. Anthony denotes in sequence all positive integers which are divisible by $2$. Bertha denotes in sequence all positive integers which are divisible by $3$. Claire denotes in sequence all positive integers which are divisible by $4$. Orderly Dora denotes all numbers written by the other three. Thereby she puts them in order by size and does not repeat a number. What is the $2017th$ number in her list? [i]¨Proposed by Richard Henner[/i]

Let a positive integer number $n$ has $k$ different prime divisors. Prove that there exists a positive integer number $x \in \left(1, \frac{n}{k}+1 \right)$ such that $x^2-x$ divides by $n$.

This problem consists of four parts. 1. Show that for any nonzero integers $m,n,$ and prime $p$, we have $v_p(mn)=v_p(m)+v_p(n).$ 2. Show that if an off prime $p$, a positive integer $k$ and integers $a,b$ satisfy $p \nmid ~^\text{'}~p|a-b$ and $p\nmid k$, then $v_p(a^k-b^k)=v_p(a-b).$ 3. Show that if $p$ is an off prime with $p|a-b$ and $p\nmid a,b$, then $v_p(a^p-b^p)=v_p(a-b)+1)$. 4. Show that if an odd prime $p$, a positive integer $k$ and integers $a,b$ satisfy $p\nmid a,b ~^\text{'}~ p|a-b$, then $v_p(a^k-b^k)=v_p(a-b)$. Proposed by LTE.

Show that for every natural number $n$ there are $n$ natural numbers $ x_1 < x_2 < ... < x_n $ such that $$\frac{1}{x_1}+\frac{1}{x_2}+...+\frac{1}{x_n}-\frac{1}{x_1x_2...x_n}\in \mathbb{N}\cup {0}$$ (15 points )

[i]In this round, problems will depend on the answers to other problems. A bolded letter is used to denote a quantity whose value is determined by another problem's answer.[/i] [u]Part I[/u] [b]p1.[/b] Let F be the answer to problem number $6$. You want to tile a nondegenerate square with side length $F$ with $1\times 2$ rectangles and $1 \times 1$ squares. The rectangles can be oriented in either direction. How many ways can you do this? [b]p2.[/b] Let [b]A[/b] be the answer to problem number $1$. Triangle $ABC$ has a right angle at $B$ and the length of $AC$ is [b]A[/b]. Let $D$ be the midpoint of $AB$, and let $P$ be a point inside triangle $ABC$ such that $PA = PC = \frac{7\sqrt5}{4}$ and $PD = \frac74$ . The length of $AB^2$ is expressible as $m/n$ for $m, n$ relatively prime positive integers. Find $m$. [b]p3.[/b] Let [b]B[/b] be the answer to problem number $2$. Let $S$ be the set of positive integers less than or equal to [b]B[/b]. What is the maximum size of a subset of $S$ whose elements are pairwise relatively prime? [b]p4.[/b] Let [b]C[/b] be the answer to problem number $3$. You have $9$ shirts and $9$ pairs of pants. Each is either red or blue, you have more red shirts than blue shirts, and you have same number of red shirts as blue pants. Given that you have [b]C[/b] ways of wearing a shirt and pants whose colors match, find out how many red shirts you own. [b]p5.[/b] Let [b]D[/b] be the answer to problem number $4$. You have two odd positive integers $a, b$. It turns out that $lcm(a, b) + a = gcd(a, b) + b =$ [b]D[/b]. Find $ab$. [b]p6.[/b] Let [b]E[/b] be the answer to problem number $5$. A function $f$ defined on integers satisfies $f(y)+f(12-y) = 10$ and $f(y) + f(8 - y) = 4$ for all integers $y$. Given that $f($ [b]E[/b] $) = 0$, compute $f(4)$. [u]Part II[/u] [b]p7.[/b] Let [b]L[/b] be the answer to problem number $12$. You want to tile a nondegenerate square with side length [b]L[/b] with $1\times 2$ rectangles and $7\times 7$ squares. The rectangles can be oriented in either direction. How many ways can you do this? [b]p8.[/b] Let [b]G[/b] be the answer to problem number $7$. Triangle $ABC$ has a right angle at $B$ and the length of $AC$ is [b]G[/b]. Let $D$ be the midpoint of $AB$, and let $P$ be a point inside triangle $ABC$ such that $PA = PC = \frac12$ and $PD = \frac{1}{2010}$ . The length of $AB^2$ is expressible as $m/n$ for $m, n$ relatively prime positive integers. Find $m$. [b]p9.[/b] Let [b]H[/b] be the answer to problem number $8$. Let $S$ be the set of positive integers less than or equal to [b]H[/b]. What is the maximum size of a subset of $S$ whose elements are pairwise relatively prime? [b]p10.[/b] Let [b]I[/b] be the answer to problem number $9$. You have $391$ shirts and $391$ pairs of pants. Each is either red or blue, you have more red shirts than blue shirts, and you have same number of red shirts as red pants. Given that you have [b]I[/b] ways of wearing a shirt and pants whose colors match, find out how many red shirts you own. [b]p11.[/b] Let [b]J[/b] be the answer to problem number $10$. You have two odd positive integers $a, b$. It turns out that $lcm(a, b) + 2a = 2 gcd(a, b) + b = $ [b]J[/b]. Find $ab$. [b]p12.[/b] Let [b]K[/b] be the answer to problem number $11$. A function $f$ defined on integers satisfies $f(y)+f(7-y) = 8$ and $f(y) + f(5 - y) = 4$ for all integers $y$. Given that $f($ [b]K[/b] $) = 453$, compute $f(2)$. PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

How many $1 < n \le 2018$ such that the set $$\{0, 1, 1+2,...,1+2+3+...+i,..., 1+2+...+n-1\}$$ is a permutation of $\{0, 1, 2, 3, 4,...,; n -1\}$ when reduced modulo $n$?

What is the smallest positive integer $n$ such that $n=x^3+y^3$ for two different positive integer tuples $(x,y)$?

Show that, exist infinite triples $(a, b, c)$ where $a, b, c$ are natural numbers, such that: $2a^2 + 3b^2 - 5c^2 = 1997$

For a positive integer $n$, denote $p(n)$ to be the number of nonnegative integer tuples $(x,y,z,w)$ such that $x+2y+2z+3w=n$. Also, denote $q(n)$ to be the number of nonnegative integer tuples $(a,b,c,d)$ such that (i) $a+b+c+d=n$ (ii) $a \ge b \ge d$ (iii) $a \ge c \ge d$ Prove that for all $n$, $p(n) = q(n)$.

Let $n$ be a positive integer. Beto writes a list of $n$ non-negative integers on the board. Then he performs a succession of moves (two steps) of the following type: First for each $i=1,2,...,n$, he counts how many numbers on the board are less than or equal to $i$. Let $a_i$ be the number obtained for each $i=1,2,...,n$. Next, he erases all the numbers from the board and writes the numbers $a_1,a_2,...,a_n$. For example, if $n=5$ and the initial numbers on the board are $0,7,2,6,2$, after the first move, the numbers on the board will bec$1,3,3,3,3$;after the second move they will be $1,1,5,5,5$, and so on. $a)$ Show that, for every $n$ and every initial configuration, there will come a time after which the numbers will no longer be modified when using this move. $b)$Find (as a function of $n$) the minimum value of $k$ such that, for any initial configuration, the moves made from move number $k$ will not change the numbers on the board.

Let $p>5$ be a prime number and $A=\{b_1,b_2,\dots,b_{\frac{p-1}{2}}\}$ be the set of all quadratic residues modulo $p$, excluding zero. Prove that there doesn't exist any natural $a,c$ satisfying $(ac,p)=1$ such that set $B=\{ab_1+c,ab_2+c,\dots,ab_{\frac{p-1}{2}}+c\}$ and set $A$ are disjoint modulo $p$. [i]This problem was proposed by Amir Hossein Pooya.[/i]

Find all $(x,y,z) \in {\mathbb{N}}^3$ such that $x^{4}-y^{4}=z^{2}$.

Find all prime numbers $p, q$ such that$$p(p+1)(p^2+1) = q^2(q^2+q+1) + 2025.$$ [i]Proposed by Md. Fuad Al Alam[/i]

Let $n \geq 2$ be a positive integer. Suppose there exist non-negative integers ${n_1},{n_2},\ldots,{n_k}$ such that $2^n - 1 \mid \sum_{i = 1}^k {{2^{{n_i}}}}$. Prove that $k \ge n$.

Determine all positive integers $n$ such that $3^{n}+1$ is divisible by $n^{2}$.

Let $f : N \to N$ such that $f(p) = 1$ for all p prime and $f(ab) =bf(a) + af(b)$ for all $a, b \in N$. Prove that if $n = p^{a_1}_1 p^{a_1}_2... p^{a_1}_k$ is the canonical distribution of $n$ and $p_i$ does not divide $a_i$ ($i = 1, 2, ..., k$) then $\frac{n}{gcd(n,f(n))}$ is square free (not divisible by a square greater than $1$).

Real numbers are placed at the vertices of an $n{}$-gon. On each side, we write the sum of the numbers on its endpoints. For which $n{}$ is it possible that the numbers on the sides form a permutation of $1, 2, 3,\ldots , n$? [i]From the folklore[/i]

Prove that $1$ plus the product of any four consecutive integers is a perfect square.

Let $ p>2 $ be a prime number. For any permutation $ \pi = ( \pi(1) , \pi(2) , \cdots , \pi(p) ) $ of the set $ S = \{ 1, 2, \cdots , p \} $, let $ f( \pi ) $ denote the number of multiples of $ p $ among the following $ p $ numbers: \[ \pi(1) , \pi(1) + \pi(2) , \cdots , \pi(1) + \pi(2) + \cdots + \pi(p) \] Determine the average value of $ f( \pi) $ taken over all permutations $ \pi $ of $ S $.

Determine all composite integers $n>1$ that satisfy the following property: if $d_1$, $d_2$, $\ldots$, $d_k$ are all the positive divisors of $n$ with $1 = d_1 < d_2 < \cdots < d_k = n$, then $d_i$ divides $d_{i+1} + d_{i+2}$ for every $1 \leq i \leq k - 2$.

Solve the equation $p^2-pq-q^3=1$ in prime numbers. [i]A. Golovanov[/i]

For a nonempty set $\, S \,$ of integers, let $\, \sigma(S) \,$ be the sum of the elements of $\, S$. Suppose that $\, A = \{a_1, a_2, \ldots, a_{11} \} \,$ is a set of positive integers with $\, a_1 < a_2 < \cdots < a_{11} \,$ and that, for each positive integer $\, n\leq 1500, \,$ there is a subset $\, S \,$ of $\, A \,$ for which $\, \sigma(S) = n$. What is the smallest possible value of $\, a_{10}$?

A list of $7$ numbers is constructed using the following procedure: each number in the list is equal to the sum of the previous number and the previous number written in reverse order. For example, if a number in the list is $23544$, the next number is $68076 = 23544 + 44532$. (It is forbidden for any number in the list to start with $0$, although the reversed numbers may start with $0$.) Decide whether it is possible to choose the first number of the list so that the seventh number is a prime number.

Determine all positive integers $n$ such that $7^n -1$ is divisible by $6^n -1$.

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]