Find the number of integers between $1$ and $200$ inclusive whose distinct prime divisors sum to $16$. (For example, the sum of the distinct prime divisors of $12$ is $2 + 3 = 5$.) In this section, the word [i]divisor [/i]is used to refer to a [i]positive divisor[/i] of an integer.

Show that a given natural number $A$ is the square of a natural number if and only if for any natural number $n$, at least one of the differences \[(A + 1)^2 - A, (A + 2)^2 - A, (A + 3)^2 - A, \cdots , (A + n)^2 - A\] is divisible by $n$.

Let $a_0,a_1,a_2...$ be a sequence of natural numbers with the following property: $a_n^2$ divides $a_{n-1} a_{n+1}$ for $\forall$ $n\in \mathbb{N}$. Prove that, if for some natural $k\geq 2$ the numbers $a_1$ and $a_k$ are coprime, then $a_1$ divides $a_0$.

For integer $n\ge 4$, find the minimal integer $f(n)$, such that for any positive integer $m$, in any subset with $f(n)$ elements of the set ${m, m+1, \ldots, m+n+1}$ there are at least $3$ relatively prime elements.

The number $2020$ has three different prime factors. What is their sum?

[b]Q.[/b] Show that for any given positive integers $k, l$, there exists infinitely many positive integers $m$, such that $i) m \geqslant k$ $ii) \text{gcd}\left(\binom{m}{k}, l\right)=1$ [i]Suggested by pigeon_in_a_hole[/i]

Let $n$ be the least positive integer greater than $1000$ for which $$\gcd(63, n+120) =21\quad \text{and} \quad \gcd(n+63, 120)=60.$$What is the sum of the digits of $n$? $\textbf{(A) } 12 \qquad\textbf{(B) } 15 \qquad\textbf{(C) } 18 \qquad\textbf{(D) } 21\qquad\textbf{(E) } 24$

Find all primes $p,$ $q,$ $r$ satisfying $p^{2}+2q^{2}=r^{2}.$

What number should we put in place of the question mark such that the following statement becomes true? $$11001_? = 54001_{10}$$ A number written in the subscript means which base the number is in.

Show that no non-zero integers $a$, $b$, $x$, $y$ satisfy $$ \begin{cases} a x - b y = 16,\\ a y + b x = 1. \end{cases} $$

Are there different mutually prime natural numbers $a$, $b$ and $c$, greater than $1$, such that $2a + 1$ is divisible by $b$, $2b + 1$ is divisible by $c$ and $2c + 1$ is divisible by $a$?

[b]p1.[/b] Find the sum of the seventeenth powers of the seventeen roots of the seventeeth degree polynomial equation $x^{17} - 17x + 17 = 0$. [b]p2.[/b] Four identical spherical cows, each of radius $17$ meters, are arranged in a tetrahedral pyramid (their centers are the vertices of a regular tetrahedron, and each one is tangent to the other three). The pyramid of cows is put on the ground, with three of them laying on it. What is the distance between the ground and the top of the topmost cow? [b]p3.[/b] If $a_n$ is the last digit of $\sum^{n}_{i=1} i$, what would the value of $\sum^{1000}_{i=1}a_i$ be? [b]p4.[/b] If there are $15$ teams to play in a tournament, $2$ teams per game, in how many ways can the tournament be organized if each team is to participate in exactly $5$ games against dierent opponents? [b]p5.[/b] For $n = 20$ and $k = 6$, calculate $$2^k {n \choose 0}{n \choose k}- 2^{k-1}{n \choose 1}{{n - 1} \choose {k - 1}} + 2^{k-2}{n \choose 2}{{n - 2} \choose {k - 2}} +...+ (-1)^k {n \choose k}{{n - k} \choose 0}$$ where ${n \choose k}$ is the number of ways to choose $k$ things from a set of $n$. [b]p6.[/b] Given a function $f(x) = ax^2 + b$, with a$, b$ real numbers such that $$f(f(f(x))) = -128x^8 + \frac{128}{3}x^6 - \frac{16}{22}x^2 +\frac{23}{102}$$ , find $b^a$. [b]p7.[/b] Simplify the following fraction $$\frac{(2^3-1)(3^3-1)...(100^3-1)}{(2^3+1)(3^3+1)...(100^3+1)}$$ [b]p8.[/b] Simplify the following expression $$\frac{\sqrt{3 + \sqrt5} + \sqrt{3 - \sqrt5}}{\sqrt{3 - \sqrt8}} -\frac{4}{ \sqrt{8 - 2\sqrt{15}}}$$ [b]p9.[/b] Suppose that $p(x)$ is a polynomial of degree $100$ such that $p(k) = k2^{k-1}$ , $k =1, 2, 3 ,... , 100$. What is the value of $p(101)$ ? [b]p10. [/b] Find all $17$ real solutions $(w, x, y, z)$ to the following system of equalities: $$ 2w + w^2x = x$$ $$ 2x + x^2y=y $$ $$ 2y + y^2z=z $$ $$ -2z+z^2w=w $$ PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Find all positive integers $a,b$ such that $\frac{a^2+b}{b^2-a}$ and $\frac{b^2+a}{a^2-b}$ are also integers.

Determine all integers $n>1$ such that every power of $n$ has an odd number of digits.

If in a $3$-digit number we replace with each other it's last two digits, and add the resulting number to the starting one, we find sum a $4$-digit number that starts with $173$. Which is the starting number?

The sum of the (decimal) digits of a natural number $n$ equals $100$, and the sum of digits of $44n$ equals $800$. Determine the sum of digits of $3n$.

Find all positive integers $x,y,z$ and $t$ such that $2^x3^y+5^z=7^t$.

Let $a, b, c$, and $d$ be four distinct integers. Prove that $(a-b)(a-c)(a-d)(b-c)(b-d)(c-d)$ is divisible by $12$.

Find all natural numbers $n$ that are smaller than $128^{97}$ and have exactly $2019$ divisors.

$(NET 4)$ A boy has a set of trains and pieces of railroad track. Each piece is a quarter of circle, and by concatenating these pieces, the boy obtained a closed railway. The railway does not intersect itself. In passing through this railway, the train sometimes goes in the clockwise direction, and sometimes in the opposite direction. Prove that the train passes an even number of times through the pieces in the clockwise direction and an even number of times in the counterclockwise direction. Also, prove that the number of pieces is divisible by $4.$

Find all prime numbers $p$ and positive integers $m$ such that $2p^2 + p + 9 = m^2.$

There are $n$ natural numbers written on the board. Every move, we could erase $a,b$ and change it to $\gcd(a,b)$ and $\text{lcm}(a,b) - \gcd(a,b)$. Prove that in finite number of moves, all numbers in the board could be made to be equal.

Let $p$ be a prime such that $p \equiv 1 (\text {mod}4)$. Evaluate \[\sum_{k=1}^{p-1} \left( \left \lfloor \frac{2k^2}{p}\right \rfloor - 2 \left \lfloor {\frac{k^2}{p}}\right \rfloor \right)\]

Prove that $5^n-3^n$ is not divisible by $2^n+65$ for any positive integer $n$.

Determine all integers $k$ such that the equation: $$\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=\frac{k}{xyz}$$ has an infinite number of integer solutions $(x,y,z)$ with gcd$(k,xyz)=1$.