Positive integers $a$, $b$ and $c$ are positive integers with greatest common divisor equal to $1$ (i.e. they have no common divisors greater than $1$), and $$\frac{ab}{a-b}=c$$ Prove that $a -b$ is a perfect square. (SL Berlov)

Prove that there exist distinct positive integers $a_1, a_2,\ldots , a_{2024}$ such that for each $i\in\{1,2,\ldots,2024\}$\[a_i\mid a_1a_2\cdots a_{i-1}a_{i+1}\cdots a_{2024}+k,\]where a) $k=1$ and b) $k=2024.$ [i]Proposed by Ishan Nath[/i]

Prove that any rational $r \in (0, 1)$ can be written uniquely in the form $$r=\frac{a_1}{1!}+\frac{a_2}{2!}+\frac{a_3}{3!}+\cdots+\frac{a_k}{k!}$$ where $a_i\text{’s}$ are nonnegative integers with $a_i\leq i-1$ for all $i$.

Solve over non-negative integers the system $$ \begin{cases} x+y+z^2=xyz, \\ z\leq min(x,y). \end{cases} $$

Let $n\ge 2$ be an integer. Given numbers $a_1, a_2, \ldots, a_n \in \{1,2,3,\ldots,2n\}$ such that $\operatorname{lcm}(a_i,a_j)>2n$ for all $1\le i<j\le n$, prove that $$a_1a_2\ldots a_n \mid (n+1)(n+2)\ldots (2n-1)(2n).$$

Prove that there exist infinitely many positive integers $ n$ such that $ p \equal{} nr,$ where $ p$ and $ r$ are respectively the semiperimeter and the inradius of a triangle with integer side lengths.

Solve the following problems - A) Find any $158$ consecutive integers such that the sum of digits for any of the numbers is not divisible by $17.$ B) Prove that, among any $159$ consecutive integers there will always be at least one integer whose sum of digits is divisible by $17.$

Let $f(x) = x^2 - x + 1$. Prove that for every natural $m>1$ the numbers $$m, f(m), f(f(m)), ...$$ are relatively prime.

* The quadratic expression $ax^2 + bx + c$ is a square (of an integer) for any integer $x$. Prove that $ax^2 + bx + c = (dx + e)^2$ for some integers d and e.

[u]Round 5[/u] [b]p13.[/b] In coordinate space, a lattice point is a point all of whose coordinates are integers. The lattice points $(x, y, z)$ in three-dimensional space satisfying $0 \le x, y, z \le 5$ are colored in n colors such that any two points that are $\sqrt3$ units apart have different colors. Determine the minimum possible value of $n$. [b]p14.[/b] Determine the number of ways to express $121$ as a sum of strictly increasing positive Fibonacci numbers. [b]p15.[/b] Let $ABCD$ be a rectangle with $AB = 7$ and $BC = 15$. Equilateral triangles $ABP$, $BCQ$, $CDR$, and $DAS$ are constructed outside the rectangle. Compute the area of quadrilateral $P QRS$. [u] Round 6[/u] Each of the three problems in this round depends on the answer to one of the other problems. There is only one set of correct answers to these problems; however, each problem will be scored independently, regardless of whether the answers to the other problems are correct. [b]p16.[/b] Let $C$ be the answer to problem $18$. Suppose that $x$ and $y$ are real numbers with $y > 0$ and $$x + y = C$$ $$x +\frac{1}{y} = -2.$$ Compute $y +\frac{1}{y}$. [b]p17.[/b] Let $A$ be the answer to problem $16$. Let $P QR$ be a triangle with $\angle P QR = 90^o$, and let $X$ be the foot of the perpendicular from point $Q$ to segment $P R$. Given that $QX = A$, determine the minimum possible area of triangle $PQR$. [b]p18.[/b] Let $B$ be the answer to problem $17$ and let $K = 36B$. Alice, Betty, and Charlize are identical triplets, only distinguishable by their hats. Every day, two of them decide to exchange hats. Given that they each have their own hat today, compute the probability that Alice will have her own hat in $K$ days. [u]Round 7[/u] [b]p19.[/b] Find the number of positive integers a such that all roots of $x^2 + ax + 100$ are real and the sum of their squares is at most $2013$. [b]p20.[/b] Determine all values of $k$ such that the system of equations $$y = x^2 - kx + 1$$ $$x = y^2 - ky + 1$$ has a real solution. [b]p21.[/b] Determine the minimum number of cuts needed to divide an $11 \times 5 \times 3$ block of chocolate into $1\times 1\times 1$ pieces. (When a block is broken into pieces, it is permitted to rotate some of the pieces, stack some of the pieces, and break any set of pieces along a vertical plane simultaneously.) [u]Round 8[/u] [b]p22.[/b] A sequence that contains the numbers $1, 2, 3, ... , n$ exactly once each is said to be a permutation of length $n$. A permutation $w_1w_2w_3... w_n$ is said to be sad if there are indices $i < j < k$ such that $w_j > w_k$ and $w_j > w_i$. For example, the permutation $3142756$ is sad because $7 > 6$ and $7 > 1$. Compute the number of permutations of length $11$ that are not sad. [b]p23.[/b] Let $ABC$ be a triangle with $AB = 39$, $BC = 56$, and $CA = 35$. Compute $\angle CAB - \angle ABC$ in degrees. [b]p24.[/b] On a strange planet, there are $n$ cities. Between any pair of cities, there can either be a one-way road, two one-way roads in different directions, or no road at all. Every city has a name, and at the source of every one-way road, there is a signpost with the name of the destination city. In addition, the one-way roads only intersect at cities, but there can be bridges to prevent intersections at non-cities. Fresh Mann has been abducted by one of the aliens, but Sophy Moore knows that he is in Rome, a city that has no roads leading out of it. Also, there is a direct one-way road leading from each other city to Rome. However, Rome is the secret police’s name for the so-described city; its official name, the name appearing on the labels of the one-way roads, is unknown to Sophy Moore. Sophy Moore is currently in Athens and she wants to head to Rome in order to rescue Fresh Mann, but she does not know the value of $n$. Assuming that she tries to minimize the number of roads on which she needs to travel, determine the maximum possible number of roads that she could be forced to travel in order to find Rome. Express your answer as a function of $n$. PS. You should use hide for answers. Rounds 1-4 have been posted [url=https://artofproblemsolving.com/community/c4h2809419p24782489]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $ p\geq 5$ be a prime number.Prove that for any partition of the set $ P\equal{}\{1,2,3,...,p\minus{}1\}$ in $ 3$ subsets there exists numbers $ x,y,z$ each belonging to a distinct subset,such that $ x\plus{}y\equiv z (mod p)$

Determine all pairs $(m, n)$ of non-negative integers that satisfy the equation $$ 20^m - 10m^2 + 1 = 19^n. $$

The last digit of number $2017^{2017} - 2013^{2015}$ is (A): $2$, (B): $4$, (C): $6$, (D): $8$, (E): None of the above.

We say that a positive integer is [i]decomposed [/i] if it is prime and also If a line is drawn separating it into two numbers, those two numbers are never composite. For example 1997 is [i]decomposed [/i] since it is prime, it is divided into: $1$, $997$; $19$, $97$; $199$, $7$ and none of those numbers are compound. How many [i]decomposed [/i] numbers are there between $2000$ and $3000$?

How many quadruples of positive integers $(a,b,m,n)$ are there such that all of the following statements hold? 1. $a,b<5000$ 2. $m,n<22$ 3. $gcd(m,n)=1$ 4. $(a^2+b^2)^m=(ab)^n$

On the board are written the natural numbers from $1$ to $2001$ inclusive. You have to delete some numbers so that among those that remain undeleted it is impossible to choose two different numbers such that the result of their multiplication is equal to one of the numbers that remain undeleted. What is the minimum number of numbers that must be deleted? For that amount, present an example showing which numbers are erased. Justify why, if fewer numbers are deleted, the desired property is not obtained.

Solve the equation $xy =1997(x + y)$ in integers.

Ten different odd primes are given. Is it possible that for any two of them, the difference of their sixteenth powers to be divisible by all the remaining ones ?

Let pairwise different positive integers $a,b, c$ with gcd$(a,b,c) = 1$ are such that $a | (b - c)^2, b | (c- a)^2, c | (a - b)^2$. Prove, that there is no non-degenerate triangle with side lengths $a, b$ and $c$.

Find the smallest possible sum $a + b + c + d + e$ where $a, b, c, d,$ and $e$ are positive integers satisfying the conditions $\star$ each of the pairs of integers $(a, b), (b, c), (c, d),$ and $(d, e)$ are [b]not[/b] relatively prime $\star$ all other pairs of the five integers [b]are[/b] relatively prime.

Let $P(x) = x^{16}-x^{15}+·...-x+ 1$, and let p be a prime such that $p-1$ is divisible by $34$ ($p = 103$ is an example). How many integers a between $1$ and $ p-1$ inclusive satisfy the property that $P(a)$ is divisible by $p$?

Determine all pairs $(a, b)$ of integers having the following property: there is an integer $d \ge 2$ such that $a^n + b^n + 1$ is divisible by $d$ for all positive integers $n$.

A positive integer $a$ is said to [i]reduce[/i] to a positive integer $b$ if when dividing $a$ by its units digits the result is $b$. For example, 2015 reduces to $\frac{2015}{5} = 403$. Find all the positive integers that become 1 after some amount of reductions. For example, 12 is one such number because 12 reduces to 6 and 6 reduces to 1.

Find all non-negative integers $m,n,p,q$ such that \[ p^mq^n = (p+q)^2 +1 . \]

Show that an integer $p > 3$ is a prime if and only if for every two nonzero integers $a,b$ exactly one of the numbers $N_1 = a+b-6ab+\frac{p-1}{6}$ , $N_2 = a+b+6ab+\frac{p-1}{6}$ is a nonzero integer.