A positive integer $n$ is called [i]mythical[/i] if every divisor of $n$ is two less than a prime. Find the unique mythical number with the largest number of divisors. [i]Proposed by Evan Chen[/i]

Let $n$ be a given number greater than 2. We consider the set $V_n$ of all the integers of the form $1 + kn$ with $k = 1, 2, \ldots$ A number $m$ from $V_n$ is called indecomposable in $V_n$ if there are not two numbers $p$ and $q$ from $V_n$ so that $m = pq.$ Prove that there exist a number $r \in V_n$ that can be expressed as the product of elements indecomposable in $V_n$ in more than one way. (Expressions which differ only in order of the elements of $V_n$ will be considered the same.)

Find the smallest positive integer $n$ for which there exist $n$ different positive integers $a_{1}, a_{2}, \cdots, a_{n}$ satisfying [list] [*] $\text{lcm}(a_1,a_2,\cdots,a_n)=1985$,[*] for each $i, j \in \{1, 2, \cdots, n \}$, $gcd(a_i,a_j)\not=1$, [*] the product $a_{1}a_{2} \cdots a_{n}$ is a perfect square and is divisible by $243$, [/list] and find all such $n$-tuples $(a_{1}, \cdots, a_{n})$.

Given a positive integer $n \geq 2$, whose canonical prime factorization is $n = p_1^{\alpha_1}p_2^{\alpha_2} \ldots p_k^{\alpha_k}$, we define the following functions: $$\varphi(n) = n\bigg(1 -\frac{1}{p_1}\bigg) \bigg(1 -\frac{1}{p_2}\bigg) \ldots \bigg(1 -\frac {1}{p_k}\bigg) ; \overline{\varphi}(n) = n\bigg(1 +\frac{1}{p_1}\bigg) \bigg(1 +\frac{1}{p_2}\bigg) \ldots \bigg(1 + \frac{1}{p_k}\bigg)$$ Consider all positive integers $n$ such that $\overline{\varphi}(n)$ is a multiple of $n + \varphi(n) $. (a) Prove that $n$ is even. (b) Determine all positive integers $n$ that satisfy this property.

If $n$ is a positive integer such that $2n$ has $28$ positive divisors and $3n$ has $30$ positive divisors, then how many positive divisors does $6n$ have? $\text{(A)}\ 32 \qquad \text{(B)}\ 34 \qquad \text{(C)}\ 35 \qquad \text{(D)}\ 36\qquad \text{(E)}\ 38$

How many $ 3$-digit positive integers have digits whose product equals $ 24$? $ \textbf{(A)}\ 12 \qquad \textbf{(B)}\ 15 \qquad \textbf{(C)}\ 18 \qquad \textbf{(D)}\ 21 \qquad \textbf{(E)}\ 24$

What is the largest prime factor of 8091?

If $x$ is the cube of a positive integer and $d$ is the number of positive integers that are divisors of $x$, then $d$ could be $ \textbf{(A)}\ 200\qquad\textbf{(B)}\ 201\qquad\textbf{(C)}\ 202\qquad\textbf{(D)}\ 203\qquad\textbf{(E)}\ 204 $

Define a sequence $(a_n)$ recursively by $a_1=0, a_2=2, a_3=3$ and $a_n=\max_{0<d<n} a_d \cdot a_{n-d}$ for $n \ge 4$. Determine the prime factorization of $a_{19702020}$.

What is the smallest positive integer that is neither prime nor square and that has no prime factor less than 50? $\textbf{(A)}\hspace{.05in}3127 \qquad \textbf{(B)}\hspace{.05in}3133 \qquad \textbf{(C)}\hspace{.05in}3137 \qquad \textbf{(D)}\hspace{.05in}3139 \qquad \textbf{(E)}\hspace{.05in}3149 $

Let $n>6$ be a perfect number, and let $n=p_1^{e_1}\cdot\cdot\cdot p_k^{e_k}$ be its prime factorisation with $1<p_1<\dots <p_k$. Prove that $e_1$ is an even number. A number $n$ is [i]perfect[/i] if $s(n)=2n$, where $s(n)$ is the sum of the divisors of $n$. (Proposed by Javier Rodrigo, Universidad Pontificia Comillas)

Let $n$ be a positive integer, with prime factorization $$n = p_1^{e_1}p_2^{e_2} \cdots p_r^{e_r}$$ for distinct primes $p_1, \ldots, p_r$ and $e_i$ positive integers. Define $$rad(n) = p_1p_2\cdots p_r,$$ the product of all distinct prime factors of $n$. Find all polynomials $P(x)$ with rational coefficients such that there exists infinitely many positive integers $n$ with $P(n) = rad(n)$.

Prove that there exist infinitely many positive integers which cannot be written in form $a^{d(a)}+b^{d(b)}$ for some positive integers $a$ and $b$ For positive integer $d(a)$ denotes number of positive divisors of $a$ [i]Proposed by Borna Vukorepa[/i]

Find all integers $n$, $n \ge 1$, such that $n \cdot 2^{n+1}+1$ is a perfect square.

At least how many numbers must be deleted from the product $1 \times 2 \times \dots \times 46 \times 47$ in order to make it a perfect square?

For how many positive integers $n \le 500$ is $n!$ divisible by $2^{n-2}$? [i]Proposed by Eugene Chen[/i]

Prove that for positive integers $ x,y,z$ the number $ x^2 \plus{} y^2 \plus{} z^2$ is not divisible by $ 3(xy \plus{} yz \plus{} zx)$.

Given a natural $n>1$ and its prime fatorization $n=p_1^{\alpha 1}p_2^{\alpha_2} \cdots p_k^{\alpha_k}$, its [i]false derived[/i] is defined by $$f(n)=\alpha_1p_1^{\alpha_1-1}\alpha_2p_2^{\alpha_2-1}...\alpha_kp_k^{\alpha_k-1}.$$ Prove that there exist infinitely many naturals $n$ such that $f(n)=f(n-1)+1$.

For every positive integer $n$, let $s(n)$ be the sum of the exponents of $71$ and $97$ in the prime factorization of $n$; for example, $s(2021) = s(43 \cdot 47) = 0$ and $s(488977) = s(71^2 \cdot 97) = 3$. If we define $f(n)=(-1)^{s(n)}$, prove that the limit \[ \lim_{n \to +\infty} \frac{f(1) + f(2) + \cdots+ f(n)}{n} \] exists and determine its value.

Is it possible to place $1995$ different natural numbers along a circle so that for any two of these numbers, the ratio of the greatest to the least is a prime? I feel that my solution's wording and notation is awkward (and perhaps unnecessarily complicated), so please feel free to critique it: [hide] Suppose that we do have such a configuration $a_{1},a_{2},...a_{1995}$. WLOG, $a_{2}=p_{1}a_{1}$. Then \[\frac{a_{2}}{a_{3}}= p_{2}, \frac{1}{p_{2}}\] \[\frac{a_{3}}{a_{4}}= p_{3}, \frac{1}{p_{3}}\] \[... \] \[\frac{a_{1995}}{a_{1}}= p_{1995}, \frac{1}{p_{1995}}\] Multiplying these all together, \[\frac{a_{2}}{a_{1}}= \frac{\prod p_{k}}{\prod p_{j}}= p_{1}\] Where $\prod p_{k}$ is some product of the elements in a subset of $\{ p_{2},p_{3}, ...p_{1995}\}$. We clear denominators to get \[p_{1}\prod p_{j}= \prod p_{k}\] Now, by unique prime factorization, the set $\{ p_{j}\}\cup \{ p_{1}\}$ is equal to the set $\{ p_{k}\}$. However, since there are a total of $1995$ primes, this is impossible. We conclude that no such configuration exists. [/hide]

What is the maximum value of $n$ for which there is a set of distinct positive integers $k_1,k_2,\ldots,k_n$ for which \[k_1^2+k_2^2+\ldots+k_n^2=2002?\] $\textbf{(A) }14\qquad\textbf{(B) }15\qquad\textbf{(C) }16\qquad\textbf{(D) }17\qquad\textbf{(E) }18$

For every positive integer $n$ with prime factorization $n = \prod_{i = 1}^{k} p_i^{\alpha_i}$, define \[\mho(n) = \sum_{i: \; p_i > 10^{100}} \alpha_i.\] That is, $\mho(n)$ is the number of prime factors of $n$ greater than $10^{100}$, counted with multiplicity. Find all strictly increasing functions $f: \mathbb{Z} \to \mathbb{Z}$ such that \[\mho(f(a) - f(b)) \le \mho(a - b) \quad \text{for all integers } a \text{ and } b \text{ with } a > b.\] [i]Proposed by Rodrigo Sanches Angelo, Brazil[/i]

How many zeros are at the end of the product \[25\times 25\times 25\times 25\times 25\times 25\times 25\times 8\times 8\times 8?\] $\text{(A)}\ 3 \qquad \text{(B)}\ 6 \qquad \text{(C)}\ 9 \qquad \text{(D)}\ 10 \qquad \text{(E)}\ 12$

Define $ n_a!$ for $ n$ and $ a$ positive to be \[ n_a ! \equal{} n (n\minus{}a)(n\minus{}2a)(n\minus{}3a)...(n\minus{}ka)\] where $ k$ is the greatest integer for which $ n>ka$. Then the quotient $ 72_8!/18_2!$ is equal to $ \textbf{(A)}\ 4^5 \qquad \textbf{(B)}\ 4^6 \qquad \textbf{(C)}\ 4^8 \qquad \textbf{(D)}\ 4^9 \qquad \textbf{(E)}\ 4^{12}$

The number of digits in $4^{16} 5^{25}$ (when written in the usual base 10 form) is A. 31 B. 30 C. 29 D. 28 E. 27