Alexander has chosen a natural number $N>1$ and has written down in a line,and in increasing order,all his positive divisors $d_1<d_2<\ldots <d_s$ (where $d_1=1$ and $d_s=N$).For each pair of neighbouring numbers,he has found their greater common divisor.The sum of all these $s-1$ numbers (the greatest common divisors) is equal to $N-2$.Find all possible values of $N$.

Let $m \leq n$ be positive integers and $p$ be a prime. Let $p-$expansions of $m$ and $n$ be \[m = a_0 + a_1p + \dots + a_rp^r\]\[n = b_0 + b_1p + \dots + b_sp^s\] respectively, where $a_r, b_s \neq 0$, for all $i \in \{0,1,\dots,r\}$ and for all $j \in \{0,1,\dots,s\}$, we have $0 \leq a_i, b_j \leq p-1$ . If $a_i \leq b_i$ for all $i \in \{0,1,\dots,r\}$, we write $ m \prec_p n$. Prove that \[p \nmid {{n}\choose{m}} \Leftrightarrow m \prec_p n\].

[b]p1.[/b] Prove that no matter what digits are placed in the four empty boxes, the eight-digit number $9999\Box\Box\Box\Box$ is not a perfect square. [b]p2.[/b] Prove that the number $m/3+m^2/2+m^3/6$ is integral for all integral values of $m$. [b]p3.[/b] An elevator in a $100$ store building has only two buttons: UP and DOWN. The UP button makes the elevator go $13$ floors up, and the DOWN button makes it go $8$ floors down. Is it possible to go from the $13$th floor to the $8$th floor? [b]p4.[/b] Cut the triangle shown in the picture into three pieces and rearrange them into a rectangle. (Pieces can not overlap.) [img]https://cdn.artofproblemsolving.com/attachments/4/b/ca707bf274ed54c1b22c4f65d3d0b0a5cfdc56.png[/img] [b]p5.[/b] Two players Tom and Sid play the following game. There are two piles of rocks, $7$ rocks in the first pile and $9$ rocks in the second pile. Each of the players in his turn can take either any amount of rocks from one pile or the same amount of rocks from both piles. The winner is the player who takes the last rock. Who does win in this game if Tom starts the game? [b]p6.[/b] In the next long multiplication example each letter encodes its own digit. Find these digits. $\begin{tabular}{ccccc} & & & a & b \\ * & & & c & d \\ \hline & & c & e & f \\ + & & a & b & \\ \hline & c & f & d & f \\ \end{tabular}$ PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

$(FRA 6)$ Consider the integer $d = \frac{a^b-1}{c}$, where $a, b$, and $c$ are positive integers and $c \le a.$ Prove that the set $G$ of integers that are between $1$ and $d$ and relatively prime to $d$ (the number of such integers is denoted by $\phi(d)$) can be partitioned into $n$ subsets, each of which consists of $b$ elements. What can be said about the rational number $\frac{\phi(d)}{b}?$

Given prime number $p$. $a_1,a_2 \cdots a_k$ ($k \geq 3$) are integers not divible by $p$ and have different residuals when divided by $p$. Let \[ S_n= \{ n \mid 1 \leq n \leq p-1, (na_1)_p < \cdots < (na_k)_p \} \] Here $(b)_p$ denotes the residual when integer $b$ is divided by $p$. Prove that $|S|< \frac{2p}{k+1}$.

Find the natural numbers $ n\ge 2 $ which have the property that the ring of integers modulo $ n $ has exactly an element that is not a sum of two squares.

Compute the number of positive four-digit multiples of $11$ whose sum of digits (in base ten) is divisible by $11$.

Given a polynomial $P(x)$ with a) natural coefficients; b) integer coefficients; Let us denote with $a_n$ the sum of the digits of $P(n)$ value. Prove that there is a number encountered in the sequence $a_1, a_2, ... , a_n, ...$ infinite times.

The arithmetic function $\nu$ is defined by $$\nu (n) = \begin{cases}0, \,\,\,\,\, n=1 \\ k, \,\,\,\,\, n=p_1^{a_1} p_2^{a_2} ... p_k^{a_k}\end{cases}$$, where $n=p_1^{a_1} p_2^{a_2} ... p_k^{a_k}$ represents the prime factorization of the number. Prove that for any naturals $m,n$, $$\tau (n^m) = \sum_{d | n} m^{\nu (d)}.$$ [i](PP-nine)[/i]

When rolling a certain unfair six-sided die with faces numbered $1, 2, 3, 4, 5$, and $6$, the probability of obtaining face $F$ is greater than $\frac{1}{6}$, the probability of obtaining the face opposite is less than $\frac{1}{6}$, the probability of obtaining any one of the other four faces is $\frac{1}{6}$, and the sum of the numbers on opposite faces is $7$. When two such dice are rolled, the probability of obtaining a sum of $7$ is $\frac{47}{288}$. Given that the probability of obtaining face $F$ is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers, find $m+n$.

Let $ P$ be a regular polygon. A regular sub-polygon of $ P$ is a subset of vertices of $ P$ with at least two vertices such that divides the circumcircle to equal arcs. Prove that there is a subset of vertices of $ P$ such that its intersection with each regular sub-polygon has even number of vertices.

Given a finite set $ P$ of prime numbers, prove that there exists a positive integer $ x$ such that it can be written in the form $ a^p \plus{} b^p$ ($ a,b$ are positive integers), for each $ p\in P$, and cannot be written in that form for each $ p$ not in $ P$.

Show that there exists a positive integer $N$ such that for all integers $a>N$, there exists a contiguous substring of the decimal expansion of $a$, which is divisible by $2011$. Note. A contiguous substring of an integer $a$ is an integer with a decimal expansion equivalent to a sequence of consecutive digits taken from the decimal expansion of $a$.

[b]p1.[/b] If $x^2 = 7$, what is $x^4 + x^2 + 1$? [b]p2.[/b] Richard and Alex are competing in a $150$-meter race. If Richard runs at a constant speed of $5$ meters per second and Alex runs at a constant speed of $3$ meters per second, how many more seconds does it take for Alex to finish the race? [b]p3.[/b] David and Emma are playing a game with a chest of $100$ gold coins. They alternate turns, taking one gold coin if the chest has an odd number of gold coins or taking exactly half of the gold coins if the chest has an even number of gold coins. The game ends when there are no more gold coins in the chest. If Emma goes first, how many gold coins does Emma have at the end? [b]p4.[/b] What is the only $3$-digit perfect square whose digits are all different and whose units digit is $5$? [b]p5.[/b] In regular pentagon $ABCDE$, let $F$ be the midpoint of $\overline{AB}$, $G$ be the midpoint of $\overline{CD}$, and $H$ be the midpoint of $\overline{AE}$. What is the measure of $\angle FGH$ in degrees? [b]p6.[/b] Water enters at the left end of a pipe at a rate of $1$ liter per $35$ seconds. Some of the water exits the pipe through a leak in the middle. The rest of the water exits from the right end of the pipe at a rate of $1$ liter per $36$ seconds. How many minutes does it take for the pipe to leak a liter of water? [b]p7.[/b] Carson wants to create a wire frame model of a right rectangular prism with a volume of $2022$ cubic centimeters, where strands of wire form the edges of the prism. He wants to use as much wire as possible. If Carson also wants the length, width, and height in centimeters to be distinct whole numbers, how many centimeters of wire does he need to create the prism? [b]p8.[/b] How many ways are there to fill the unit squares of a $3 \times 5$ grid with the digits $1$, $2$, and $3$ such that every pair of squares that share a side differ by exactly $1$? [b]p9.[/b] In pentagon ABCDE, $AB = 54$, $AE = 45$, $DE = 18$, $\angle A = \angle C = \angle E$, $D$ is on line segment $\overline{BE}$, and line $BD$ bisects angle $\angle ABC$, as shown in the diagram below. What is the perimeter of pentagon $ABCDE$? [img]https://cdn.artofproblemsolving.com/attachments/2/0/7c25837bb10b128a1c7a292f6ce8ce3e64b292.png[/img] [b]p10.[/b] If $x$ and $y$ are nonzero real numbers such that $\frac{7}{x} + \frac{8}{y} = 91$ and $\frac{6}{x} + \frac{10}{y} = 89$, what is the value of $x + y$? [b]p11.[/b] Hilda and Marianne play a game with a shued deck of $10$ cards, numbered from $1$ to $10$. Hilda draws five cards, and Marianne picks up the five remaining cards. Hilda observes that she does not have any pair of consecutive cards - that is, no two cards have numbers that differ by exactly $1$. Additionally, the sum of the numbers on Hilda's cards is $1$ less than the sum of the numbers on Marianne's cards. Marianne has exactly one pair of consecutive cards - what is the sum of this pair? [b]p12.[/b] Regular hexagon $AUSTIN$ has side length $2$. Let $M$ be the midpoint of line segment $\overline{ST}$. What is the area of pentagon $MINUS$? [b]p13.[/b] At a collector's store, plushes are either small or large and cost a positive integer number of dollars. All small plushes cost the same price, and all large plushes cost the same price. Two small plushes cost exactly one dollar less than a large plush. During a shopping trip, Isaac buys some plushes from the store for 59 dollars. What is the smallest number of dollars that the small plush could not possibly cost? [b]p14.[/b] Four fair six-sided dice are rolled. What is the probability that the median of the four outcomes is $5$? [b]p15.[/b] Suppose $x_1, x_2,..., x_{2022}$ is a sequence of real numbers such that: $x_1 + x_2 = 1$ $x_2 + x_3 = 2$ $...$ $x_{2021} + x_{2022} = 2021$ If $x_1 + x_{499} + x_{999} + x_{1501} = 222$, then what is the value of $x_{2022}$? [b]p16.[/b] A cone has radius $3$ and height $4$. An infinite number of spheres are placed in the cone in the following way: sphere $C_0$ is placed inside the cone such that it is tangent to the base of the cone and to the curved surface of the cone at more than one point, and for $i \ge 1$, sphere $C_i$ is placed such that it is externally tangent to sphere $C_{i-1}$ and internally tangent to more than one point of the curved surface of the cone. If $V_i$ is the volume of sphere $C_i$, compute $V_0 + V_1 + V_2 + ... $ . [img]https://cdn.artofproblemsolving.com/attachments/b/4/b43e40bb0a5974dd9d656691c14b4ae268b5b5.png[/img] [b]p17.[/b] Call an ordered pair, $(x, y)$, relatable if $x$ and $y$ are positive integers where $y$ divides $3600$, $x$ divides $y$ and $\frac{y}{x}$ is a prime number. For every relatable ordered pair, Leanne wrote down the positive difference of the two terms of the pair. What is the sum of the numbers she wrote down? [b]p18.[/b] Let $r, s$, and $t$ be the three roots of $P(x) = x^3 - 9x - 9$. Compute the value of $(r^3 + r^2 - 10r - 8)(s^3 + s^2 - 10s - 8)(t^3 + t^2 - 10t - 8)$. [b]p19.[/b] Compute the number of ways to color the digits $0, 1, 2, 3, 4, 5, 6, 7, 8$ and $9$ red, blue, or green such that: (a) every prime integer has at least one digit that is not blue, and (b) every composite integer has at least one digit that is not green. Note that $0$ is not composite. For example, since $12$ is composite, either the digit $1$, the digit $2$, or both must be not green. [b]p20.[/b] Pentagon $ABCDE$ has $AB = DE = 4$ and $BC = CD = 9$ with $\angle ABC = \angle CDE = 90^o$, and there exists a circle tangent to all five sides of the pentagon. What is the length of segment $\overline{AE}$? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $n_1,\ldots,n_k$ be positive integers, and define $d_1=1$ and $d_i=\frac{(n_1,\ldots,n_{i-1})}{(n_1,\ldots,n_{i})}$, for $i\in \{2,\ldots,k\}$, where $(m_1,\ldots,m_{\ell})$ denotes the greatest common divisor of the integers $m_1,\ldots,m_{\ell}$. Prove that the sums \[\sum_{i=1}^k a_in_i\] with $a_i\in\{1,\ldots,d_i\}$ for $i\in\{1,\ldots,k\}$ are mutually distinct $\mod n_1$.

For each positive integer $n$ we consider the sequence of $2004$ integers$$\left [n+\sqrt{n}\right ],\left [n+1+\sqrt{n+1}\right ],\left [n+2+\sqrt{n+2}\right ],\ldots ,\left [n+2003+\sqrt{n+2003}\right ]$$Determine the smallest integer $n$ such that the $2004$ numbers in the sequence are $2004$ consecutive integers. Clarification: The brackets indicate the integer part.

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 that can be written in the following way $\frac{m^2 + 20mn + n^2}{m^3 + n^3}$ Also, $m,n$ are relatively prime positive integers.

Oleksiy placed positive integers in the cells of the $8\times 8$ chessboard. For each pair of adjacent-by-side cells, Fedir wrote down the product of the numbers in them and added all the products. Oleksiy wrote down the sum of the numbers in each pair of adjacent-by-side cells and multiplied all the sums. It turned out that the last digits of both numbers are equal to $1$. Prove that at least one of the boys made a mistake in the calculation. For example, for a square $3\times 3$ and the arrangement of numbers shown below, Fedir would write the following numbers: $2, 6, 8, 24, 15, 35, 2, 6, 8, 20, 18, 42$, and their sum ends with a digit $6$; Oleksiy would write the following numbers: $3, 5, 6, 10, 8, 12, 3, 5, 6, 9, 9, 13$, and their product ends with a digit $0$. \begin{tabular}{| c| c | c |} \hline 1 & 2 & 3 \\ \hline 2 & 4 & 6 \\ \hline 3 & 5 & 7 \\ \hline \end{tabular} [i]Proposed by Oleksiy Masalitin and Fedir Yudin[/i]

For which positive integers $n$, one can find a non-integer rational number $x$ such that $$x^n+(x+1)^n$$ is an integer?

Consider the polynomials \[f(p) = p^{12} - p^{11} + 3p^{10} + 11p^3 - p^2 + 23p + 30;\] \[g(p) = p^3 + 2p + m.\] Find all integral values of $m$ for which $f$ is divisible by $g$.

Given a fixed positive integer $a\geq 9$. Prove: There exist finitely many positive integers $n$, satisfying: (1)$\tau (n)=a$ (2)$n|\phi (n)+\sigma (n)$ Note: For positive integer $n$, $\tau (n)$ is the number of positive divisors of $n$, $\phi (n)$ is the number of positive integers $\leq n$ and relatively prime with $n$, $\sigma (n)$ is the sum of positive divisors of $n$.

Let $a$ be a rational number and let $n$ be a positive integer. Prove that the polynomial $X^{2^n}(X+a)^{2^n}+1$ is irreducible in the ring $\mathbb{Q}[X]$ of polynomials with rational coefficients. [i]Proposed by Vincent Jugé, École Polytechnique, Paris.[/i]

Find all prime numbers $ p $ such that $ 4p^2 + 1 $ and $ 6p^2 + 1 $ are also prime numbers.

$n$ is a positive integer, $F_n=2^{2^{n}}+1$. Prove that for $n \geq 3$, there exists a prime factor of $F_n$ which is larger than $2^{n+2}(n+1)$.