Solve the following equation in $\mathbb{Z}$: \[3^{2a + 1}b^2 + 1 = 2^c\]

Let $\mathbb N$ denote the set of positive integers, and for a function $f$, let $f^k(n)$ denote the function $f$ applied $k$ times. Call a function $f : \mathbb N \to \mathbb N$ [i]saturated[/i] if \[ f^{f^{f(n)}(n)}(n) = n \] for every positive integer $n$. Find all positive integers $m$ for which the following holds: every saturated function $f$ satisfies $f^{2014}(m) = m$. [i]Proposed by Evan Chen[/i]

Let $S$ be the set of all positive integers $n$ such that the sum of all factors of $n$, including $1$ and $n$, is $120$. Compute the sum of all numbers in $S$. [i]Proposed by Evin Liang[/i]

Write $102$ as the sum of the largest number of distinct primes.

Prove that the sum of digits of the number $1972^n$ is not bounded from above when $n$ tends to infinity.

The sequence $ (a_n)$ is defined by: $ a_0\equal{}a_1\equal{}1$ and $ a_{n\plus{}1}\equal{}14a_n\minus{}a_{n\minus{}1}$ for all $ n\ge 1$. Prove that $ 2a_n\minus{}1$ is a perfect square for any $ n\ge 0$.

Solve in positive integers the following equation; $xy(x+y-10)-3x^2-2y^2+21x+16y=60$

Write the numbers from $1$ to $16$ in the cells of a of a $4 \times 4$ square so that: 1. Each cell contains exactly one number; 2. Each number is written exactly once; 3. For any two cells that are symmetrical with respect to any of the perpendicular bisectors of sides of the original $4 \times 4$ square, the sum of numbers in them is a prime number The figure below shows examples of such pairs of cells, sums of numbers in which have to be prime. [img]https://i.ibb.co/fqX05dY/Kyiv-MO-2024-Round-1-8-2.png[/img] [i]Proposed by Mykhailo Shtandenko[/i]

Ten adults enter a room, remove their shoes, and toss their shoes into a pile. Later, a child randomly pairs each left shoe with a right shoe without regard to which shoes belong together. The probability that for every positive integer $k<5,$ no collection of $k$ pairs made by the child contains the shoes from exactly $k$ of the adults is $\tfrac{m}{n},$ where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

$100$ fractions are written on a board, their numerators are numbers from $1$ to $100$ (each once) and denominators are also numbers from $1$ to $100$ (also each once). It appears that the sum of these fractions equals to $a/2$ for some odd $a$. Prove that it is possible to interchange numerators of two fractions so that sum becomes a fraction with odd denominator. [i]N. Agakhanov, I. Bogdanov [/i]

Let $n$ be a positive integer and let $k_0,k_1, \dots,k_{2n}$ be nonzero integers such that $k_0+k_1 +\dots+k_{2n}\neq 0$. Is it always possible to a permutation $(a_0,a_1,\dots,a_{2n})$ of $(k_0,k_1,\dots,k_{2n})$ so that the equation \begin{align*} a_{2n}x^{2n}+a_{2n-1}x^{2n-1}+\dots+a_0=0 \end{align*} has not integer roots?

Find all pairs $x,y$ of natural numbers that satisfy the equation $$x^2-xy+2x-3y=1997$$

Call a positive integer $x$ to be [i]remote from squares and cubes [/i] if each integer $k$ satisfies both $|x - k^2| > 10^6$ and $|x - k^3| > 10^6$. Prove that there exist infinitely many positive integer $n$ such that $2^n$ is remote from squares and cubes.

For any positive integer $n$, show that there exists a polynomial $P(x)$ of degree $n$ with integer coefficients such that $P(0),P(1), \ldots, P(n)$ are all distinct powers of $2$.

Consider a sequence $\{a_i\}^\infty_{i\ge1}$ of positive integers. For all positvie integers $n$ prove that there exists infinitely many positive integers $k$ such that there is no pair $(m,t)$ of positive integers where $m>n$ and $$kn+a_n=tm(m+1)+a_m$$

Let $S \subset Z^+$ be a set of positive integers with the following property: for any $a, b \in S$, if $a \ne b$ then $a + b$ is a perfect square. Given that $2009 \in S$ and $2087 \in S$, what is the maximum number of elements in $S$?

[b]p25.[/b] You are given that $1000!$ has $2568$ decimal digits. Call a permutation $\pi$ of length $1000$ good if $\pi(2i) > \pi (2i - 1)$ for all $1 \le i \le 500$ and $\pi (2i) > \pi (2i + 1)$ for all $1 \le i \le 499$. Let $N$ be the number of good permutations. Estimate $D$, the number of decimal digits in $N$. You will get $\max \left( 0, 25 - \left\lceil \frac{|D-X|}{10} \right\rceil \right)$ points, where $X$ is the true answer. [b]p26.[/b] A year is said to be [i]interesting [/i] if it is the product of $3$, not necessarily distinct, primes (for example $2^2 \cdot 5$ is interesting, but $2^2 \cdot 3 \cdot 5$ is not). How many interesting years are there between $ 5000$ and $10000$, inclusive? For an estimate of $E$, you will get $\max \left( 0, 25 - \left\lceil \frac{|E-X|}{10} \right\rceil \right)$ points, where $X$ is the true answer. [b]p27.[/b] Sam chooses $1000$ random lattice points $(x, y)$ with $1 \le x, y \le 1000$ such that all pairs $(x, y)$ are distinct. Let $N$ be the expected size of the maximum collinear set among them. Estimate $\lfloor 100N \rfloor$. Let $S$ be the answer you provide and $X$ be the true value of $\lfloor 100N \rfloor$. You will get $\max \left( 0, 25 - \left\lceil \frac{|S-X|}{10} \right\rceil \right)$ points for your estimate. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

In the sequence $ (a_n)$ with general term $ a_n \equal{} n^3 \minus{} (2n \plus{} 1)^2$, does there exist a term that is divisible by 2006?

Find all triples of natural numbers $(x,y,z)$ for which: $xyz=x!+y^x+y^z+z!$.

$11.$ Let $a,$ $b,$ and $c$ be real numbers such that $a-7b+8c=4.$ and $8a+4b-c=7.$ What is the value of $a^2-b^2+c^2 ?$

Prove that $\overline{a0... 09}$ (in which $a > 0$ is a digit and there is at least one zero) is not a perfect square. (VA Senderov)

We call a positive integer [i]interesting [/i] if the number and the number with its digits written in reverse order both leave remainder $2$ in division by $4$. a) Determine if $2018$ is an interesting number. b) For every positive integer $n$, find how many interesting $n$-digit numbers there are.

[list=a] [*] Prove that for any positive integers $a,b,c$, there exists a positive integer $N$ such that $$(N+a^2)(N+b^2)(N+c^2)$$ is a perfect square. [*] Prove that there exist five distinct positive integers $a,b,c,d,e$ for which there exists a positive integer $N$ such that $$(N+a^2)(N+b^2)(N+c^2)(N+d^2)(N+e^2)$$ is a perfect square. [/list] [i]Luminița Popescu[/i]

Consider all $6$-digit numbers of the form $abccba$ where $b$ is odd. Determine the number of all such $6$-digit numbers that are divisible by $7$.

A $x>2$ real number is given. Bob has got $1997$ labels and writes one of the numbers $"x^0, x^1, x^2 ,\dotsm x^{1995}, x^{1996}"$ each labels such that all labels has distinct numbers. Bob puts some labels to right pocket, some labels to left pocket. Prove that sum of numbers of the right pocket never equal to sum of numbers of the left pocket.