Let $n > 1$ be an odd integer. Prove that a necessary and sufficient condition for the existence of positive integers $x$ and $y$ satisfying $$\frac{4}{n}=\frac{1}{x}+\frac{1}{y}$$ is that $n$ has a prime divisor of the form $4k - 1$.

Let $a,b,c,d,p$ and $q$ be positive integers satisfying $ad-bc=1$ and $\frac{a}{b}>\frac{p}{q}>\frac{c}{d}$. Prove that: $(a)$ $q\ge b+d$ $(b)$ If $q=b+d$, then $p=a+c$.

Josh is older than Fred. Josh notices that if he switches the two digits of his age (an integer), he gets Fred’s age. Moreover, the difference between the squares of their ages is a square of an integer. How old are Josh and Fred?

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 $p$

Define a [b]ring[/b] in the plane to be the set of points at a distance of at least $r$ and at most $R$ from a specific point $O$, where $r<R$ are positive real numbers. Rings are determined by the three parameters $(O, R, r)$. The area of a ring is labeled $S$. A point in the plane for which both its coordinates are integers is called an integer point. [b]a)[/b] For each positive integer $n$, show that there exists a ring not containing any integer point, for which $S>3n$ and $R<2^{2^n}$. [b]b)[/b] Show that each ring satisfying $100\cdot R<S^2$ contains an integer point.

Find all positive integers $n$ such that the set $\{n,n+1,n+2,n+3,n+4,n+5\}$ can be partitioned into two subsets so that the product of the numbers in each subset is equal.

Let $ a$ be the greatest positive root of the equation $ x^3 \minus{} 3 \cdot x^2 \plus{} 1 \equal{} 0.$ Show that $ \left[a^{1788} \right]$ and $ \left[a^{1988} \right]$ are both divisible by 17. Here $ [x]$ denotes the integer part of $ x.$

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.)

$p, q, r$ are distinct prime numbers which satisfy $$2pqr + 50pq = 7pqr + 55pr = 8pqr + 12qr = A$$ for natural number $A$. Find all values of $A$.

The natural numbers are colored green or blue so that: $\bullet$ The sum of a green and a blue is blue; $\bullet$ The product of a green and a blue is green. How many ways are there to color the natural numbers with these rules, so that $462$ are blue and $2016$ are green?

Find all functions $f: \mathbb{N} \rightarrow \mathbb{Z}$ such that $|f(k)| \leq k$ for all positive integers $k$ and there is a prime number $p>2024$ which satisfies both of the following conditions: $1)$ For all $a \in \mathbb{N}$ we have $af(a+p) = af(a)+pf(a),$ $2)$ For all $a \in \mathbb{N}$ we have $p|a^{\frac{p+1}{2}}-f(a).$ [i]Authored by Nikola Velov[/i]

Let $k\geq 2$,$n_1,n_2,\cdots ,n_k\in \mathbb{N}_+$,satisfied $n_2|2^{n_1}-1,n_3|2^{n_2}-1,\cdots ,n_k|2^{n_{k-1}}-1,n_1|2^{n_k}-1$. Prove:$n_ 1=n_ 2=\cdots=n_k=1$.

[hide=D stands for Descartes, L stands for Leibniz]they had two problem sets under those two names[/hide] [b]L.10[/b] Given the following system of equations where $x, y, z$ are nonzero, find $x^2 + y^2 + z^2$. $$x + 2y = xy$$ $$3y + z = yz$$ $$3x + 2z = xz$$ [u]Set 4[/u] [b]L.16 / D.23[/b] Anson, Billiam, and Connor are looking at a $3D$ figure. The figure is made of unit cubes and is sitting on the ground. No cubes are floating; in other words, each unit cube must either have another unit cube or the ground directly under it. Anson looks from the left side and says, “I see a $5 \times 5$ square.” Billiam looks from the front and says the same thing. Connor looks from the top and says the same thing. Find the absolute difference between the minimum and maximum volume of the figure. [b]L.17[/b] The repeating decimal $0.\overline{MBMT}$ is equal to $\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers, and $M, B, T$ are distinct digits. Find the minimum value of $q$. [b]L.18[/b] Annie, Bob, and Claire have a bag containing the numbers $1, 2, 3, . . . , 9$. Annie randomly chooses three numbers without replacement, then Bob chooses, then Claire gets the remaining three numbers. Find the probability that everyone is holding an arithmetic sequence. (Order does not matter, so $123$, $213$, and $321$ all count as arithmetic sequences.) [b]L.19[/b] Consider a set $S$ of positive integers. Define the operation $f(S)$ to be the smallest integer $n > 1$ such that the base $2^k$ representation of $n$ consists only of ones and zeros for all $k \in S$. Find the size of the largest set $S$ such that $f(S) < 2^{2019}$. [b]L.20 / D.25[/b] Find the largest solution to the equation $$2019(x^{2019x^{2019}-2019^2+2019})^{2019} = 2019^{x^{2019}+1}.$$ [u]Set 5[/u] [b]L.21[/b] Steven is concerned about his artistic abilities. To make himself feel better, he creates a $100 \times 100$ square grid and randomly paints each square either white or black, each with probability $\frac12$. Then, he divides the white squares into connected components, groups of white squares that are connected to each other, possibly using corners. (For example, there are three connected components in the following diagram.) What is the expected number of connected components with 1 square, to the nearest integer? [img]https://cdn.artofproblemsolving.com/attachments/e/d/c76e81cd44c3e1e818f6cf89877e56da2fc42f.png[/img] [b]L.22[/b] Let x be chosen uniformly at random from $[0, 1]$. Let n be the smallest positive integer such that $3^n x$ is at most $\frac14$ away from an integer. What is the expected value of $n$? [b]L.23[/b] Let $A$ and $B$ be two points in the plane with $AB = 1$. Let $\ell$ be a variable line through $A$. Let $\ell'$ be a line through $B$ perpendicular to $\ell$. Let X be on $\ell$ and $Y$ be on $\ell'$ with $AX = BY = 1$. Find the length of the locus of the midpoint of $XY$ . [b]L.24[/b] Each of the numbers $a_i$, where $1 \le i \le n$, is either $-1$ or $1$. Also, $$a_1a_2a_3a_4+a_2a_3a_4a_5+...+a_{n-3}a_{n-2}a_{n-1}a_n+a_{n-2}a_{n-1}a_na_1+a_{n-1}a_na_1a_2+a_na_1a_2a_3 = 0.$$ Find the number of possible values for $n$ between $4$ and $100$, inclusive. [b]L.25[/b] Let $S$ be the set of positive integers less than $3^{2019}$ that have only zeros and ones in their base $3$ representation. Find the sum of the squares of the elements of $S$. Express your answer in the form $a^b(c^d - 1)(e^f - 1)$, where $a, b, c, d, e, f$ are positive integers and $a, c, e$ are not perfect powers. PS. You should use hide for answers. D.1-15 / L1-9 problems have been collected [url=https://artofproblemsolving.com/community/c3h2790795p24541357]here [/url] and D.16-30/ L10-15 [url=https://artofproblemsolving.com/community/c3h2790818p24541688]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Find all solutions of $2^n + 7 = x^2$ in which n and x are both integers . Prove that there are no other solutions.

a) Show that $m^2- m +1$ is an element of the set $\{n^2 + n +1 | n \in N\}$, for any positive integer $ m$. b) Let $p$ be a perfect square, $p> 1$. Prove that there exists positive integers $r$ and $q$ such that $$p^2 + p +1=(r^2 + r + 1)(q^2 + q + 1).$$

Let there be $131$ given distinct natural numbers, each having prime divisors not exceeding $42$. Prove that one can choose four of them whose product is a perfect square.

Distinct prime numbers $p,q,r$ satisfy the equation $$2pqr+50pq=7pqr+55pr=8pqr+12qr=A$$ for some positive integer $A.$ What is $A$?

Let $ABC$ be a triangle with sides 3, 4, and 5, and $DEFG$ be a 6-by-7 rectangle. A segment is drawn to divide triangle $ABC$ into a triangle $U_1$ and a trapezoid $V_1$ and another segment is drawn to divide rectangle $DEFG$ into a triangle $U_2$ and a trapezoid $V_2$ such that $U_1$ is similar to $U_2$ and $V_1$ is similar to $V_2$. The minimum value of the area of $U_1$ can be written in the form $m/n$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

For any odd prime $p$ and any integer $n,$ let $d_p (n) \in \{ 0,1, \dots, p-1 \}$ denote the remainder when $n$ is divided by $p.$ We say that $(a_0, a_1, a_2, \dots)$ is a [i]p-sequence[/i], if $a_0$ is a positive integer coprime to $p,$ and $a_{n+1} =a_n + d_p (a_n)$ for $n \geqslant 0.$ (a) Do there exist infinitely many primes $p$ for which there exist $p$-sequences $(a_0, a_1, a_2, \dots)$ and $(b_0, b_1, b_2, \dots)$ such that $a_n >b_n$ for infinitely many $n,$ and $b_n > a_n$ for infinitely many $n?$ (b) Do there exist infinitely many primes $p$ for which there exist $p$-sequences $(a_0, a_1, a_2, \dots)$ and $(b_0, b_1, b_2, \dots)$ such that $a_0 <b_0,$ but $a_n >b_n$ for all $n \geqslant 1?$ [I]United Kingdom[/i]

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]

a) Prove that if the sum of the non-zero digits $a_1, a_2, ... , a_n$ is a multiple of $27$, then it is possible to permute these digits in order to obtain an $n$-digit number that is a multiple of $27$. b) Prove that if the non-zero digits $a_1, a_2, ... , a_n$ have the property that every ndigit number obtained by permuting these digits is a multiple of $27$, then the sum of these digits is a multiple of $27$

Let $P(x) = 2x^3-3x^2+2$, and the sets: $$A =\{ P(n) | n \in N, n \le 1999\}, B=\{p^2+1 |p \in N\}, C=\{ q^2+2 | q \in N\}$$ Prove that the sets $A \cap B$ and $A\cap C$ have the same number of elements

Let $n$ be a positive integer divisible by $39$. What is the smallest possible sum of digits that $n$ can have (in base $10$)?

For positive integers $a, b, c$ (not necessarily distinct), suppose that $a+bc, b+ac, c+ab$ are all perfect squares. Show that $$a^2(b+c)+b^2(a+c)+c^2(a+b)+2abc$$ can be written as sum of two squares.

Prove that, if the terms of an infinite arithmetic progression of natural numbers are not all equal, they cannot all be primes.