Let $n=p_1p_2\cdots p_s$, where $p_1,\ldots,p_s$ are distinct odd prime numbers. (a) Prove that the expression $$F_n(x)=\prod\left(x^{\frac n{p_{i_1}\cdots p_{i_k}}}-1\right)^{(-1)^k},$$where the product goes over all subsets $\{p_{i_1},\ldots,p_{i_k}\}$ or $\{p_1,\ldots,p_s\}$ (including itself and the empty set), can be written as a polynomial in $x$ with integer coefficients. (b) Prove that if $p$ is a prime divisor of $F_n(2)$, then either $p\mid n$ or $n\mid p-1$.

Let $(p_1,p_2,..., p_n)$ be a random permutation of the set $\{1,2,...,n)$. If $n$ is odd, prove that the product $(p_1-1)\cdot (p_2-2)\cdot ...\cdot (p_n-n)$ is an even number. @below fixed.

Let $n$ be a positive integer. Prove that the equation \[x+y+\frac{1}{x}+\frac{1}{y}=3n\] does not have solutions in positive rational numbers.

Each of two boxes contains both black and white marbles, and the total number of marbles in the two boxes is $25.$ One marble is taken out of each box randomly. The probability that both marbles are black is $27/50,$ and the probability that both marbles are white is $m/n,$ where $m$ and $n$ are relatively prime positive integers. What is $m+n?$

Find all triples ($a$,$b$,$n$) of positive integers such that $$a^3=b^2+2^n$$

Given an integer $k>1$, show that there exist an integer an $n>1$ and distinct positive integers $a_1,a_2,\cdots a_n$, all greater than $1$, such that the sums $\sum_{j=1}^n a_j$ and $\sum_{j=1}^n \phi (a_j)$ are both $k$-th powers of some integers. (Here $\phi (m)$ denotes the number of positive integers less than $m$ and relatively prime to $m$.)

Does there exist two infinite positive integer sets $S,T$, such that any positive integer $n$ can be uniquely expressed in the form $$n=s_1t_1+s_2t_2+\ldots+s_kt_k$$ ,where $k$ is a positive integer dependent on $n$, $s_1<\ldots<s_k$ are elements of $S$, $t_1,\ldots, t_k$ are elements of $T$?

Let $(a,b)=(a_n,a_{n+1}),\forall n\in\mathbb{N}$ all be positive interger solutions that satisfies $$1\leq a\leq b$$ and $$\dfrac{a^2+b^2+a+b+1}{ab}\in\mathbb{N}$$ And the value of $a_n$ is [b]only[/b] determined by the following recurrence relation:$ a_{n+2} = pa_{n+1} + qa_n + r$ Find $(p,q,r)$.

Let $n$ be a positive integer. Prove that there exist integers $a_1, a_2,..., a_n$ such that for any integer $x$, the number $(... (((x^2 + a_1)^2 + a_2)^2 + ...)^2 + a_{n-1})^2 + a_n$ is divisible by $2n - 1$.

For $m$ and $n$ positive integers that are prime to each other, determine the possible values ​​of $$\gcd (5^m + 7^m, 5^n + 7^n)$$

In base ten, the number $100! = 100 \cdot 99 \cdot 98 \cdot 97... 2 \cdot 1$ has $158$ digits, and the last $24$ digits are all zeros. Find the number of zeros there are at the end of this same number when it is written in base $24$.

Call a fraction $\frac{a}{b}$, not necessarily in the simplest form [i]special[/i] if $a$ and $b$ are positive integers whose sum is $15$. How many distinct integers can be written as the sum of two, not necessarily different, special fractions? $\textbf{(A)}\ 9 \qquad\textbf{(B)}\ 10 \qquad\textbf{(C)}\ 11 \qquad\textbf{(D)}\ 12 \qquad\textbf{(E)}\ 13$

Find all positive integers $m$ that have some multiple of the form $x^2+5y^2+2024$, with $x$ and $y$ integers.

A sequence of positive integer numbers $a_1,...,a_{100}$ such is that $a_1=1$, and for all $n=1, 2,...,100$ number $(a_1+...+a_n) \left ( \frac{1}{a_1}+...+\frac{1}{a_n} \right )$ is integer. What is the maximum value that can take $ a_{100}$? [i] M. Turevskii [/i]

[b]p1.[/b] A boy has as many sisters as brothers. How ever, his sister has twice as many brothers as sisters. How many boys and girls are there in the family? [b]p2.[/b] Solve each of the following problems. (1) Find a pair of numbers with a sum of $11$ and a product of $24$. (2) Find a pair of numbers with a sum of $40$ and a product of $400$. (3) Find three consecutive numbers with a sum of $333$. (4) Find two consecutive numbers with a product of $182$. [b]p3.[/b] $2008$ integers are written on a piece of paper. It is known that the sum of any $100$ numbers is positive. Show that the sum of all numbers is positive. [b]p4.[/b] Let $p$ and $q$ be prime numbers greater than $3$. Prove that $p^2 - q^2$ is divisible by $24$. [b]p5.[/b] Four villages $A,B,C$, and $D$ are connected by trails as shown on the map. [img]https://cdn.artofproblemsolving.com/attachments/4/9/33ecc416792dacba65930caa61adbae09b8296.png[/img] On each path $A \to B \to C$ and $B \to C \to D$ there are $10$ hills, on the path $A \to B \to D$ there are $22$ hills, on the path $A \to D \to B$ there are $45$ hills. A group of tourists starts from $A$ and wants to reach $D$. They choose the path with the minimal number of hills. What is the best path for them? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Find the smallest positive integer $N$ such that there are no different sets $A, B$ that satisfy the following conditions. (Here, $N$ is not a power of $2$. That is, $N \neq 1, 2^1, 2^2, \dots$.) [list] [*] $A, B \subseteq \{1, 2^1, 2^2, 2^3, \dots, 2^{2023}\} \cup \{ N \}$ [*] $|A| = |B| \geq 1$ [*] Sum of elements in $A$ and sum of elements in $B$ are equal. [/list]

Find all relatively prime integers $a,b$ such that $$a^2+a=b^3+b$$

[b]p1.[/b] A square is tiled by smaller squares as shown in the figure. Find the area of the black square in the middle if the perimeter of the square $ABCD$ is $14$ cm. [img]https://cdn.artofproblemsolving.com/attachments/1/1/0f80fc5f0505fa9752b5c9e1c646c49091b4ca.png[/img] [b]p2.[/b] If $a, b$, and $c$ are numbers so that $a + b + c = 0$ and $a^2 + b^2 + c^2 = 1$. Compute $a^4 + b^4 + c^4$. [b]p3.[/b] A given fraction $\frac{a}{b}$ ($a, b$ are positive integers, $a \ne b$) is transformed by the following rule: first, $1$ is added to both the numerator and the denominator, and then the numerator and the denominator of the new fraction are each divided by their greatest common divisor (in other words, the new fraction is put in simplest form). Then the same transformation is applied again and again. Show that after some number of steps the denominator and the numerator differ exactly by $1$. [b]p4.[/b] A goat uses horns to make the holes in a new $30\times 60$ cm large towel. Each time it makes two new holes. Show that after the goat repeats this $61$ times the towel will have at least two holes whose distance apart is less than $6$ cm. [b]p5.[/b] You are given $555$ weights weighing $1$ g, $2$ g, $3$ g, $...$ , $555$ g. Divide these weights into three groups whose total weights are equal. [b]p6.[/b] Draw on the regular $8\times 8$ chessboard a circle of the maximal possible radius that intersects only black squares (and does not cross white squares). Explain why no larger circle can satisfy the condition. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Find all positive integers $b$ for which there exists a positive integer $a$ with the following properties: - $a$ is not a divisor of $b$. - $a^a$ is a divisor of $b^b$

For $A\subset\mathbb Z$ and $a,b\in\mathbb Z$. We define $aA+b: =\{ax+b|x\in A\}$. If $a\neq0$ then we calll $aA+b$ and $A$ to similar sets. In this question the Cantor set $C$ is the number of non-negative integers that in their base-3 representation there is no $1$ digit. You see \[C=(3C)\dot\cup(3C+2)\ \ \ \ \ \ (1)\] (i.e. $C$ is partitioned to sets $3C$ and $3C+2$). We give another example $C=(3C)\dot\cup(9C+6)\dot\cup(3C+2)$. A representation of $C$ is a partition of $C$ to some similiar sets. i.e. \[C=\bigcup_{i=1}^{n}C_{i}\ \ \ \ \ \ (2)\] and $C_{i}=a_{i}C+b_{i}$ are similar to $C$. We call a representation of $C$ a primitive representation iff union of some of $C_{i}$ is not a set similar and not equal to $C$. Consider a primitive representation of Cantor set. Prove that a) $a_{i}>1$. b) $a_{i}$ are powers of 3. c) $a_{i}>b_{i}$ d) (1) is the only primitive representation of $C$.

How many integer pairs $(x,y)$ satisfies $x^2+y^2=9999(x-y)$?

For a nonnegative integer $n$ define $\operatorname{rad}(n)=1$ if $n=0$ or $n=1$, and $\operatorname{rad}(n)=p_1p_2\cdots p_k$ where $p_1<p_2<\cdots <p_k$ are all prime factors of $n$. Find all polynomials $f(x)$ with nonnegative integer coefficients such that $\operatorname{rad}(f(n))$ divides $\operatorname{rad}(f(n^{\operatorname{rad}(n)}))$ for every nonnegative integer $n$.

Find all polynomials $P(x)$ with integer coefficients such that for all real numbers $s$ and $t$, if $P(s)$ and $P(t)$ are both integers, then $P(st)$ is also an integer.

2011 non-zero integers are given. It is known that the sum of any one of them with the product of the remaining 2010 numbers is negative. Prove that if all numbers are split arbitrarily into two groups, the sum of the two products will also be negative. (Authors: N. Agahanov & I. Bogdanov)

Let $p$ be a prime such that $p^2$ divides $2^{p-1}-1$. Prove that for all positive integers $n$ the number $\left(p-1\right)\left(p!+2^n\right)$ has at least $3$ different prime divisors. [i]Aleksandar Ivanov[/i]