Let $A$ be a set of natural numbers, for which for $\forall n\in \mathbb{N}$ exactly one of the numbers $n$, $2n$, and $3n$ is an element of $A$. If $2\in A$, show whether $13824\in A$.

Given a subset of $\{1,2,...,n\}$, we define its [i]alternating sum [/i] in the following way: we order the elements of the subset in descending order and, starting with the largest, we alternately add and subtract the successive numbers. For example, the [i]alternating sum[/i] of the set $\{1,3,4,6,8\}$ is $8-6+4-3+1 = 4$. Determines the sum of the alternating sums of all subsets of $\{1,2,...,10\}$ with an odd number of elements.

$(FRA 2)$ Let $n$ be an integer that is not divisible by any square greater than $1.$ Denote by $x_m$ the last digit of the number $x^m$ in the number system with base $n.$ For which integers $x$ is it possible for $x_m$ to be $0$? Prove that the sequence $x_m$ is periodic with period $t$ independent of $x.$ For which $x$ do we have $x_t = 1$. Prove that if $m$ and $x$ are relatively prime, then $0_m, 1_m, . . . , (n-1)_m$ are different numbers. Find the minimal period $t$ in terms of $n$. If n does not meet the given condition, prove that it is possible to have $x_m = 0 \neq x_1$ and that the sequence is periodic starting only from some number $k > 1.$

Given an integer $a_0$, we define a sequence of real numbers $a_0, a_1, . . .$ using the relation $$a^2_i = 1 + ia^2_{i-1},$$ for $i \ge 1$. An index $j$ is called [i]good [/i] if $a_j$ can be an integer for some $a_0$. Determine the sum of the indices $j$ which lie in the interval $[0, 99]$ and which are not good.

[b]p1.[/b] The number $100$ is written as a sum of distinct positive integers. Determine, with proof, the maximum number of terms that can occur in the sum. [b]p2.[/b] Inside an equilateral triangle of side length $s$ are three mutually tangent circles of radius $1$, each one of which is also tangent to two sides of the triangle, as depicted below. Find $s$. [img]https://cdn.artofproblemsolving.com/attachments/4/3/3b68d42e96717c83bd7fa64a2c3b0bf47301d4.png[/img] [b]p3.[/b] Color a $4\times 7$ rectangle so that each of its $28$ unit squares is either red or green. Show that no matter how this is done, there will be two columns and two rows, so that the four squares occurring at the intersection of a selected row with a selected column all have the same color. [b]p4.[/b] (a) Show that the $y$-intercept of the line through any two distinct points of the graph of $f(x) = x^2$ is $-1$ times the product of the $x$-coordinates of the two points. (b) Find all real valued functions with the property that the $y$-intercept of the line through any two distinct points of its graph is $-1$ times the product of the $x$-coordinates. Prove that you have found all such functions and that all functions you have found have this property. [b]p5.[/b] Let $n$ be a positive integer. We consider sets $A \subseteq \{1, 2,..., n\}$ with the property that the equation $x+y=z$ has no solution with $x\in A$, $y \in A$, $z \in A$. (a) Show that there is a set $A$ as described above that contains $[(n + l)/2]$ members where $[x]$ denotes the largest integer less than or equal to $x$. (b) Show that if $A$ has the property described above, then the number of members of $A$ is less than or equal to $[(n + l)/2]$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Find all pairs of integers $(c, d)$, both greater than 1, such that the following holds: For any monic polynomial $Q$ of degree $d$ with integer coefficients and for any prime $p > c(2c+1)$, there exists a set $S$ of at most $\big(\tfrac{2c-1}{2c+1}\big)p$ integers, such that \[\bigcup_{s \in S} \{s,\; Q(s),\; Q(Q(s)),\; Q(Q(Q(s))),\; \dots\}\] contains a complete residue system modulo $p$ (i.e., intersects with every residue class modulo $p$).

Let $k$ be a natural number.Find all the couples of natural numbers $(n,m)$ such that : $(2^k)!=2^n*m$

There are $200$ numbers on a blackboard: $ 1! , 2! , 3! , 4! , ... ... , 199! , 200!$. Julia erases one of the numbers. When Julia multiplies the remaining $199$ numbers, the product is a perfect square. Which number was erased?

$S={1,2,...,1000}$ and $T'=\left\{ 1001-t|t \in T\right\}$. A set $P$ satisfies the following three conditions: $1.$ All elements of $P$ are a subset of $S$. $2. A,B \in P \Rightarrow A \cap B \neq \O$ $3. A \in P \Rightarrow A' \in P$ Find the maximum of $|P|$.

Show that for any $n$ we can find a set $X$ of $n$ distinct integers greater than 1, such that the average of the elements of any subset of $X$ is a square, cube or higher power.

In a sequence of natural numbers, the first number is $a$, and each subsequent number is the smallest number coprime to all the previous ones and greater than all of them. Prove that in this sequence from some place all numbers will be primes.

For every $ n\in\mathbb{N}$ let $ d(n)$ denote the number of (positive) divisors of $ n$. Find all functions $ f: \mathbb{N}\to\mathbb{N}$ with the following properties: [list][*] $ d\left(f(x)\right) \equal{} x$ for all $ x\in\mathbb{N}$. [*] $ f(xy)$ divides $ (x \minus{} 1)y^{xy \minus{} 1}f(x)$ for all $ x$, $ y\in\mathbb{N}$.[/list] [i]Proposed by Bruno Le Floch, France[/i]

Suppose $x$ is a random real number between $1$ and $4$, and $y$ is a random real number between $1$ and $9$. If the expected value of \[ \left\lceil \log_2 x \right\rceil - \left\lfloor \log_3 y \right\rfloor \] can be expressed as $\frac mn$ where $m$ and $n$ are relatively prime positive integers, compute $100m + n$. [i]Proposed by Lewis Chen[/i]

Define a function $f(n)$ from the positive integers to the positive integers such that $f(f(n))$ is the number of positive integer divisors of $n$. Prove that if $p$ is a prime, then $f(p)$ is prime.

(a) Positive rational numbers $a, b,$ and $c$ have the property that $\frac{a}{b} + \frac{b}{c} + \frac{c}{a}$ is an integer. Is it possible for $\frac{a}{c} + \frac{c}{b} + \frac{b}{a}$ to also be an integer except for the trivial solution? (b) Positive real numbers $a, b,$ and $c$ have the property that $\frac{a}{b} + \frac{b}{c} + \frac{c}{a}$ is an integer. Is it possible for $\frac{a}{c} + \frac{c}{b} + \frac{b}{a}$ to also be an integer except for the trivial solution? [i](Andrew Brahms, USA)[/i]

Call a sequence of positive integers $\{a_n\}$ good if for any distinct positive integers $m,n$, one has $$\gcd(m,n) \mid a_m^2 + a_n^2 \text{ and } \gcd(a_m,a_n) \mid m^2 + n^2.$$ Call a positive integer $a$ to be $k$-good if there exists a good sequence such that $a_k = a$. Does there exists a $k$ such that there are exactly $2019$ $k$-good positive integers?

Let $n$ be a positive integer. (a) Prove that $1^5 +3^5 +5^5 +...+(2n-1)^5$ is divisible by $n$. (b) Prove that $1^3 +3^3 +5^3 +...+(2n-1)^3$ is divisible by $n^2$.

Find all pairs $(a,b)$ of positive integers such that $a!+b$ and $b!+a$ are both powers of $5$. [i]Nikola Velov, North Macedonia[/i]

Consider the sequence of integers $ \left( a_n\right)_{n\ge 0} $ defined as $$ a_n=\left\{\begin{matrix}n^6-2017, & 7|n\\ \frac{1}{7}\left( n^6-2017\right) , & 7\not | n\end{matrix}\right. . $$ Determine the largest length a string of consecutive terms from this sequence sharing a common divisor greater than $ 1 $ may have.

Determine all positive integers $ n\geq 2$ that satisfy the following condition: for all $ a$ and $ b$ relatively prime to $ n$ we have \[a \equiv b \pmod n\qquad\text{if and only if}\qquad ab\equiv 1 \pmod n.\]

Suppose $N$ is any positive integer. Add the digits of $N$ to obtain a smaller integer. Repeat this process of digit-addition till you get a single digit numbem. Find the number of positive integers $N \le 1000$, such that the final single-digit number $n$ is equal to $5$. Example: $N = 563\to (5 + 6 + 3) = 14 \to(1 + 4) = 5$ will be counted as one such integer.

a) Show that the number $(2k + 1)^3 - (2k - 1)^3$, $k \in Z$, is the sum of three perfect squares. b) Represent the number $(2n + 1)^3 -2$, $n \in N^*$, as the sum of $3n- 1$ perfect squares greater than $1$.

In a $99\times 101$ table , cubes of natural numbers, as shown in figure . Prove that the sum of all numbers in the table are divisible by $200$. [img]https://cdn.artofproblemsolving.com/attachments/3/e/dd3d38ca00a36037055acaaa0c2812ae635dcb.png[/img]

Show that there exist infinitely many composite positive integers $n$ such that $n$ divides $2^{\frac{n-1}{2}}+1$

How many multiples of $2005$ are factors of $(2005)^2$?