Let $\phi(n)$ denote the number of positive integers less than or equal to $n$ that are coprime to $n$. Find the sum of all $1<n<100$ such that $\phi(n)\mid n$.

Let $ A$ be a set of real numbers such that $ A$ has at least four elements. Suppose $ A$ has the property that $ a^2 \plus{} bc$ is a rational number for all distinct numbers $ a,b,c$ in $ A$. Prove that there exists a positive integer $ M$ such that $ a\sqrt{M}$ is a rational number for every $ a$ in $ A$.

Let $\mathbb{Z} _{>0}$ be the set of positive integers. Find all functions $f: \mathbb{Z} _{>0}\rightarrow \mathbb{Z} _{>0}$ such that \[ m^2 + f(n) \mid mf(m) +n \] for all positive integers $m$ and $n$.

For a positive integer $n$, define the set $S_n$ as $S_n =\{(a, b)|a, b \in N, lcm[a, b] = n\}$ . Let $f(n)$ be the sum of $\phi (a)\phi (b)$ for all $(a, b) \in S_n$. If a prime $p$ relatively prime to $n$ is a divisor of $f(n)$, prove that there exists a prime $q|n$ such that $p|q^2 - 1$.

Prove that two natural numbers whose digits are all ones are relatively prime if and only if the numbers of their digits are relatively prime.

p1. Given $f(x)=\frac{1+x}{1-x}$ , for $x \ne 1$ . Defined $p @ q = \frac{p+q}{1+pq}$ for all positive rational numbers $p$ and $q$. Note the sequence with $a_1,a_2,a_3,...$ with $a_1=2 @3$, $a_{n}=a_{n-1}@ (n+2)$ for $n \ge 2$. Determine $f(a_{233})$ and $a_{233}$ p2. It is known that $ a$ and $ b$ are positive integers with $a > b > 2$. Is $\frac{2^a+1}{2^b-1}$ an integer? Write down your reasons. p3. Given a cube $ABCD.EFGH$ with side length $ 1$ dm. There is a square $PQRS$ on the diagonal plane $ABGH$ with points $P$ on $HG$ and $Q$ on $AH$ as shown in the figure below. Point $T$ is the center point of the square $PQRS$. The line $HT$ is extended so that it intersects the diagonal line $BG$ at $N$. Point $M$ is the projection of $N$ on $BC$. Determine the volume of the truncated prism $DCM.HGN$. [img]https://cdn.artofproblemsolving.com/attachments/f/6/22c26f2c7c66293ad7065a3c8ce3ac2ffd938b.png[/img] 4. Nine pairs of husband and wife want to take pictures in a three-line position with the background of the Palembang Ampera Bridge. There are $4$ people in the front row, $6$ people in the middle row, and $ 8$ people in the back row. They agreed that every married couple must be in the same row, and every two people next to each other must be a married couple or of the same sex. Specify the number of different possible arrangements of positions. p5. p5. A hotel provides four types of rooms with capacity, rate, and number of rooms as presented in the following table. [b] type of room, capacity of persons/ room, day / rate (Rp.), / number of rooms [/b][img]https://cdn.artofproblemsolving.com/attachments/3/c/e9e1ed86887e692f9d66349a82eaaffc730b46.jpg[/img] A group of four families wanted to stay overnight at the hotel. Each family consists of husband and wife and their unmarried children. The number of family members by gender is presented in the following table. [b]family / man / woman/ total[/b] [img]https://cdn.artofproblemsolving.com/attachments/4/6/5961b130c13723dc9fa4e34b43be30c31ee635.jpg[/img] The group leader enforces the following provisions. I. Each husband and wife must share a room and may not share a room with other married couples. II. Men and women may not share the same room unless they are from the same family. III. At least one room is occupied by all family representatives (“representative room”) IV. Each family occupies at most $3$ types of rooms. V. No rooms are occupied by more than one family except representative rooms. You are asked to arrange a room for the group so that the total cost of lodging is as low as possible. Provide two possible alternative room arrangements for each family and determine the total cost.

Let be two natural primes $ 1\le q \le p. $ Prove that $ \left( \sqrt{p^2+q} +p\right)^2 $ is irrational and its fractional part surpasses $ 3/4. $

Prove that for all $n\in\mathbb{Z}^+$, we have \[ \sum\limits_{p=1}^n\sum\limits_{q=1}^p\left\lfloor -\frac{1+\sqrt{8q+(2p-1)^2}}{2}\right\rfloor =-\frac{n(n+1)(n+2)}{3} \]

For integer $n\geq4$, find the smallest integer $f(n)$, such that for any positive integer $m$, in any subset with $f(n)$ elements of the set $\{m, m+1, \cdots, m+n-1\}$ there are at least three elements that are relatively prime .

Let $\mathbb{P}$ be the set of all prime numbers. Find all functions $f:\mathbb{P}\rightarrow\mathbb{P}$ such that: $$f(p)^{f(q)}+q^p=f(q)^{f(p)}+p^q$$ holds for all $p,q\in\mathbb{P}$. [i]Proposed by Dorlir Ahmeti, Albania[/i]

If $a,b\neq 0$ are real numbers such that $a^2b^2(a^2b^2+4)=2(a^6+b^6)$ , then show that $a,b$ can’t be both of them rational .

Let $m$ and $n$ be arbitrary positive integers, and $a, b, c$ be different natural numbers of the form $2^m.5^n$. Determine the number of all equations of the form $ax^2-2bx+c=0$ if it is known that each equation has only one real solution.

Numbers $1$ through $2014$ are written on a board. A valid operation is to erase two numbers $a$ and $b$ on the board and replace them with the greatest common divisor and the least common multiple of $a$ and $b$. Prove that, no matter how many operations are made, the sum of all the numbers that remain on the board is always larger than $2014$ $\times$ $\sqrt[2014]{2014!}$

Let $a, b, c$ are natural numbers such that $a^{2}+b^{2}=c^{2}$ and $c-b=1$ Prove that $(i)$ $a$ is odd. $(ii)$ $b$ is divisible by $4$ $(iii)$ $a^{b}+b^{a}$ is divisible by $c$

Let $a$ be an integer and $n$ a positive integer . Show that the sum : $$\sum_{k=1}^{n} a^{(k,n)}$$ is divisible by $n$ , where $(x,y)$ is the greatest common divisor of the numbers $x$ and $y$ .

For any positive integer $n$, let $D_n$ denote the set of all positive divisors of $n$, and let $f_i(n)$ denote the size of the set $$F_i(n) = \{a \in D_n | a \equiv i \pmod{4} \}$$where $i = 0, 1, 2, 3$. Determine the smallest positive integer $m$ such that $f_0(m) + f_1(m) - f_2(m) - f_3(m) = 2017$.

Let $p$ is a prime and $p\equiv 2\pmod 3$. For $\forall a\in\mathbb Z$, if $$p\mid \prod\limits_{i=1}^p(i^3-ai-1),$$then $a$ is called a "GuGu" number. How many "GuGu" numbers are there in the set $\{1,2,\cdots ,p\}?$ (We are allowed to discuss now. It is after 00:00 Feb 14 Beijing Time)

A thousand integer divisions are made: $2018$ is divided by each of the integers from $ 1$ to $1000$. Thus, a thousand integer quotients are obtained with their respective remainders. Which of these thousand remainders is the bigger?

Let $n>2$ be a positive integer and $b=2^{2^n}$. Let $a$ be an odd positive integer such that $a\le b \le 2a$. Show that $a^2+b^2-ab$ is not a square.

Let $n \ge 3$ be a positive integer. Determine, in terms of $n$, how many triples of sets $(A,B,C)$ satisfy the conditions: $\bullet$ $A, B$ and $C$ are pairwise disjoint , that is, $A \cap B = A \cap C= B \cap C= \emptyset$. $\bullet$ $A \cup B \cup C= \{ 1 , 2 , ... , n \}$. $\bullet$ The sum of the elements of $A$, the sum of the elements of $B$ and the sum of the elements of $C$ leave the same remainder when divided by $3$. Note: One or more of the sets may be empty.

Determine the number of positive odd and even factor of $ 5^6\minus{}1$.

[b]2A. [/b] Two triangles $ABC$ and $ABD$ share a common side. $ABC$ is drawn such that its entire area lies inside the larger triangle $ABD$. If $AB = 20$, side $AD$ meets side $AB$ at a right angle, and point $C$ is between points $A$ and $D$, then find the area outside of triangle $ABC$ but within $ABD$, given that both triangles have integral side lengths and $AB$ is the smallest side of either triangle. $ABC$ and $ABD$ are both primitive right triangles. [i] (1 point)[/i] [b]2B.[/b] Find the sum of all positive integral factors of the correct answer to [b]2A[/b]. [i](2 points)[/i] [b]2C.[/b] Let $B$ be the sum of the digits of the correct answer to [b]2B[/b] above. If the solution to the functional equation $21*f(x) - 7*f(1/x) = Bx$ is of the form $(Ax^2 + C) / Dx$, find $C$, given that $A$, $C$, and $D$ are relatively prime (they don’t share a common prime factor). [i](3 points)[/i] [hide=ANSWER KEY]2A.780 2B. 2352 2C. 3[/hide]

Given a positive integer $m$. Let $$A_l = (4l+1)(4l+2)...(4(5^m+1)l)$$ for any positive integer $l$. Prove that there exist infinite number of positive integer $l$ which $$5^{5^ml}\mid A_l\text{ and } 5^{5^ml+1}\nmid A_l$$ and find the minimum value of $l$ satisfying the above condition.

Compute the remainder when $98!$ is divided by $101$.

a) Show that the product of all differences of possible couples of six given positive integers is divisible by $960$ b) Show that the product of all differences of possible couples of six given positive integers is divisible by $34560$ PS. a) original from Albania b) modified by problem selecting committee