Find all the prime numbers $p$ and $q$ such that $ p^2+q=37q^2+p $. Clarification: $1$ is not a prime number.

For $p$, $q$, $l$ strictly positive real numbers, consider the following problem: for $y \geq 0$ fixed, determine the values $x \geq 0$ such that $x^p - lx^q \leq y$. Denote by $S(y)$ the set of solutions of this problem. Prove that if one has $p < q$, $\epsilon \in (0, l^\frac{1}{p-q})$, $0 \leq x \leq \epsilon$ and $x \in S(y)$, then $$x\leq ky^{\delta},\ \text{where}\ k=\epsilon\left(\epsilon^p-l\epsilon^q\right)^{-\frac{1}{p}}\ \text{and}\ \delta=\frac{1}{p}$$

Prove that for any positive integer $n > 2$ we can find $n$ distinct positive integers, the sum of whose reciprocals is equal to $1$.

For any two positive integers $n>m$ prove the following inequality: $$[m,n]+[m+1,n+1]\geq \dfrac{2nm}{\sqrt{m-n}}$$ As always, $[x,y]$ means the least common multiply of $x,y$. [I]Proposed by A. Golovanov[/i]

Find all pairs of positive integers $a,b$ with $$a^a+b^b \mid (ab)^{|a-b|}-1.$$

Let $a_1, a_2,\cdots
,a_n$ be positive integers and let $a$ be an integer greater than $1$ and divisible by the product $a_1a_2\cdots a_n$. Prove that $a^{n+1} + a-1$ is not divisible by the product $(a + a_1 - 1)(a + a_2 - 1) \cdots
(a + a_n - 1)$.

What is the largest prime factor of $7999488$?

Determine all pairs $(x, y)$ of positive integers such that \[\sqrt[3]{7x^2-13xy+7y^2}=|x-y|+1.\] [i]Proposed by Titu Andreescu, USA[/i]

Let $m,n$ be positive integers of distinct parities and such that $m<n<5m$. Show that there exists a partition with two element subsets of the set $\{ 1,2,3,\ldots ,4mn\}$ such that the sum of numbers in each set is a perfect square. [i]Dinu Șerbănescu[/i]

Determine all pairs of non-negative integers $(m, n)$ with m ≥n, such that $(m+n)^3$ divides $2n(3m^2+n^2)+8$

[b]p1.[/b] Two players play the following game. On the lowest left square of an $8 \times 8$ chessboard there is a rook (castle). The first player is allowed to move the rook up or to the right by an arbitrary number of squares. The second layer is also allowed to move the rook up or to the right by an arbitrary number of squares. Then the first player is allowed to do this again, and so on. The one who moves the rook to the upper right square wins. Who has a winning strategy? [b]p2.[/b] Find the smallest positive whole number that ends with $17$, is divisible by $17$, and the sum of its digits is $17$. [b]p3.[/b] Three consecutive $2$-digit numbers are written next to each other. It turns out that the resulting $6$-digit number is divisible by $17$. Find all such numbers. [b]p4.[/b] Let $ABCD$ be a convex quadrilateral (a quadrilateral $ABCD$ is called convex if the diagonals $AC$ and $BD$ intersect). Suppose that $\angle CBD = \angle CAB$ and $\angle ACD = \angle BDA$ . Prove that $\angle ABC = \angle ADC$. [b]p5.[/b] A circle of radius $1$ is cut into four equal arcs, which are then arranged to make the shape shown on the picture. What is its area? [img]https://cdn.artofproblemsolving.com/attachments/f/3/49c3fe8b218ab0a5378ecc635b797a912723f9.png[/img] PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Prove that the cube of any positive integer greater than 1 can be represented as a difference of the squares of two positive integers.

We define the following sequences: • Sequence $A$ has $a_n = n$. • Sequence $B$ has $b_n = a_n$ when $a_n \not\equiv 0$ (mod 3) and $b_n = 0$ otherwise. • Sequence $C$ has $c_n =\sum_{i=1}^{n} b_i$ .• Sequence $D$ has $d_n = c_n$ when $c_n \not\equiv 0$ (mod 3) and $d_n = 0$ otherwise. • Sequence $E$ has $e_n =\sum_{i=1}^{n}d_i$ Prove that the terms of sequence E are exactly the perfect cubes.

For each prime $p$, construct a graph $G_p$ on $\{1,2,\ldots p\}$, where $m\neq n$ are adjacent if and only if $p$ divides $(m^{2} + 1-n)(n^{2} + 1-m)$. Prove that $G_p$ is disconnected for infinitely many $p$

Given are positive integers $a, b$ satisfying $a \geq 2b$. Does there exist a polynomial $P(x)$ of degree at least $1$ with coefficients from the set $\{0, 1, 2, \ldots, b-1 \}$ such that $P(b) \mid P(a)$?

High school graduate Igor Petrov, who dreamed of becoming a diplomat, took the entrance exam in mathematics to Moscow University. Igor remembered all the problems offered during the exam, but forgot some numerical data in one. This is the task: “When multiplying two natural numbers, the difference of which is $10$, an error was made: the hundreds digit in the product was increased by $2$. When dividing the resulting (incorrect) product by the smaller of the factors, the result was quotient $k$ and remainder $r$.. Find the numbers that needed to be multiplied.” . The values of $k$ and $r$ were given in the condition, but Igor forgot them. However, he remembered that the problem had two answers. What could the numbers $ k$ and $r$ be equal to (they are both integers and positive)? [i]Note. The problem in question was proposed at one of the humanities faculties of Moscow State University in 1991. [/i]

Prove that for each positive integer n,the equation $x^{2}+15y^{2}=4^{n}$ has at least $n$ integer solution $(x,y)$

Given a sequence of prime numbers $p_1, p_2,\cdots$ , with the following property: $p_{n+2}$ is the largest prime divisor of $p_n+p_{n+1}+2018$ Show that the set $\{p_i\}_{i\in \mathbb{N}}$ is finite.

Determine the number of positive integers $n$ for which $(n+15)(n+2010)$ is a perfect square.

Ana writes an infinite list of numbers using the following procedure. The first number of the list is a positive integer $a$ chosen by Ana. From there, each number in the list is obtained by calculating the sum of all the integers from $1$ to the last number written. For example, if $a = 3$, Ana's list starts as $3, 6, 21, 231, \dots$ because $1 + 2 + 3 = 6$, $1 + 2 + 3 + 4 + 5 + 6 = 21$ and $1 + 2 + 3 + \dots + 21 = 231$. Is it possible for all the numbers in Ana's list to be even?

Let $ n > 1$ be an integer, and $ n$ can divide $ 2^{\phi(n)} \plus{} 3^{\phi(n)} \plus{} \cdots \plus{} n^{\phi(n)},$ let $ p_{1},p_{2},\cdots,p_{k}$ be all distinct prime divisors of $ n$. Show that $ \frac {1}{p_{1}} \plus{} \frac {1}{p_{2}} \plus{} \cdots \plus{} \frac {1}{p_{k}} \plus{} \frac {1}{p_{1}p_{2}\cdots p_{k}}$ is an integer. ( where $ \phi(n)$ is defined as the number of positive integers $ \leq n$ that are relatively prime to $ n$.)

[b] Problem 5. [/b]Denote with $d(a,b)$ the numbers of the divisors of natural $a$, which are greater or equal to $b$. Find all natural $n$, for which $d(3n+1,1)+d(3n+2,2)+\ldots+d(4n,n)=2006.$ [i]Ivan Landgev[/i]

Let $m,n$ are positive integers. a)Prove that $(m,n)=2\sum_{k=0}^{m-1}[\frac{kn}{m}]+m+n-mn$. b)If $m,n\geq 2$, prove that $\sum_{k=0}^{m-1}[\frac{kn}{m}]=\sum_{k=0}^{n-1}[\frac{km}{n}]$.

[b]p1.[/b] Show that for every $n \ge 6$, a square in the plane may be divided into $n$ smaller squares, not necessarily all of the same size. [b]p2.[/b] Let $n$ be the $4018$-digit number $111... 11222...2225$, where there are $2008$ ones and $2009$ twos. Prove that $n$ is a perfect square. (Giving the square root of $n$ is not sufficient. You must also prove that its square is $n$.) [b]p3.[/b] Let $n$ be a positive integer. A game is played as follows. The game begins with $n$ stones on the table. The two players, denoted Player I and Player II (Player I goes first), alternate in removing from the table a nonzero square number of stones. (For example, if $n = 26$ then in the first turn Player I can remove $1$ or $4$ or $9$ or $16$ or $25$ stones.) The player who takes the last stone wins. Determine if the following sentence is TRUE or FALSE and prove your answer: There are infinitely many starting values n such that Player II has a winning strategy. (Saying that Player II has a winning strategy means that no matter how Player I plays, Player II can respond with moves that lead to a win for Player II.) [b]p4.[/b] Consider a convex quadrilateral $ABCD$. Divide side $AB$ into $8$ equal segments $AP_1$, $P_1P_2$, $...$ , $P_7B$. Divide side $DC$ into $8$ equal segments $DQ_1$, $Q_1Q_2$, $...$ , $Q_7C$. Similarly, divide each of sides $AD$ and $BC$ into $8$ equal segments. Draw lines to form an $8 \times 8$ “checkerboard” as shown in the picture. Color the squares alternately black and white. (a) Show that each of the $7$ interior lines $P_iQ_i$ is divided into $8$ equal segments. (b) Show that the total area of the black regions equals the total area of the white regions. [img]https://cdn.artofproblemsolving.com/attachments/1/4/027f02e26613555181ed93d1085b0e2de43fb6.png[/img] [b]p5.[/b] Prove that exactly one of the following two statements is true: A. There is a power of $10$ that has exactly $2008$ digits in base $2$. B. There is a power of $10$ that has exactly $2008$ digits in base $5$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Three students write on the blackboard next to each other three two-digit squares. In the end, they observe that the 6-digit number thus obtained is also a square. Find this number!