Let $n>1$ be an integer, and let $k$ be the number of distinct prime divisors of $n$. Prove that there exists an integer $a$, $1<a<\frac{n}{k}+1$, such that $n \mid a^2-a$.

Find all the natural numbers $m,n$ such that $1+5 \cdot 2^m=n^2$.

$ S$ is a non-empty subset of the set $ \{ 1, 2, \cdots, 108 \}$, satisfying: (1) For any two numbers $ a,b \in S$ ( may not distinct), there exists $ c \in S$, such that $ \gcd(a,c)\equal{}\gcd(b,c)\equal{}1$. (2) For any two numbers $ a,b \in S$ ( may not distinct), there exists $ c' \in S$, $ c' \neq a$, $ c' \neq b$, such that $ \gcd(a, c') > 1$, $ \gcd(b,c') >1$. Find the largest possible value of $ |S|$.

Let $n$ be a positive integer and let $d_{1},d_{2},,\ldots ,d_{k}$ be its divisors, such that $1=d_{1}<d_{2}<\ldots <d_{k}=n$. Find all values of $n$ for which $k\geq 4$ and $n=d_{1}^{2}+d_{2}^{2}+d_{3}^{2}+d_{4}^{2}$.

Let $p$ be an odd prime number. Around a circular table, $p$ students sit. We give $p$ pieces of candy to those students in the following manner. The first candy we give to an arbitrary student. Then, going around clockwise, we skip two students and give the next student a piece of candy, then we skip 4 students and give another piece of candy to the next student... In general in the $k-$th turn we skip $2k$ students and give the next student a piece of candy. We do this until we don't give out all $p$ pieces of candy. $a)$ How many students won't get any pieces of candy? $b)$ How many pairs of neighboring students (those students who sit next to each other on the table) both got at least a piece of candy?

Let $n > 1$ be a given integer. Prove that infinitely many terms of the sequence $(a_k )_{k\ge 1}$, defined by \[a_k=\left\lfloor\frac{n^k}{k}\right\rfloor,\] are odd. (For a real number $x$, $\lfloor x\rfloor$ denotes the largest integer not exceeding $x$.) [i]Proposed by Hong Kong[/i]

Find the remainder when $$\prod_{i = 1}^{1903} (2^i + 5)$$ is divided by $1000$.

Let $m,n > 1$ be integer numbers. Solve in positive integers $x^n+y^n = 2^m$.

Calculate the product of all positive integers less than $100$ and having exactly three positive divisors. Show that this product is a square.

Find all quadruple $ (m,n,p,q) \in \mathbb{Z}^4$ such that \[ p^m q^n \equal{} (p\plus{}q)^2 \plus{} 1.\]

For each integer $n \ge 100$ we define $T(n)$ to be the number obtained from $n$ by moving the two leading digits to the end. For example, $T(12345) = 34512$ and $T(100) = 10$. Find all integers $n \ge 100$ for which $n + T(n) = 10n$.

Let $n > 1$ be a positive integer and $p$ a prime number such that $n | (p - 1) $and $p | (n^6 - 1)$. Prove that at least one of the numbers $p- n$ and $p + n$ is a perfect square.

For a prime $ p$ and a given integer $ n$ let $ \nu_p(n)$ denote the exponent of $ p$ in the prime factorisation of $ n!$. Given $ d \in \mathbb{N}$ and $ \{p_1,p_2,\ldots,p_k\}$ a set of $ k$ primes, show that there are infinitely many positive integers $ n$ such that $ d\mid \nu_{p_i}(n)$ for all $ 1 \leq i \leq k$. [i]Author: Tejaswi Navilarekkallu, India[/i]

Define a regular $n$-pointed star to be the union of $n$ line segments $P_1P_2, P_2P_3,\ldots, P_nP_1$ such that $\bullet$ the points $P_1, P_2,\ldots, P_n$ are coplanar and no three of them are collinear, $\bullet$ each of the $n$ line segments intersects at least one of the other line segments at a point other than an endpoint, $\bullet$ all of the angles at $P_1, P_2,\ldots, P_n$ are congruent, $\bullet$ all of the $n$ line segments $P_2P_3,\ldots, P_nP_1$ are congruent, and $\bullet$ the path $P_1P_2, P_2P_3,\ldots, P_nP_1$ turns counterclockwise at an angle of less than 180 degrees at each vertex. There are no regular 3-pointed, 4-pointed, or 6-pointed stars. All regular 5-pointed stars are similar, but there are two non-similar regular 7-pointed stars. How many non-similar regular 1000-pointed stars are there?

Find all the integers $x,y$ satisfying equation $x^2+x=y^4+y^3+y^2+y$.

find all pairs of relatively prime natural numbers $ (m,n) $ in such a way that there exists non constant polynomial f satisfying \[ gcd(a+b+1, mf(a)+nf(b) > 1 \] for every natural numbers $ a $ and $ b $

$S= \{1,4,8,9,16,...\} $is the set of perfect integer power. ( $S=\{ n^k| n, k \in Z, k \ge 2 \}$. )We arrange the elements in $S$ into an increasing sequence $\{a_i\}$ . Show that there are infinite many $n$, such that $9999|a_{n+1}-a_n$

Given a positive integer $ n $, let $ A, B $ be two co-prime positive integers such that $$ \frac{B}{A} = \left(\frac{n\left(n+1\right)}{2}\right)!\cdot\prod\limits_{k=1}^{n}{\frac{k!}{\left(2k\right)!}} $$ Prove that $ A $ is a power of $ 2 $.

Let $n > 6$ be a perfect number. Let $p_1^{a_1} \cdot p_2^{a_2} \cdot ... \cdot p_k^{a_k}$ be the prime factorisation of $n$, where we assume that $p_1 < p_2 <...< p_k$ and $a_i > 0$ for all $ i = 1,...,k$. Prove that $a_1$ is even. Remark: An integer $n \ge 2$ is called a perfect number if the sum of its positive divisors, excluding $ n$ itself, is equal to $n$. For example, $6$ is perfect, as its positive divisors are $\{1, 2, 3, 6\}$ and $1+2+3=6$.

[b]p1.[/b] Fun facts! We know that $1008^2-1007^2 = 1008+1007$ and $1009^2-1008^2 = 1009+1008$. Now compute the following: $$1010^2 - 1009^2 - 1.$$ [b]p2.[/b] Let $m$ be the smallest positive multiple of $2018$ such that the fraction $m/2019$ can be simplified. What is the number $m$? [b]p3.[/b] Given that $n$ satisfies the following equation $$n + 3n + 5n + 7n + 9n = 200,$$ find $n$. [b]p4.[/b] Grace and Somya each have a collection of coins worth a dollar. Both Grace and Somya have quarters, dimes, nickels and pennies. Serena then observes that Grace has the least number of coins possible to make one dollar and Somya has the most number of coins possible. If Grace has $G$ coins and Somya has $S$ coins, what is $G + S$? [b]p5.[/b] What is the ones digit of $2018^{2018}$? [b]p6.[/b] Kaitlyn plays a number game. Each time when Kaitlyn has a number, if it is even, she divides it by $2$, and if it is odd, she multiplies it by $5$ and adds $1$. Kaitlyn then takes the resulting number and continues the process until she reaches $1$. For example, if she begins with $3$, she finds the sequence of $6$ numbers to be $$3, 3 \cdot 5 + 1 = 16, 16/2 = 8, 8/2 = 4, 4/2 = 2, 2/2 = 1.$$ If Kaitlyn's starting number is $51$, how many numbers are in her sequence, including the starting number and the number $1$? [b]p7.[/b] Andrew likes both geometry and piano. His piano has $88$ keys, $x$ of which are white and $y$ of which are black. Each white key has area $3$ and each black key has area $11$. If the keys of his piano have combined area $880$, how many black keys does he have? [b]p8.[/b] A six-sided die contains the numbers $1$, $2$, $3$, $4$, $5$, and $6$ on its faces. If numbers on opposite faces of a die always sum to $7$, how many distinct dice are possible? (Two dice are considered the same if one can be rotated to obtain the other.) [b]p9.[/b] In $\vartriangle ABC$, $AB$ is $12$ and $AC$ is $15$. Alex draws the angle bisector of $BAC$, $AD$, such that $D$ is on $BC$. If $CD$ is $10$, then the area of $\vartriangle ABC$ can be expressed in the form $\frac{m \sqrt{n}}{p}$ where $m, p$ are relatively prime and $n$ is not divisible by the square of any prime. Find $m + n + p$. [b]p10.[/b] Find the smallest positive integer that leaves a remainder of $2$ when divided by $5$, a remainder of $3$ when divided by $6$, a remainder of $4$ when divided by $7$, and a remainder of $5$ when divided by $8$. [b]p11.[/b] Chris has a bag with $4$ marbles. Each minute, Chris randomly selects a marble out of the bag and flips a coin. If the coin comes up heads, Chris puts the marble back in the bag, while if the coin comes up tails, Chris sets the marble aside. What is the expected number of seconds it will take Chris to empty the bag? [b]p12.[/b] A real fixed point $x$ of a function $f(x)$ is a real number such that $f(x) = x$. Find the absolute value of the product of the real fixed points of the function $f(x) = x^4 + x - 16$. [b]p13.[/b] A triangle with angles $30^o$, $75^o$, $75^o$ is inscribed in a circle with radius $1$. The area of the triangle can be expressed as $\frac{a+\sqrt{b}}{c}$ where $b$ is not divisible by the square of any prime. Find $a + b + c$. [b]p14.[/b] Dora and Charlotte are playing a game involving flipping coins. On a player's turn, she first chooses a probability of the coin landing heads between $\frac14$ and $\frac34$ , and the coin magically flips heads with that probability. The player then flips this coin until the coin lands heads, at which point her turn ends. The game ends the first time someone flips heads on an odd-numbered flip. The last player to flip the coin wins. If both players are playing optimally and Dora goes first, let the probability that Charlotte win the game be $\frac{a}{b}$ . Find $a \cdot b$. [b]p15.[/b] Jonny is trying to sort a list of numbers in ascending order by swapping pairs of numbers. For example, if he has the list $1$, $4$, $3$, $2$, Jonny would swap $2$ and $4$ to obtain $1$, $2$, $3$, $4$. If Jonny is given a random list of $400$ distinct numbers, let $x$ be the expected minimum number of swaps he needs. Compute $\left \lfloor \frac{x}{20} \right \rfloor$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Find all functions $f: \mathbb{N} \to \mathbb{N}$ such that $$\gcd(f(x),y)f(xy)=f(x)f(y)$$ for all positive integers $x, y$.

The positive integers $a$ and $b$ are such that $a+k$ is divisible by $b+k$ for all positive integers numbers $k<b$. Prove that $a-k$ is divisible by $b-k$ for all positive integers $k<b$.

1. (a) For which positive integers n, does 2n divide the sum of the first n positive integers? (b) Determine, with proof, those positive integers n (if any) which have the property that 2n + 1 divides the sum of the first n positive integers.

Prove that there exist infinitely many positive integers $n$, for which at least one of the numbers $2^{2^n}+1$ and $2018^{2^n}+1$ is composite.

Let $a,b,$ be positive integers and $a>b>2$. Prove that $\frac{2^a+1}{2^b-1}$ is never an integer