For a positive integer $N>1$ with unique factorization $N=p_1^{\alpha_1}p_2^{\alpha_2}\dotsb p_k^{\alpha_k}$, we define \[\Omega(N)=\alpha_1+\alpha_2+\dotsb+\alpha_k.\] Let $a_1,a_2,\dotsc, a_n$ be positive integers and $p(x)=(x+a_1)(x+a_2)\dotsb (x+a_n)$ such that for all positive integers $k$, $\Omega(P(k))$ is even. Show that $n$ is an even number.

The year $1978$ was “peculiar” in that the sum of the numbers formed with the first two digits and the last two digits is equal to the number formed with the middle two digits, i.e., $19+78=97$. What was the last previous peculiar year, and when will the next one occur?

Do there exist infinitely many triplets of positive integers $(a, b, c)$ such that the following two conditions hold: 1. $\gcd(a, b, c) = 1$; 2. $a+b+c, a^2+b^2+c^2$ and $abc$ are all perfect squares? [i](Proposed by Ivan Chan Guan Yu)[/i]

Let $d(k)$ be the number of positive divisors of integer $k$. A number $n$ is called [i]balanced[/i] if $d(n-1) \leq d(n) \leq d(n+1)$ or $d(n-1) \geq d(n) \geq d(n+1)$. Show that there are infinitely many balanced numbers.

For positive real numbers $x$ and $y$, define their special mean to be average of their arithmetic and geometric means. Find the total number of pairs of integers $(x, y)$, with $x \le y$, from the set of numbers $\{1,2,...,2016\}$, such that the special mean of $x$ and $y$ is a perfect square.

Let $n$ and $k$ be natural numbers and $p$ a prime number. Prove that if $k$ is the exact exponent of $p$ in $2^{2^n}+1$ (i.e. $p^k$ divides $2^{2^n}+1$, but $p^{k+1}$ does not), then $k$ is also the exact exponent of $p$ in $2^{p-1}-1$.

Find the greatest positive integer $m$ with the following property: For every permutation $a_1, a_2, \cdots, a_n,\cdots$ of the set of positive integers, there exists positive integers $i_1<i_2<\cdots <i_m$ such that $a_{i_1}, a_{i_2}, \cdots, a_{i_m}$ is an arithmetic progression with an odd common difference.

Given that $p$ and $p^4 + 34$ are both prime numbers, compute $p$.

Prove that if the number $A = 111 \cdots 1$ ($n$ digits) is prime, then $n$ is prime. Is the converse true?

For each $m\in \mathbb N$ we define $rad\ (m)=\prod p_i$, where $m=\prod p_i^{\alpha_i}$. [b]abc Conjecture[/b] Suppose $\epsilon >0$ is an arbitrary number, then there exist $K$ depinding on $\epsilon$ that for each 3 numbers $a,b,c\in\mathbb Z$ that $gcd (a,b)=1$ and $a+b=c$ then: \[ max\{|a|,|b|,|c|\}\leq K(rad\ (abc))^{1+\epsilon} \] Now prove each of the following statements by using the $abc$ conjecture : a) Fermat's last theorem for $n>N$ where $N$ is some natural number. b) We call $n=\prod p_i^{\alpha_i}$ strong if and only $\alpha_i\geq 2$. c) Prove that there are finitely many $n$ such that $n,\ n+1,\ n+2$ are strong. d) Prove that there are finitely many rational numbers $\frac pq$ such that: \[ \Big| \sqrt[3]{2}-\frac pq \Big|<\frac{2^ {1384}}{q^3} \]

The non-negative integers $x,y$ satisfy $\sqrt{x}+\sqrt{x+60}=\sqrt{y}$. Find the largest possible value for $x$.

Prove that there are no $2003$ odd positive integers whose product equals their sum. Is the previous proposition true for $2001$ odd positive integers?

For each $ k\in\mathbb{N} ,k\le 2000, $ Let $ r_k $ be the remainder of the division of $ k $ by $ 4, $ and $ r'_k $ be the remainder of the division of $ k $ by $ 3. $ Prove that there is an unique $ m\in\mathbb{N} ,m\le 1999 $ such that $$ r_1+r_2+\cdots +r_m=r'_{m+1} +r'_{m+2} +\cdots r'_{2000} . $$ [i]Mircea Fianu[/i]

Find the positive integers $n$ that are not divisible by $3$ if the number $2^{n^2-10}+2133$ is a perfect cube. [hide="Note"] [color=#BF0000]The wording of this problem is perhaps not the best English. As far as I am aware, just solve the diophantine equation $x^3=2^{n^2-10}+2133$ where $x,n \in \mathbb{N}$ and $3\nmid n$.[/color][/hide]

For each positive integer $n$, determine the digits of units and hundreds of the decimal representation of the number $$\frac{1 + 5^{2n+1}}{6}$$

Let $\{b_n\}_{n\geq 1}^{\infty}$ be a sequence of positive integers. The sequence $\{a_n\}_{n\geq 1}^{\infty}$ is defined as follows: $a_1$ is a fixed positive integer and \[a_{n+1}=a_n^{b_n}+1 ,\qquad \forall n\geq 1.\] Find all positive integers $m\geq 3$ with the following property: If the sequence $\{a_n\mod m\}_{n\geq 1 }^{\infty}$ is eventually periodic, then there exist positive integers $q,u,v$ with $2\leq q\leq m-1$, such that the sequence $\{b_{v+ut}\mod q\}_{t\geq 1}^{\infty}$ is purely periodic.

Natural $a,b,c$ are pairwise prime. There is infinite table with one integer number in every cell. Sum of numbers in every $a \times a$, every $b \times b$, every $c \times c$ squares is even. Is it true, that every number in table must be even?

Three natural numbers greater than or equal to $2$ are written, not necessarily different, and from them a sequence is constructed using the following procedure: in each step, if the penultimate number written is $a$, the penultimate one is $b$ and the last one is $c$, it is written $x$ such that $$x\cdot c=a+b+186.$$Determine all the possible values of the three numbers initially written so that when the process continues indefinitely all the written numbers are natural numbers greater than or equal to $2$.

Find the smallest nomial of this sequence that $a_1=1993^{1994^{1995}}$ and \[ a_{n+1}=\begin{cases}\frac{a_n}{2}&\text{if $n$ is even}\\a_n+7 &\text{if $n$ is odd.} \end{cases} \]

Prove that, least $30$% of the natural numbers $n$ between $1$ and $1000000$ the first digit of $2^n$ is $1$.

In parallelogram $ABCD$, points $E$ and $F$ on segments $AD$ and $CD$ are such that $\angle BCE=\angle BAF$. Points $K$ and $L$ on segments $AD$ and $CD$ are such that $AK=ED$ and $CL=FD$. Prove that $\angle BKD=\angle BLD$.

$(MOR 1)$ Denote by $[x]$ the greatest integer not exceeding $x$. For all real $k > 1$, define two sequences: \[a_n(k) = [nk]\text{ and } b_n(k) =\left[\frac{nk}{k - 1}\right]\] If $A(k) = \{a_n(k) : n \in\mathbb{N}\}$ and $B(k) = \{b_n(k) : n \in \mathbb{N}\}$, prove that $A(k)$ and $B(k)$ form a partition of $\mathbb{N}$ if and only if $k$ is irrational.

Find all integers $n$ for which there exists a table with $n$ rows, $2022$ columns, and integer entries, such that subtracting any two rows entry-wise leaves every remainder modulo $2022$. [i]Proposed by Tony Wang[/i]

Define a [i]beautiful number[/i] to be an integer of the form $a^n$, where $a\in\{3,4,5,6\}$ and $n$ is a positive integer. Prove that each integer greater than $2$ can be expressed as the sum of pairwise distinct beautiful numbers. [i]Proposed by Matthew Babbitt[/i]

Let $a, b$ and $c$ be nonzero digits. Let $p$ be a prime number which divides the three digit numbers $\overline{abc}$ and $\overline{cba}.$ Show that $p$ divides at least one of the numbers $a+b+c, a-b+c$ and $a-c.$