Find all positive integers $ a$ and $ b$ such that $ \frac{a^{4}\plus{}a^{3}\plus{}1}{a^{2}b^{2}\plus{}ab^{2}\plus{}1}$ is an integer.

Call a positive integer [i]balanced [/i] if the number of its distinct prime factors is equal to the number of its digits in the decimal representation; for example, the number $385 = 5 \cdot 7 \cdot 11$ is balanced, while $275 = 5^2 \cdot 11$ is not. Prove that there exist only a finite number of balanced numbers.

Let $ x \in \mathbb{Q}^+$. Prove that there exits $\alpha_1,\alpha_2,...,\alpha_k \in \mathbb{N}$ and pairwe distinct such that \[x= \sum_{i=1}^{k} \frac{1}{\alpha_i}\]

We write $\{a,b,c\}$ for the set of three different positive integers $a, b$, and $c$. By choosing some or all of the numbers a, b and c, we can form seven nonempty subsets of $\{a,b,c\}$. We can then calculate the sum of the elements of each subset. For example, for the set $\{4,7,42\}$ we will find sums of $4, 7, 42,11, 46, 49$, and $53$ for its seven subsets. Since $7, 11$, and $53$ are prime, the set $\{4,7,42\}$ has exactly three subsets whose sums are prime. (Recall that prime numbers are numbers with exactly two different factors, $1$ and themselves. In particular, the number $1$ is not prime.) What is the largest possible number of subsets with prime sums that a set of three different positive integers can have? Give an example of a set $\{a,b,c\}$ that has that number of subsets with prime sums, and explain why no other three-element set could have more.

Given positive integers $m$ and $n$, prove that there is a positive integer $c$ such that the numbers $cm$ and $cn$ have the same number of occurrences of each non-zero digit when written in base ten.

Is there a power of $2$ such that it is possible to rearrange the digits, giving another power of $2$?

A natural number is called [i]good[/i] if it can be represented as sum of two coprime natural numbers, the first of which decomposes into odd number of primes (not necceserily distinct) and the second to even. Prove that there exist infinity many $n$ with $n^4$ being good.

Find all $n\in \mathbb{N} $ so that 7 divides $5^n + 1$

Let $k\geq 2$ and $n_1, \dots, n_k \in \mathbf{Z}^+$. If $n_2 | (2^{n_1} -1)$, $n_3 | (2^{n_2} -1)$, $\dots$, $n_k | (2^{n_{k-1}} -1)$, $n_1 | (2^{n_k} -1)$, show that $n_1 = \dots = n_k =1$.

[b]p1.[/b] A [i]multimagic [/i] square is a $3 \times 3$ array of distinct positive integers with the property that the product of the $3$ numbers in each row, each column, and each of the two diagonals of the array is always the same. (a) Prove that the numbers $1, 2, 3, . . . , 9$ cannot be used to form a multimagic square. (b) Give an example of a multimagic square. [b]p2.[/b] A sequence $a_1, a_2, a_3, ... , a_n$ of real numbers is called an arithmetic progression if $$a_1 - a_2 = a_2 - a_3 = ... = a_{n-1} - a_n.$$ Prove that there exist distinct positive integers $n_1, n_2, n_3, ... , n_{2014}$ such that $$\frac{1}{n_1},\frac{1}{n_2}, ... ,\frac{1}{n_{2014}}$$ is an arithmetic progression. [b]p3.[/b] Let $\lfloor x \rfloor$ be the largest integer that is less than or equal to $x$. For example, $\lfloor 3.9 \rfloor = 3$ and $\lfloor 4\rfloor = 4$. Determine (with proof) all real solutions of the equation $$x^2 - 25 \lfloor x\rfloor + 100 = 0.$$ [b]p4.[/b] An army has $10$ cannons and $8$ carts. Each cart can carry at most one cannon. It takes one day for a cart to cross the desert. What is the least number of days that it takes to get the cannons across the desert? (Cannons can be left part way and picked up later during the procedure.) Prove that the amount of time that your solution requires to move the cannons across the desert is the smallest possible. [b]p5.[/b] Let $C$ be a convex polygon with $4031$ sides. Let $p$ be the length of its perimeter and let $d$ be the sum of the lengths of its diagonals. Show that $$\frac{d}{p}> 2014.$$ PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

$10000$ nonzero digits are written in a $100$-by-$100$ table, one digit per cell. From left to right, each row forms a $100$-digit integer. From top to bottom, each column forms a $100$-digit integer. So the rows and columns form $200$ integers (each with $100$ digits), not necessarily distinct. Prove that if at least $199$ of these $200$ numbers are divisible by $2013$, then all of them are divisible by $2013$.

[b]p1.[/b] Solve: $INK + INK + INK + INK + INK + INK = PEN$ ($INK$ and $PEN$ are $3$-digit numbers, and different letters stand for different digits). [b]p2. [/b]Two people play a game. They put $3$ piles of matches on the table: the first one contains $1$ match, the second one $3$ matches, and the third one $4$ matches. Then they take turns making moves. In a move, a player may take any nonzero number of matches FROM ONE PILE. The player who takes the last match from the table loses the game. a) The player who makes the first move can win the game. What is the winning first move? b) How can he win? (Describe his strategy.) [b]p3.[/b] The planet Naboo is under attack by the imperial forces. Three rebellion camps are located at the vertices of a triangle. The roads connecting the camps are along the sides of the triangle. The length of the first road is less than or equal to $20$ miles, the length of the second road is less than or equal to $30$ miles, and the length of the third road is less than or equal to $45$ miles. The Rebels have to cover the area of this triangle with a defensive field. What is the maximal area that they may need to cover? [b]p4.[/b] Money in Wonderland comes in $\$5$ and $\$7$ bills. What is the smallest amount of money you need to buy a slice of pizza that costs $\$ 1$ and get back your change in full? (The pizza man has plenty of $\$5$ and $\$7$ bills.) For example, having $\$7$ won't do, since the pizza man can only give you $\$5$ back. [b]p5.[/b] (a) Put $5$ points on the plane so that each $3$ of them are vertices of an isosceles triangle (i.e., a triangle with two equal sides), and no three points lie on the same line. (b) Do the same with $6$ points. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $x_n=\binom{2n}{n}$ for all $n\in\mathbb{Z}^+$. Prove there exist infinitely many finite sets $A,B$ of positive integers, satisfying $A \cap B = \emptyset $, and \[\frac{{\prod\limits_{i \in A} {{x_i}} }}{{\prod\limits_{j\in B}{{x_j}} }}=2012.\]

Let $A=(a_{ij})$ be the $n\times n$ matrix, where $a_{ij}$ is the remainder of the division of $i^j+j^i$ by $3$ for $i,j=1,2,\ldots,n$. Find the greatest $n$ for which $\det A\ne0$.

Let $n$ be a positive integer and let $a, b, c, d$ be integers such that $n | a + b + c + d$ and $n | a^2 + b^2 + c^2 + d^2. $ Show that $$n | a^4 + b^4 + c^4 + d^4 + 4abcd.$$

In a family there are four children of different ages, each age being a positive integer not less than $2$ and not greater than $16$. A year ago the square of the age of the eldest child was equal to the sum of the squares of the ages of the remaining children. One year from now the sum of the squares of the youngest and the oldest will be equal to the sum of the squares of the other two. How old is each child?

[b]6.1[/b] Three shooters - Anilov, Borisov and Vorobiev - made $6$ each shots at one target and scored equal points. It is known that Anilov scored $43$ points in the first three shots, and Borisov scored $43$ points with the first shot knocked out 3 points. How many points did each shooter score per shot? if there was one hit in 50, two in 25, three in 20, three in 10, two in 5, in 3 - two, in 2 - two, in 1 - three? [img]https://cdn.artofproblemsolving.com/attachments/a/1/4abb71f7bccc0b9d2e22066ec17c31ef139d6a.png[/img] [b]6.2 / 7.4 [/b]Prove that a $10 \times 10$ chessboard cannot be covered with $ 25$ figures like [img]https://cdn.artofproblemsolving.com/attachments/0/4/89aafe1194628332ec13ad1c713bb35cbefff7.png[/img]. [b]6.3[/b] The squares of a chessboard contain natural numbers such that each is equal to the arithmetic mean of its neighbors. Sum of numbers standing in the corners of the board is $16$. Find the number standing on the field $e2$. [b]6.4 [/b] There is a table $ 100 \times 100$. What is the smallest number of letters which can be arranged in its cells so that no two are identical the letters weren't next to each other? [b]6.5[/b] The pioneer detachment is lined up in a rectangle. In each rank the tallest is noted, and from these pioneers the most short. In each row, the lowest one is noted, and from them is selected the tallest. Which of these two pioneers is taller? (This means that the two pioneers indicated are the highest of the low and the lowest of tall - must be different) [b]6.6[/b] Find the product of three numbers whose sum is equal to the sum of their squares, equal to the sum of their cubes and equal to $1$. PS. You should use hide for answers.Collected [url=https://artofproblemsolving.com/community/c3983461_1964_leningrad_math_olympiad]here[/url].

Let $a_1, \dots, a_n, b_1, \dots, b_n$ be $2n$ positive integers such that the $n+1$ products \[a_1 a_2 a_3 \cdots a_n, b_1 a_2 a_3 \cdots a_n, b_1 b_2 a_3 \cdots a_n, \dots, b_1 b_2 b_3 \cdots b_n\] form a strictly increasing arithmetic progression in that order. Determine the smallest possible integer that could be the common difference of such an arithmetic progression.

If $p$ and $q$ are natural numbers so that \[ \frac{p}{q}=1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+ \ldots -\frac{1}{1318}+\frac{1}{1319}, \] prove that $p$ is divisible with $1979$.

The sequence $\{x_n\}_{n\ge1}$ is defined by $x_1 = 7$, $x_{n+1} = 2x^2_n - 1$, for all positive integers $n$. Prove that for all positive integers $n$ the number $x_n$ cannot be divisible by $2003$.

Find the largest integer $n$, where $2009^n$ divides $2008^{2009^{2010}} + 2010^{2009^{2008}}$ .

Show that it is possible to express $1$ as a sum of $6$, and as a sum of $9$ reciprocals of odd positive integers. Generalize the problem.

Solve in integers the following equation $\frac{1}{2}(x + y)(y + z)(z + x) + (x + y + z)^3 = 1 - xyz$.

For each positive integer $x$, let $\varphi(x)$ be the number of integers $1 \leq k \leq x$ that do not have prime factors in common with $x$. Determine all positive integers $n$ such that there are distinct positive integers $a_1,a_2, \ldots, a_n$ so that the set: $$S = \{a_1, a_2, \ldots, a_n, \varphi(a_1), \varphi(a_2), \ldots, \varphi(a_n)\}$$ Have exactly $2n$ consecutive integers (in some order).

Let $p$ be a prime number and $n \ge 2$ a positive integer, such that $p | (n^6 -1)$. Prove that $n > \sqrt{p}-1$.