Find all numbers $n$ that can be expressed in the form $n=k+2\lfloor\sqrt{k}\rfloor+2$ for some nonnegative integer $k$.

A hyper-primitive root is a k-tuple $ (a_{1},a_{2},\dots,a_{k})$ and $ (m_{1},m_{2},\dots,m_{k})$ with the following property: For each $ a\in\mathbb N$, that $ (a,m) \equal{} 1$, has a unique representation in the following form: \[ a\equiv a_{1}^{\alpha_{1}}a_{2}^{\alpha_{2}}\dots a_{k}^{\alpha_{k}}\pmod{m}\qquad 1\leq\alpha_{i}\leq m_{i}\] Prove that for each $ m$ we have a hyper-primitive root.

For each $n\in\mathbb{N}$ let $d_n$ denote the gcd of $n$ and $(2019-n)$. Find value of $d_1+d_2+\cdots d_{2018}+d_{2019}$

Let $d(n)$ denote the number of positive divisors of the natural number $n$. Find all the natural numbers $n$ such that $$d(n) = \frac{n}{5}$$.

Determine the greatest integer $N$ such that $N$ is a divisor of $n^{13}-n$ for all integers $n$.

Madam Mim has a deck of $52$ cards, stacked in a pile with their backs facing up. Mim separates the small pile consisting of the seven cards on the top of the deck, turns it upside down, and places it at the bottom of the deck. All cards are again in one pile, but not all of them face down; the seven cards at the bottom do, in fact, face up. Mim repeats this move until all cards have their backs facing up again. In total, how many moves did Mim make $?$

For each positive integer $ n$,define $ f(n)\equal{}lcm(1,2,...,n)$. (a)Prove that for every $ k$ there exist $ k$ consecutive positive integers on which $ f$ is constant. (b)Find the maximum possible cardinality of a set of consecutive positive integers on which $ f$ is strictly increasing and find all sets for which this maximum is attained.

Let $A$ be the greatest possible value of a product of positive integers that sums to $2014$. Compute the sum of all bases and exponents in the prime factorization of $A$. For example, if $A=7\cdot 11^5$, the answer would be $7+11+5=23$.

Find all pairs of positive integers $(x,y) $ for which $x^3 + y^3 = 4(x^2y + xy^2 - 5) .$

Let $(F_n )_{n\ge o}$ be the sequence of Fibonacci numbers: $F_0 = 0$, $F_1 = 1$ and $F_{n+2} = F_{n+1}+F_n$ , for every $n \ge 0$. Prove that for any prime $p \ge 3$, $p$ divides $F_{2p} - F_p$ .

Positive integers $a,b,c$ satisfy the equations $a^2=b^3+ab$ and $c^3=a+b+c$. Prove that $a=bc$.

Find all intengers n such that $2^n - 1$ is a multiple of 3 and $(2^n - 1)/3$ is a divisor of $4m^2 + 1$ for some intenger m.

Let $n$ be a positive integer with $k$ digits. A number $m$ is called an $alero$ of $n$ if there exist distinct digits $a_1$, $a_2$, $\dotsb$, $a_k$, all different from each other and from zero, such that $m$ is obtained by adding the digit $a_i$ to the $i$-th digit of $n$, and no sum exceeds 9. For example, if $n$ $=$ $2024$ and we choose $a_1$ $=$ $2$, $a_2$ $=$ $1$, $a_3$ $=$ $5$, $a_4$ $=$ $3$, then $m$ $=$ $4177$ is an alero of $n$, but if we choose the digits $a_1$ $=$ $2$, $a_2$ $=$ $1$, $a_3$ $=$ $5$, $a_4$ $=$ $6$, then we don't obtain an alero of $n$, because $4$ $+$ $6$ exceeds $9$. Find the smallest $n$ which is a multiple of $2024$ that has an alero which is also a multiple of $2024$.

Determine all pairs $ (m,n)$ of natural numbers for which $ m^2\equal{}nk\plus{}2$ where $ k\equal{}\overline{n1}$. EDIT. [color=#FF0000]It has been discovered the correct statement is with $ k\equal{}\overline{1n}$.[/color]

Prove that any rational number can be represented as a product of four rational numbers whose sum is zero.

Given arbitrary positive integer $ a$ larger than $ 1$, show that for any positive integer $ n$, there always exists a n-degree integral coefficient polynomial $ p(x)$, such that $ p(0)$, $ p(1)$, $ \cdots$, $ p(n)$ are pairwise distinct positive integers, and all have the form of $ 2a^k\plus{}3$, where $ k$ is also an integer.

Let $p> 3$ be a prime number. Determine the number of all ordered sixes $(a, b, c, d, e, f)$ of positive integers whose sum is $3p$ and all fractions $\frac{a + b}{c + d},\frac{b + c}{d + e},\frac{c + d}{e + f},\frac{d + e}{f + a},\frac{e + f}{a + b}$ have integer values.

How many positive integers less than $10,000$ have an even number of even digits and an odd number of odd digits ? (Assume no number starts with zero.)

Find all integer solutions to $y^2(x^2+y^2-2xy-x-y)=(x+y)^2(x-y)$.

Let $a, n \in \mathbb{Z}^{*}_{+}$. $a$ is defined inductively in the base $n$-[i]recursive[/i]. We first write $a$ in the base $n$, e.g., as a sum of terms of the form $k_tn^t$, with $0 \le k_t < n$. For each exponent $t$, we write $t$ in the base $n$-[i]recursive[/i], until all the numbers in the representation are less than $n$. For instance, $1309 = 3^6 + 2.3^5 + 1.3^4 + 1.3^2 + 1.3 + 1$ $ = 3^{2.3} + 2.3^{3+2} + 1.3^{3+1} + 1.3^2 + 1$ Let $x_1 \in \mathbb{Z}$ arbitrary. We define $x_n$ recursively, as following: if $x_{n-1} > 0$, we write $x_{n-1}$ in the base $n$-[i]recursive[/i] and we replace all the numbers $n$ for $n+1$ (even the exponents!), so we obtain the successor of $x_n$. If $x_{n-1} = 0$, then $x_n = 0$. Example: $x_1 = 2^{2^{2} + 2 + 1} + 2^{2+1} + 2 + 1$ $\Rightarrow x_2 = 3^{3^{3} + 3 + 1} + 3^{3+1} + 3$ $\Rightarrow x_3 = 4^{4^{4} + 4 + 1} + 4^{4+1} + 3$ $\Rightarrow x_4 = 5^{5^{5} + 5 + 1} + 5^{5+1} + 2$ $\Rightarrow x_5 = 6^{6^{6} + 6 + 1} + 6^{6+1} + 1$ $\Rightarrow x_6 = 7^{7^{7} + 7 + 1} + 7^{7+1}$ $\Rightarrow x_7 = 8^{8^{8} + 8 + 1} + 7.8^8 + 7.8^7 + 7.8^6 + ... + 7$ $.$ $.$ $.$ Prove that $\exists N : x_N = 0$.

Let $n$ be a positive integer. Show that there exists a one-to-one function $\sigma : \{1,2,...,n\} \to \{1,2,...,n\}$ such that $$\sum_{k=1}^{n} \frac{k}{(k+\sigma(k))^2} < \frac{1}{2}.$$

The [i]popularity [/i] of a positive integer $n$ is the number of positive integer divisors of $n$. For example, $1$ has popularity $1$, and $12$ has popularity $6$. For each number $n$ between $1$ and $30$ inclusive, Cathy writes the number $n$ on $k$ pieces of paper, where $k$ is the popularity of $n$. Cathy then picks a piece of paper at random. Compute the probability that she will pick an even integer.

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?

Let $a > 1$ be a positive integer and $d > 1$ be a positive integer coprime to $a$. Let $x_1=1$, and for $k\geq 1$, define $$x_{k+1} = \begin{cases} x_k + d &\text{if } a \text{ does not divide } x_k \\ x_k/a & \text{if } a \text{ divides } x_k \end{cases}$$ Find, in terms of $a$ and $d$, the greatest positive integer $n$ for which there exists an index $k$ such that $x_k$ is divisible by $a^n$.

$3.$ Positive integers $a$ and $b$ are such that $a+b=\frac{a}{b}+\frac{b}{a}.$ What is the value of $a^2+b^2 ?$