Find all triples $(a, b, p)$ of positive integers, where $p$ is a prime number, such that $a^p - b^p = 2013$.

Let $f:Z \to N -\{0\}$ such that: $f(x + y)f(x-y) = (f(x)f(y))^2$ and $f(1)\ne 1$. Provethat $\log_{f(1)}f(z)$ is a perfect square for every integer $z$.

For a positive integer $a$, define $F_1 ^{(a)}=1$, $F_2 ^{(a)}=a$ and for $n>2$, $F_n ^{(a)}=F_{n-1} ^{(a)}+F_{n-2} ^{(a)}$. A positive integer is fibonatic when it is equal to $F_n ^{(a)}$ for a positive integer $a$ and $n>3$. Prove that there are infintely many not fibonatic integers.

For each positive integer $k$, denote by $\tau(k)$ the number of all positive divisors of $k$, including $1$ and $k$. Let $a$ and $b$ be positive integers such that $\tau(\tau(an)) = \tau(\tau(bn))$ for all positive integers $n$. Prove that $a=b$.

We call a positive integer $n\ge 4$[i] beautiful[/i] if there exists some permutation $$\{x_1,x_2,\dots ,x_{n-1}\}$$ of $\{1,2,\dots ,n-1\}$ such that $\{x^1_1,\ x^2_2,\ \dots,x^{n-1}_{n-1}\}$ gives all the residues $\{1,2,\dots, n-1\}$ modulo $n$. Prove that if $n$ is beautiful then $n=2p,$ for some prime number $p.$

In the country of Postonia, one wants to have only two values of stamps. These values should be integers greater than $1$ with the difference $2$, and should have the property that one can combine the stamps for any postage which is greater than or equal to the sum of these two values. What values can be chosen?

Suppose that $d(n)$ is the number of positive divisors of natural number $n$. Prove that there is a natural number $n$ such that $$ \forall i\in \mathbb{N} , i \le 1402: \frac{d(n)}{d(n \pm i)} >1401 $$ [i]Proposed by Navid Safaei and Mohammadamin Sharifi [/i]

Find all polynomials $f\in Z[X],$ such that for each odd prime $p$ $$f(p)|(p-3)!+\frac{p+1}{2}.$$

A four-digit number has the property that the units digit equals the tens digit increased by $1$, the hundreds digit equals twice the tens digit, and the thousands digit is at least twice the units. Determine this four-digit number, knowing that it is twice a prime number.

A real number $a$ is given. The sequence $n_{1}< n_{2}< n_{3}< ...$ consists of all the positive integral $n$ such that $\{na\}< \frac{1}{10}$. Prove that there are at most three different numbers among the numbers $n_{2}-n_{1}$, $n_{3}-n_{2}$, $n_{4}-n_{3}$, $\ldots$. [i]A corollary of a theorem from ergodic theory[/i]

Let $A$ be a set of positive integers having the following property: for each positive integer $n$ exactly one of the three numbers $n, 2n$ and $3n$ is an element of $A$. Furthermore, it is given that $2 \in A$. Prove that $13824 \notin A$.

The smallest number greater than 2 that leaves a remainder of 2 when divided by 3, 4, 5, or 6 lies between what numbers? $\textbf{(A)}\hspace{.05in}40\text{ and }50 \qquad \textbf{(B)}\hspace{.05in}51\text{ and }55 \qquad \textbf{(C)}\hspace{.05in}56\text{ and }60 \qquad \textbf{(D)}\hspace{.05in} \text{61 and 65}\qquad \textbf{(E)}\hspace{.05in} \text{66 and 99}$

Prove that the two last digits of $9^{9^{9}}$ and $9^{9^{9^{9}}}$ are the same in decimal representation.

p1. Nurbaya's rectangular courtyard will be covered by a number of paving blocks in the form of a regular hexagon or its pieces like the picture below. The length of the side of the hexagon is $ 12$ cm. [img]https://cdn.artofproblemsolving.com/attachments/6/1/281345c8ee5b1e80167cc21ad39b825c1d8f7b.png[/img] Installation of other paving blocks or pieces thereof so that all fully covered page surface. To cover the entire surface The courtyard of the house required $603$ paving blocks. How many paving blocks must be cut into models $A, B, C$, and $D$ for the purposes of closing. If $17$ pieces of model $A$ paving blocks are needed, how many the length and width of Nurbaya's yard? Count how much how many pieces of each model $B, C$, and $D$ paving blocks are used. p2. Given the square $PQRS$. If one side lies on the line $y = 2x - 17$ and its two vertices lie on the parabola $y = x^2$, find the maximum area of possible squares $PQRS$ . p3. In the triangular pyramid $T.ABC$, the points $E, F, G$, and $H$ lie at , respectively $AB$, $AC$, $TC$, and $TB$ so that $EA : EB = FA : FC = HB : HT = GC : GT = 2:1$. Determine the ratio of the volumes of the two halves of the divided triangular pyramid by the plane $EFGH$. p4. We know that $x$ is a non-negative integer and $y$ is an integer. Define all pair $(x, y)$ that satisfy $1 + 2^x + 2^{2x + 1} = y^2$. p5. The coach of the Indonesian basketball national team will select the players for become a member of the core team. The coach will judge five players $A, B, C, D$ and $E$ in one simulation (or trial) match with total time $80$ minute match. At any time there is only one in five players that is playing. There is no limit to the number of substitutions during the match. Total playing time for each player $A, B$, and $C$ are multiples of $5$ minutes, while the total playing time of each players $D$ and $E$ are multiples of $7$ minutes. How many ways each player on the field based on total playing time?

Let $a,b$ be two positive integers and $b^2+a-1|a^2+b-1$. Prove that $b^2+a-1$ has at least two prime divisors.

Define [i]glueing[/i] of positive integers as writing their base ten representations one after another and interpreting the result as the base ten representation of a single positive integer. Find all positive integers $k$ for which there exists an integer $N_k$ with the following property: for all $n \ge N_k$, we can glue the numbers $1,2,\dots,n$ in some order so that the result is a number divisible by $k$. [i]Remark[/i]. The base ten representation of a positive integer never starts with zero. [i]Example[/i]. Glueing $15, 14, 7$ in this order makes $15147$.

Find all polynomials $f\in \mathbb{Z}[X]$ such that if $p$ is prime then $f(p)$ is also prime.

For each positive integer $n$, define $f(n)$ to be the least positive integer for which the following holds: For any partition of $\{1,2,\dots, n\}$ into $k>1$ disjoint subsets $A_1, \dots, A_k$, [u]all of the same size[/u], let $P_i(x)=\prod_{a\in A_i}(x-a)$. Then there exist $i\neq j$ for which \[\deg(P_i(x)-P_j(x))\geq \frac{n}{k}-f(n)\] a) Prove that there is a constant $c$ so that $f(n)\le c\cdot \sqrt{n}$ for all $n$. b) Prove that for infinitely many $n$, one has $f(n)\ge \ln(n)$.

Find whether the number of powers of 2, which have a digit sum smaller than $2019^{2019}$, is finite or infinite.

[b]a) [/b]Is the number $ 1111\cdots11$ (with $ 2010$ ones) a prime number? [b]b)[/b] Prove that every prime factor of $ 1111\cdots11$ (with $ 2011$ ones) is of the form $ 4022j\plus{}1$ where $ j$ is a natural number.

Prove tha number $19 \cdot 8^n +17$ is composite for every positive integer $n$

Determine all pairs $(a,b)$ of positive integers for which there exist positive integers $g$ and $N$ such that $$\gcd (a^n+b,b^n+a)=g$$ holds for all integers $n\geqslant N.$ (Note that $\gcd(x, y)$ denotes the greatest common divisor of integers $x$ and $y.$) [i]Proposed by Valentio Iverson, Indonesia[/i]

Find the number of rational solutions of the following equations (i.e., rational $x$ and $y$ satisfy the equations) $$x^2+y^2=2$$$$x^2+y^2=3$$$\textbf{(A)}~2\text{ and }2$ $\textbf{(B)}~2\text{ and }0$ $\textbf{(C)}~2\text{ and infinitely many}$ $\textbf{(D)}~\text{Infinitely many and }0$

Let $p{}$ be a prime number. There are $p{}$ integers $a_0,\ldots,a_{p-1}$ around a circle. In one move, it is allowed to select some integer $k{}$ and replace the existing numbers via the operation $a_i\mapsto a_i-a_{i+k}$ where indices are taken modulo $p{}.$ Find all pairs of natural numbers $(m, n)$ with $n>1$ such that for any initial set of $p{}$ numbers, after performing any $m{}$ moves, the resulting $p{}$ numbers will all be divisible by $n{}.$ [i]Proposed by P. Kozhevnikov[/i]

Find all integers $n$ for which $n^7-41$ is the square of an integer