Given an integer $h > 1$. Let's call a positive common fraction (not necessarily irreducible) [i]good[/i] if the sum of its numerator and denominator is equal to $h$. Let's say that a number $h$ is [i]remarkable[/i] if every positive common fraction whose denominator is less than $h$ can be expressed in terms of good fractions (not necessarily various) using the operations of addition and subtraction. Prove that $h$ is remarkable if and only if it is prime. (Recall that an common fraction has an integer numerator and a natural denominator.)

Let $n >3$ be an odd positive integer and $n=\prod_{i=1}^k p_i^{\alpha_i}$ where $p_i$ are primes and $\alpha_i$ are positive integers. We know that \[m=n(1-\frac{1}{p_1})(1-\frac{1}{p_2})(1-\frac{1}{p_3}) \cdots (1-\frac{1}{p_n}).\] Prove that there exists a prime $P$ such that $P|2^m -1$ but $P \nmid n.$

For an arbitrary positive integer $m$, not divisible by $3$, consider the permutation $x \mapsto 3x \pmod{m}$ on the set $\{ 1,2,\dotsc ,m-1\}$. This permutation can be decomposed into disjointed cycles; for instance, for $m=10$ the cycles are $(1\mapsto 3\to 9,\mapsto 7,\mapsto 1)$, $(2\mapsto 6\mapsto 8\mapsto 4\mapsto 2)$ and $(5\mapsto 5)$. For which integers $m$ is the number of cycles odd?

Find all odd integers $n$ for which $\frac{2^{\phi (n)}-1}{n}$ is a perfect square.

A deck of $n > 1$ cards is given. A positive integer is written on each card. The deck has the property that the arithmetic mean of the numbers on each pair of cards is also the geometric mean of the numbers on some collection of one or more cards. For which $n$ does it follow that the numbers on the cards are all equal? [i]Proposed by Oleg Košik, Estonia[/i]

Does there exist a $6$-digit number $A$ such that none of its $500 000$ multiples $A$, $2A$, $3A$, ..., $500 000A$ ends in $6$ identical digits? (S Tokarev)

The least common multiple of $a$ and $b$ is $12$, and the least common multiple of $b$ and $c$ is $15$. What is the least possible value of the least common multiple of $a$ and $c$? $\textbf{(A) }20\qquad\textbf{(B) }30\qquad\textbf{(C) }60\qquad\textbf{(D) }120\qquad \textbf{(E) }180$

Let $x_0, x_1, \dots , x_{n_0-1}$ be integers, and let $d_1, d_2, \dots, d_k$ be positive integers with $n_0 = d_1 > d_2 > \cdots > d_k$ and $\gcd (d_1, d_2, \dots , d_k) = 1$. For every integer $n \ge n_0$, define \[ x_n = \left\lfloor{\frac{x_{n-d_1} + x_{n-d_2} + \cdots + x_{n-d_k}}{k}}\right\rfloor. \] Show that the sequence $\{x_n\}$ is eventually constant.

Let $(a_{n})_{n\geq 1}$ be a sequence defined by $a_{n}=2^{n}+49$. Find all values of $n$ such that $a_{n}=pg, a_{n+1}=rs$, where $p,q,r,s$ are prime numbers with $p<q, r<s$ and $q-p=s-r$.

Suppose $ n$ is a product of four distinct primes $ a,b,c,d$ such that: $ (i)$ $ a\plus{}c\equal{}d;$ $ (ii)$ $ a(a\plus{}b\plus{}c\plus{}d)\equal{}c(d\minus{}b);$ $ (iii)$ $ 1\plus{}bc\plus{}d\equal{}bd$. Determine $ n$.

Find all triplets of integers $(a,b,c)$ such that the number $$N = \frac{(a-b)(b-c)(c-a)}{2} + 2$$ is a power of $2016$. (A power of $2016$ is an integer of form $2016^n$,where n is a non-negative integer.)

Determine the largest positive integer $n$ that divides $p^6 - 1$ for all primes $p > 7$.

In January Jeff’s investment went up by three quarters. In February it went down by one quarter. In March it went up by one third. In April it went down by one fifth. In May it went up by one seventh. In June Jeff’s investment fell by $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. If Jeff’s investment was worth the same amount at the end of June as it had been at the beginning of January, find $m + n$.

Let $d$ be a positive integer. Show that for every integer $S$, there exist a positive integer $n$ and a sequence $a_1, ..., a_n \in \{-1, 1\}$ such that $S = a_1(1 + d)^2 + a_2(1 + 2d)^2 + ... + a_n(1 + nd)^2$.

Define a positive integer $n$ to be [i]almost-cubic [/i] if it becomes a perfect cube upon concatenating the digit $5$. For example, $12$ is almost-cubic because $125 = 5^3$. Find the remainder when the sum of all almost-cubic $n < 10^8$ is divided by $1000$.

Given coprime positive integers $p,q>1$, call all positive integers that cannot be written as $px+qy$(where $x,y$ are non-negative integers) [i]bad[/i], and define $S(p,q)$ to be the sum of all bad numbers raised to the power of $2019$. Prove that there exists a positive integer $n$, such that for any $p,q$ as described, $(p-1)(q-1)$ divides $nS(p,q)$.

In a particular European city, there are only $7$ day tickets and $30$ day tickets to the public transport. The former costs $7.03$ euro and the latter costs $30$ euro. Aina the Algebraist decides to buy at once those tickets that she can travel by the public transport the whole three year (2014-2016, 1096 days) visiting in the city. What is the cheapest solution?

Let $a_1,a_2,a_3,a_4$ be positive integers, with the property that it is impossible to assign them around a circle where all the neighbors are coprime. Let $i,j,k\in\{1,2,3,4\}$ with $i \neq j$, $j\neq k$, and $k\neq i $. Determine the maximum number of triples $(i,j,k)$ for which $$ ({\rm gcd}(a_i,a_j))^2|a_k. $$

Given $a$, $b$ are positive integers greater than $1$, and for every positive integer $n$, $b^{n}-1$ divides $a^{n}-1$. Define the polynomial $p_{n}(x)$ as follows: $p_0{x}=-1$, $p_{n+1}(x)=b^{n+1}(x-1)p_{n}(bx)-a(b^{n+1}-1)p_{n}(x)$, for $n \ge 0$. Prove that there exist integers $C$ and positive integer $k$ such that $p_{k}(x)=Cx^k$.

How many positive integers $n$ are there such that there are $20$ positive integers that are less than $n$ and relatively prime with $n$? $ \textbf{(A)}\ 1 \qquad \textbf{(B)}\ 2 \qquad \textbf{(C)}\ 3 \qquad \textbf{(D)}\ 4 \qquad \textbf{(E)}\ \text{None}$

What is the greatest multiple of $9$ that can be formed by using each of the digits in the set $\{1, 3,5, 7, 9\}$ at most once.

[u]Set 1[/u] [b]B1.[/b] Evaluate $2 + 0 - 2 \times 0$. [b]B2.[/b] It takes four painters four hours to paint four houses. How many hours does it take forty painters to paint forty houses? [b]B3.[/b] Let $a$ be the answer to this question. What is $\frac{1}{2-a}$? [u]Set 2[/u] [b]B4.[/b] Every day at Andover is either sunny or rainy. If today is sunny, there is a $60\%$ chance that tomorrow is sunny and a $40\%$ chance that tomorrow is rainy. On the other hand, if today is rainy, there is a $60\%$ chance that tomorrow is rainy and a $40\%$ chance that tomorrow is sunny. Given that today is sunny, the probability that the day after tomorrow is sunny can be expressed as n%, where n is a positive integer. What is $n$? [b]B5.[/b] In the diagram below, what is the value of $\angle DD'Y$ in degrees? [img]https://cdn.artofproblemsolving.com/attachments/0/8/6c966b13c840fa1885948d0e4ad598f36bee9d.png[/img] [b]B6.[/b] Christina, Jeremy, Will, and Nathan are standing in a line. In how many ways can they be arranged such that Christina is to the left of Will and Jeremy is to the left of Nathan? Note: Christina does not have to be next to Will and Jeremy does not have to be next to Nathan. For example, arranging them as Christina, Jeremy, Will, Nathan would be valid. [u]Set 3[/u] [b]B7.[/b] Let $P$ be a point on side $AB$ of square $ABCD$ with side length $8$ such that $PA = 3$. Let $Q$ be a point on side $AD$ such that $P Q \perp P C$. The area of quadrilateral $PQDB$ can be expressed in the form $m/n$ for relatively prime positive integers $m$ and $n$. Compute $m + n$. [b]B8.[/b] Jessica and Jeffrey each pick a number uniformly at random from the set $\{1, 2, 3, 4, 5\}$ (they could pick the same number). If Jessica’s number is $x$ and Jeffrey’s number is $y$, the probability that $x^y$ has a units digit of $1$ can be expressed as $m/n$ , where $m$ and $n$ are relatively prime positive integers. Find $m + n$. [b]B9.[/b] For two points $(x_1, y_1)$ and $(x_2, y_2)$ in the plane, we define the taxicab distance between them as $|x_1 - x_2| + |y_1 - y_2|$. For example, the taxicab distance between $(-1, 2)$ and $(3,\sqrt2)$ is $6-\sqrt2$. What is the largest number of points Nathan can find in the plane such that the taxicab distance between any two of the points is the same? [u]Set 4[/u] [b]B10.[/b] Will wants to insert some × symbols between the following numbers: $$1\,\,\,2\,\,\,3\,\,\,4\,\,\,6$$ to see what kinds of answers he can get. For example, here is one way he can insert $\times$ symbols: $$1 \times 23 \times 4 \times 6 = 552.$$ Will discovers that he can obtain the number $276$. What is the sum of the numbers that he multiplied together to get $276$? [b]B11.[/b] Let $ABCD$ be a parallelogram with $AB = 5$, $BC = 3$, and $\angle BAD = 60^o$ . Let the angle bisector of $\angle ADC$ meet $AC$ at $E$ and $AB$ at $F$. The length $EF$ can be expressed as $m/n$, where $m$ and $n$ are relatively prime positive integers. What is $m + n$? [b]B12.[/b] Find the sum of all positive integers $n$ such that $\lfloor \sqrt{n^2 - 2n + 19} \rfloor = n$. Note: $\lfloor x \rfloor$ denotes the greatest integer less than or equal to $x$. [u]Set 5[/u] [b]B13.[/b] This year, February $29$ fell on a Saturday. What is the next year in which February $29$ will be a Saturday? [b]B14.[/b] Let $f(x) = \frac{1}{x} - 1$. Evaluate $$f\left( \frac{1}{2020}\right) \times f\left( \frac{2}{2020}\right) \times f\left( \frac{3}{2020}\right) \times \times ... \times f\left( \frac{2019}{2020}\right) .$$ [b]B15.[/b] Square $WXYZ$ is inscribed in square $ABCD$ with side length $1$ such that $W$ is on $AB$, $X$ is on $BC$, $Y$ is on $CD$, and $Z$ is on $DA$. Line $W Y$ hits $AD$ and $BC$ at points $P$ and $R$ respectively, and line $XZ$ hits $AB$ and $CD$ at points $Q$ and $S$ respectively. If the area of $WXYZ$ is $\frac{13}{18}$ , then the area of $PQRS$ can be expressed as $m/n$ for relatively prime positive integers $m$ and $n$. What is $m + n$? PS. You had better use hide for answers. Last sets have been posted [url=https://artofproblemsolving.com/community/c4h2777424p24371574]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Prove that for every positive integer $m$, every prime $p$ and every positive integer $j \le p^{m-1}$, $p^m$ divides $${p^m \choose p^j }- {p^{m-1} \choose j}$$

Alex has a piece of cheese. He chooses a positive number $a\neq 1$ and cuts the piece into several pieces one by one. Every time he chooses a piece and cuts it in the same ratio $1:a.$ His goal is to divide the cheese into two piles of equal masses. Can he do it?

Let $\sigma (n)$ shows the number of positive divisors of $n$. Let $s(n)$ be the number of positive divisors of $n+1$ such that for every divisor $a$, $a-1$ is also a divisor of $n$. Find the maximum value of $2s(n)- \sigma (n) $.