Azambuja writes a rational number $q$ on a blackboard. One operation is to delete $q$ and replace it by $q+1$; or by $q-1$; or by $\frac{q-1}{2q-1}$ if $q \neq \frac{1}{2}$. The final goal of Azambuja is to write the number $\frac{1}{2018}$ after performing a finite number of operations. [b]a)[/b] Show that if the initial number written is $0$, then Azambuja cannot reach his goal. [b]b)[/b] Find all initial numbers for which Azambuja can achieve his goal.

Let $ a_1,a_2,a_3,\dots$ be infinite sequence of positive integers satisfying the following conditon: for each prime number $ p$, there are only finite number of positive integers $ i$ such that $ p|a_i$. Prove that that sequence contains a sub-sequence $ a_{i_1},a_{i_2},a_{i_3},\dots$, with $ 1 \le i_1<i_2<i_3<\dots$, such that for each $ m \ne n$, $ \gcd(a_{i_m},a_{i_n})\equal{}1$.

Determine the integers $a, b, c$ for which $$\frac{a+1}{3}=\frac{b+2}{4}=\frac{5}{c+3}$$

[url=https://artofproblemsolving.com/community/c893771h1861957p12597232]8.1[/url] The point $E$ lies on the base $[AD]$ of the trapezoid $ABCD$. The perimeters of the triangles $ABE, BCE$ and $CDE$ are equal. Prove that $|BC| = |AD|/2$ [b]8.2[/b] In a convex pentagon, the lengths of all sides are equal. Find the point on the longest diagonal from which all sides are visible underneath angles not exceeding a right angle. [url=https://artofproblemsolving.com/community/c893771h1862007p12597620]8.3[/url] Every city in the certain state is connected by airlines with no more than with three other ones, but one can get from every city to every other city changing a plane once only or directly. What is the maximal possible number of the cities? [url=https://artofproblemsolving.com/community/c893771h1861966p12597273]8.4*/7.4*[/url] (asterisk problems in separate posts) [url=https://artofproblemsolving.com/community/c893771h1862002p12597605]8.5[/url] Four different three-digit numbers starting with the same digit have the property that their sum is divisible by three of them without a remainder. Find these numbers. [url=https://artofproblemsolving.com/community/c893771h1861967p12597280]8.6[/url] Given a finite sequence of zeros and ones, which has two properties: a) if in some arbitrary place in the sequence we select five digits in a row and also select five digits in any other place in a row, then these fives will be different (they may overlap); b) if you add any digit to the right of the sequence, then property (a) will no longer hold true. Prove that the first four digits of our sequence coincide with the last four. PS. You should use hide for answers.Collected [url=https://artofproblemsolving.com/community/c3988085_1969_leningrad_math_olympiad]here[/url].

Find the largest number $n$ such that $(2004!)!$ is divisible by $((n!)!)!$.

Find all natural solutions of $3^{x}= 2^{x}y+1.$

A nine-digit telephone number [i]abcdefghi [/i] is called [i]memorizable [/i] if the sequence of four initial digits [i]abcd [/i] is repeated in the sequence of the final five digits [i]efghi[/i]. How many [i]memorizable [/i] numbers of nine digits exist?

Let $p \ge 2$ be a prime number and $a \ge 1$ a positive integer with $p \neq a$. Find all pairs $(a,p)$ such that: $a+p \mid a^2+p^2$

Find all the pairs of positive integers $(x,p)$ such that p is a prime, $x \leq 2p$ and $x^{p-1}$ is a divisor of $ (p-1)^{x}+1$.

For each of the $9$ positive integers $n,2n,3n,\dots , 9n$ Alice take the first decimal digit (from the left) and writes it onto a blackboard. She selected $n$ so that among the nine digits on the blackboard there is the least possible number of different digits. What is this number of different digits equals to?

Let $P (n)$ and $Q (n)$ be two polynomials (not constant) whose coefficients are integers not negative. For each positive integer $n$, define $x_n = 2016^{P (n)} + Q (n)$. Prove that there exist infinite primes $p$ for which there is a positive integer $m$, squarefree, such that $p | x_m$. Clarification: A positive integer is squarefree if it is not divisible by the square of any prime number.

[b]p1.[/b] A number of Exonians took a math test. If all of their scores were positive integers and the mean of their scores was $8.6$, find the minimum possible number of students. [b]p2.[/b] Find the least composite positive integer that is not divisible by any of $3, 4$, and $5$. [b]p3.[/b] Five checkers are on the squares of an $8\times 8$ checkerboard such that no two checkers are in the same row or the same column. How many squares on the checkerboard share neither a row nor a column with any of the five checkers? [b]p4.[/b] Let the operation $x@y$ be $y - x$. Compute $((... ((1@2)@3)@ ...@ 2013)@2014)@2015$. [b]p5.[/b] In a town, each family has either one or two children. According to a recent survey, $40\%$ of the children in the town have a sibling. What fraction of the families in the town have two children? [b]p6.[/b] Equilateral triangles $ABE$, $BCF$, $CDG$ and $DAH$ are constructed outside the unit square $ABCD$. Eliza wants to stand inside octagon $AEBFCGDH$ so that she can see every point in the octagon without being blocked by an edge. What is the area of the region in which she can stand? [b]p7.[/b] Let $S$ be the string $0101010101010$. Determine the number of substrings containing an odd number of $1$'s. (A substring is defined by a pair of (not necessarily distinct) characters of the string and represents the characters between, inclusively, the two elements of the string.) [b]p8.[/b] Let the positive divisors of $n$ be $d_1, d_2, ...$ in increasing order. If $d_6 = 35$, determine the minimum possible value of $n$. [b]p9.[/b] The unit squares on the coordinate plane that have four lattice point vertices are colored black or white, as on a chessboard, shown on the diagram below. [img]https://cdn.artofproblemsolving.com/attachments/6/4/f400d52ae9e8131cacb90b2de942a48662ea8c.png[/img] For an ordered pair $(m, n)$, let $OXZY$ be the rectangle with vertices $O = (0, 0)$, $X = (m, 0)$, $Z = (m, n)$ and $Y = (0, n)$. How many ordered pairs $(m, n)$ of nonzero integers exist such that rectangle $OXZY$ contains exactly $32$ black squares? [b]p10.[/b] In triangle $ABC$, $AB = 2BC$. Given that $M$ is the midpoint of $AB$ and $\angle MCA = 60^o$, compute $\frac{CM}{AC}$ . PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Determine if there exist $101$ positive integers (not necessarily distinct) such that their product is equal to the sum of all their pairwise LCM.

$x,y$ are naturals. $GCM(x^7,y^4)*GCM(x^8,y^5)=xy$ Prove that $xy$ is cube

Alice and Bob are independently trying to figure out a secret password to Cathy’s bitcoin wallet. Both of them have already figured out that: $\bullet$ it is a $4$-digit number whose first digit is $5$. $\bullet$ it is a multiple of $9$; $\bullet$ The larger number is more likely to be a password than a smaller number. Moreover, Alice figured out the second and the third digits of the password and Bob figured out the third and the fourth digits. They told this information to each other but not actual digits. After that the conversation followed: Alice: ”I have no idea what the number is.” Bob: ”I have no idea too.” After that both of them knew which number they should try first. Identify this number

Define a sequence of positive integers $\{a_n\}$ through the recursive formula: $a_{n+1}=a_n+b_n(n\ge 1)$,where $b_n$ is obtained by rearranging the digits of $a_n$ (in decimal representation) in reverse order (for example,if $a_1=250$,then $b_1=52,a_2=302$,and so on). Can $a_7$ be a prime?

Find all $f: \mathbb{N}\rightarrow \mathbb{N}$, for which $f(f(n)+m)=n+f(m+2014)$ for $\forall$ $m,n\in \mathbb{N}$.

a) The product of $n$ integers equals $n$, and their sum is zero. Prove that $n$ is divisible by $4$. b) Let $n$ is divisible by $4$. Prove that there exist $n$ integers such, that their product equals $n$, and their sum is zero.

Let $a, b, c$ be nonzero integers, with $1$ as their only positive common divisor, such that $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}= 0$. Find the number of such triples $(a, b, c)$ with $50 \ge |a| \ge |b| \ge |c| 1$.

An arithmetic progression is formed by $9$ positive integers such that the product of these $9$ terms is a multiple of $3$. Prove that said product is also multiple of $81$.

Prove that: there exists a positive constant $K$, and an integer series $\{a_n\}$, satisfying: $(1)$ $0<a_1<a_2<\cdots <a_n<\cdots $; $(2)$ For any positive integer $n$, $a_n<1.01^n K$; $(3)$ For any finite number of distinct terms in $\{a_n\}$, their sum is not a perfect square.

Find all integers $n > 2$ for which $(2n)! = (n-2)!n!(n+2)!$ .

$a,b,c$ are non-negative integers. Solve: $a!+5^b=7^c$ [i]Proposed by Serbia[/i]

Let $n = 2^k$. Prove that we can select $n$ integers from any $2n-1$ integers such that their sum is divisible by $n$.

A natural number $n$ has exactly $1995$ units in its binary representation. Show that $n!$ is divisible by $2^{n-1995}$.