ is equal to the sum of the digits of $n$. (c) Prove that for any natural number $n$, it is possible to find $n$ consecutive numbers such that none of them is prime.

Let $ p$ be a prime number. Solve in $ \mathbb{N}_0\times\mathbb{N}_0$ the equation $ x^3\plus{}y^3\minus{}3xy\equal{}p\minus{}1$.

Let $A$ be a $2\,x\,2$ matrix with real numbers. Prove that if $A^3=\mathbb{O}$ then $A^2=\mathbb{O}$.

Prove that $[\sqrt{n}+\sqrt{n+1}]=[\sqrt{4n+1}]$ for all $n \in N$.

On a $(4n + 2)\times (4n + 2)$ square grid, a turtle can move between squares sharing a side.The turtle begins in a corner square of the grid and enters each square exactly once, ending in the square where she started. In terms of $n$, what is the largest positive integer $k$ such that there must be a row or column that the turtle has entered at least $k$ distinct times?

Let $n$ be a positive integer and $M = \{1, 2, 3, 4, 5, 6\}$. Two persons $A$ and $B$ play in the following Way: $A$ writes down a digit from $M$, $B$ appends a digit from $M$, and so it becomes alternately one digit from $M$ is appended until the $2n$-digit decimal representation of a number has been created. If this number is divisible by $9$, $B$ wins, otherwise $A$ wins. For which $n$ can $A$ and for which $n$ can $B$ force the win?

Let $ABC$ be a triangle and let $\Omega$ denote the circumcircle of $ABC$. The foot of altitude from $A$ to $BC$ is $D$. The foot of altitudes from $D$ to $AB$ and $AC$ are $K;L$ , respectively. Let $KL$ intersect $\Omega$ at $X;Y$, and let $AD$ intersect $\Omega$ at $Z$. Prove that $D$ is the incenter of triangle $XYZ$

An equilateral triangle $ABC$ is in between two parallel lines $x, y$ that pass through points $A$ and $B$ respectively. Given that $C$ is twice as far from $y$ as $x$, the acute angle that $CA$ makes with $x$ is $\theta$. Then $(\tan \theta)^2$ is of the form $\frac{p}{q}$ where $p, q$ are relatively prime positive integers. Find $p + q$.

In a plane three points $P,Q,R,$ not on a line, are given. Let $k, l, m$ be positive numbers. Construct a triangle $ABC$ whose sides pass through $P, Q,$ and $R$ such that $P$ divides the segment $AB$ in the ratio $1 : k$, $Q$ divides the segment $BC$ in the ratio $1 : l$, and $R$ divides the segment $CA$ in the ratio $1 : m.$

Of the integers $a$, $b$, and $c$ that satisfy $0 < c < b < a$ and $$a^3 - b^3 - c^3 - abc + 1 = 2022,$$ let $c'$ be the least value of $ c$ appearing in any solution, let $a'$ be the least value of $a$ appearing in any solution with $c = c'$, and let $b'$ be the value of $b$ in the solution where $c = c'$ and $a = a'$. Find $a' + b' + c'$.

Let $P = (-4, 0)$ and $Q = (4, 0)$ be two points on the $x$-axis of the Cartesian coordinate plane, and let $X$ and $Y$ be points on the $x$-axis and $y$-axis, respectively, such that over all $Z$ on line $\overleftrightarrow{XY}$, the perimeter of $\triangle ZPQ$ has a minimum value of $25$. What is the smallest possible value of $XY^2$?

Let $a, b$ and $c$ be natural numbers. Determine the smallest value that the following expression can take: $$\frac{a}{gcd\,\,(a + b, a - c)} + \frac{b}{gcd\,\,(b + c, b - a)} + \frac{c}{gcd\,\,(c + a, c - b)}.$$ . Remark: $gcd \,\, (6, 0) = 6$ and $gcd\,\,(3, -6) = 3$.

Let $ r$ and $ s$ two orthogonal lines that does not lay on the same plane. Let $ AB$ be their common perpendicular, where $ A\in{}r$ and $ B\in{}s$(*).Consider the sphere of diameter $ AB$. The points $ M\in{r}$ and $ N\in{s}$ varies with the condition that $ MN$ is tangent to the sphere on the point $ T$. Find the locus of $ T$. Note: The plane that contains $ B$ and $ r$ is perpendicular to $ s$.

Let $m,n,k$ be positive integers and $1+m+n \sqrt 3=(2+ \sqrt 3)^{2k+1}$. Prove that $m$ is a perfect square.

A list of numbers $a_1,a_2,\ldots,a_m$ contains an arithmetic trio $a_i, a_j, a_k$ if $i < j < k$ and $2a_j = a_i + a_k$. Let $n$ be a positive integer. Show that the numbers $1, 2, 3, \ldots, n$ can be reordered in a list that does not contain arithmetic trios.

a)Prove that $\frac{1}{2}+\frac{1}{3}+...+\frac{1}{{{2}^{m}}}<m$, for any $m\in {{\mathbb{N}}^{*}}$. b)Let ${{p}_{1}},{{p}_{2}},...,{{p}_{n}}$ be the prime numbers less than ${{2}^{100}}$. Prove that $\frac{1}{{{p}_{1}}}+\frac{1}{{{p}_{2}}}+...+\frac{1}{{{p}_{n}}}<10$

Let $ A$ represent the set of all functions $ f : \mathbb{N} \rightarrow \mathbb{N}$ such that for all $ k \in \overline{1, 2007}$, $ f^{[k]} \neq \mathrm{Id}_{\mathbb{N}}$ and $ f^{[2008]} \equiv \mathrm{Id}_{\mathbb{N}}$. a) Prove that $ A$ is non-empty. b) Find, with proof, whether $ A$ is infinite. c) Prove that all the elements of $ A$ are bijective functions. (Denote by $ \mathbb{N}$ the set of the nonnegative integers, and by $ f^{[k]}$, the composition of $ f$ with itself $ k$ times.)

A positive integer $a$ is called a [i]double number[/i] if it has an even number of digits (in base 10) and its base 10 representation has the form $a = a_1a_2 \cdots a_k a_1 a_2 \cdots a_k$ with $0 \le a_i \le 9$ for $1 \le i \le k$, and $a_1 \ne 0$. For example, $283283$ is a double number. Determine whether or not there are infinitely many double numbers $a$ such that $a + 1$ is a square and $a + 1$ is not a power of $10$.

On the same side of a straight line three circles are drawn as follows: a circle with a radius of $4$ inches is tangent to the line, the other two circles are equal, and each is tangent to the line and to the other two circles. The radius of the equal circles is: $ \textbf{(A)}\ 24 \qquad\textbf{(B)}\ 20\qquad\textbf{(C)}\ 18\qquad\textbf{(D)}\ 16\qquad\textbf{(E)}\ 12 $

Points $P$ and $Q$ on sides $AB$ and $AC$ of triangle $ABC$ are such that $PB = QC$. Prove that $PQ < BC$.

Let $ABCDE$ be a convex pentagon with $AB = BC =CA$ and $CD = DE = EC$. Let $T$ be the centroid of $\vartriangle ABC$, and $N$ be the midpoint of $AE$. Compute $\angle NT D$

Let $K$, $L$, $M$, $N$ be the midpoints of sides $BC$, $CD$, $DA$, $AB$ respectively of a convex quadrilateral $ABCD$. The common points of segments $AK$, $BL$, $CM$, $DN$ divide each of them into three parts. It is known that the ratio of the length of the medial part to the length of the whole segment is the same for all segments. Does this yield that $ABCD$ is a parallelogram?

Let $d(n)$ be the number of positive divisors of a positive integer $n$ (including $1$ and $n$). Find all values of $n$ such that $n + d(n) = d(n)^2$.

Let $n$ be a given positive integer. Sisyphus performs a sequence of turns on a board consisting of $n + 1$ squares in a row, numbered $0$ to $n$ from left to right. Initially, $n$ stones are put into square $0$, and the other squares are empty. At every turn, Sisyphus chooses any nonempty square, say with $k$ stones, takes one of these stones and moves it to the right by at most $k$ squares (the stone should say within the board). Sisyphus' aim is to move all $n$ stones to square $n$. Prove that Sisyphus cannot reach the aim in less than \[ \left \lceil \frac{n}{1} \right \rceil + \left \lceil \frac{n}{2} \right \rceil + \left \lceil \frac{n}{3} \right \rceil + \dots + \left \lceil \frac{n}{n} \right \rceil \] turns. (As usual, $\lceil x \rceil$ stands for the least integer not smaller than $x$. )