Find all $f : \mathbb{N} \to \mathbb{N} $ such that $f(a) + f(b)$ divides $2(a + b - 1)$ for all $a, b \in \mathbb{N}$. Remark: $\mathbb{N} = \{ 1, 2, 3, \ldots \} $ denotes the set of the positive integers.

For an $n \times n$ table filled with natural numbers, we say it is a [i]divisor table[/i] if: - the numbers in the $i$-th row are exactly all the divisors of some natural number $r_i$, - the numbers in the $j$-th column are exactly all the divisors of some natural number $c_j$, - $r_i \ne r_j$ for every $i \ne j$. A prime number $p$ is given. Determine the smallest natural number $n$, divisible by $p$, such that there exists an $n \times n$ divisor table, or prove that such $n$ does not exist. [i]Proposed by Pavle Martinović[/i]

Given $s,t$ are non-zero integers, $(x,y) $ is an integer pair , A transformation is to change pair $(x,y)$ into pair $(x+t,y-s)$ . If the two integers in a certain pair becoems relatively prime after several tranfomations , then we call the original integer pair "a good pair" . (1) Is $(s,t)$ a good pair ? (2) Prove :for any $s$ and $t$ , there exists pair $(x,y)$ which is " a good pair".

Consider the following $3\times 2$ array formed by using the numbers $1,2,3,4,5,6$, $$\begin{pmatrix} a_{11}& a_{12}\\a_{21}& a_{22}\\ a_{31}& a_{32}\end{pmatrix}=\begin{pmatrix}1& 6\\2& 5\\ 3& 4\end{pmatrix}.$$ Observe that all row sums are equal, but the sum of the square of the squares is not the same for each row. Extend the above array to a $3\times k$ array $(a_{ij})_{3\times k}$ for a suitable $k$, adding more columns, using the numbers $7,8,9,\dots ,3k$ such that $$\sum_{j=1}^k a_{1j}=\sum_{j=1}^k a_{2j}=\sum_{j=1}^k a_{3j}~~\text{and}~~\sum_{j=1}^k (a_{1j})^2=\sum_{j=1}^k (a_{2j})^2=\sum_{j=1}^k (a_{3j})^2$$

Can two perfect cubes fit between two consecutive perfect squares? In other words, do there exist positive integers $a$, $b$, $n$ such that $n^2 < a^3 < b^3 < (n + 1)^2$? [i](3 points)[/i]

Find the number of integers $n$ with $1 \le n \le 2017$ so that $(n-2)(n-0)(n-1)(n-7)$ is an integer multiple of $1001$.

[i]20 problems for 20 minutes.[/i] [b]p1.[/b] Evaluate $=\frac{1}{2 \cdot 3 \cdot 4}+\frac{1}{3 \cdot 4 \cdot 5}$. [b]p2.[/b] A regular hexagon and a regular $n$-sided polygon have the same perimeter. If the ratio of the side length of the hexagon to the side length of the $n$-sided polygon is $2 : 1$, what is $n$? [b]p3.[/b] How many nonzero digits are there in the decimal representation of $2 \cdot 10\cdot 500 \cdot 2500$? [b]p4.[/b] When the numerator of a certain fraction is increased by $2012$, the value of the fraction increases by $2$. What is the denominator of the fraction? [b]p5.[/b] Sam did the computation $1 - 10 \cdot a + 22$, where $a$ is some real number, except he messed up his order of operations and computed the multiplication last; that is, he found the value of $(1 - 10) \cdot (a + 22)$ instead. Luckily, he still ended up with the right answer. What is $a$? [b]p6.[/b] Let $n! = n \cdot(n-1) \cdot\cdot\cdot 2 \cdot 1$. For how many integers $n$ between $1$ and $100$ inclusive is $n!$ divisible by $36$? [b]p7.[/b] Simplify the expression $\sqrt{\frac{3 \cdot 27^3}{27 \cdot 3^3}}$ [b]p8.[/b] Four points $A,B,C,D$ lie on a line in that order such that $\frac{AB}{CB}=\frac{AD}{CD}$ . Let $M$ be the midpoint of segment $AC$. If $AB = 6$, $BC = 2$, compute $MB \cdot MD$. [b]p9.[/b] Allan has a deck with $8$ cards, numbered $1$, $1$, $2$, $2$, $3$, $3$, $4$, $4$. He pulls out cards without replacement, until he pulls out an even numbered card, and then he stops. What is the probability that he pulls out exactly $2$ cards? [b]p10.[/b] Starting from the sequence $(3, 4, 5, 6, 7, 8, ... )$, one applies the following operation repeatedly. In each operation, we change the sequence $$(a_1, a_2, a_3, ... , a_{a_1-1}, a_{a_1} , a_{a_1+1},...)$$ to the sequence $$(a_2, a_3, ... , a_{a_1} , a_1, a_{a_1+1}, ...) .$$ (In other words, for a sequence starting with$ x$, we shift each of the next $x-1$ term to the left by one, and put x immediately to the right of these numbers, and keep the rest of the terms unchanged. For example, after one operation, the sequence is $(4, 5, 3, 6, 7, 8, ... )$, and after two operations, the sequence becomes $(5, 3, 6, 4, 7, 8,... )$. How many operations will it take to obtain a sequence of the form $(7, ... )$ (that is, a sequence starting with $7$)? [b]p11.[/b] How many ways are there to place $4$ balls into a $4\times 6$ grid such that no column or row has more than one ball in it? (Rotations and reflections are considered distinct.) [b]p12.[/b] Point $P$ lies inside triangle $ABC$ such that $\angle PBC = 30^o$ and $\angle PAC = 20^o$. If $\angle APB$ is a right angle, find the measure of $\angle BCA$ in degrees. [b]p13.[/b] What is the largest prime factor of $9^3 - 4^3$? [b]p14.[/b] Joey writes down the numbers $1$ through $10$ and crosses one number out. He then adds the remaining numbers. What is the probability that the sum is less than or equal to $47$? [b]p15.[/b] In the coordinate plane, a lattice point is a point whose coordinates are integers. There is a pile of grass at every lattice point in the coordinate plane. A certain cow can only eat piles of grass that are at most $3$ units away from the origin. How many piles of grass can she eat? [b]p16.[/b] A book has 1000 pages numbered $1$, $2$, $...$ , $1000$. The pages are numbered so that pages $1$ and $2$ are back to back on a single sheet, pages $3$ and $4$ are back to back on the next sheet, and so on, with pages $999$ and $1000$ being back to back on the last sheet. How many pairs of pages that are back to back (on a single sheet) share no digits in the same position? (For example, pages $9$ and $10$, and pages $89$ and $90$.) [b]p17.[/b] Find a pair of integers $(a, b)$ for which $\frac{10^a}{a!}=\frac{10^b}{b!}$ and $a < b$. [b]p18.[/b] Find all ordered pairs $(x, y)$ of real numbers satisfying $$\begin{cases} -x^2 + 3y^2 - 5x + 7y + 4 = 0 \\ 2x^2 - 2y^2 - x + y + 21 = 0 \end{cases}$$ [b]p19.[/b] There are six blank fish drawn in a line on a piece of paper. Lucy wants to color them either red or blue, but will not color two adjacent fish red. In how many ways can Lucy color the fish? [b]p20.[/b] There are sixteen $100$-gram balls and sixteen $99$-gram balls on a table (the balls are visibly indistinguishable). You are given a balance scale with two sides that reports which side is heavier or that the two sides have equal weights. A weighing is defined as reading the result of the balance scale: For example, if you place three balls on each side, look at the result, then add two more balls to each side, and look at the result again, then two weighings have been performed. You wish to pick out two different sets of balls (from the $32$ balls) with equal numbers of balls in them but different total weights. What is the minimal number of weighings needed to ensure this? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Given an integer $n$. We rearrange the digits of $n$ to get another number $m$. Prove that it is impossible to get $m+n = 999999999$.

Let $f(x) = 3x + 2.$ Prove that there exists $m \in \mathbb{N}$ such that $f^{100}(m)$ is divisible by $1988$.

$(SWE 6)$ Prove that there are infinitely many positive integers that cannot be expressed as the sum of squares of three positive integers.

Let $p$ - a prime, where $p> 11$. Prove that there exists a number $k$ such that the product $p \cdot k$ can be written in the decimal system with only ones.

Prove that every rational number is representable as $x^4+y^4-z^4-t^4$ with rational $x,y,z,t$.

For all positive integers $n$ and $k$, define $F(n,k) = \sum_{r = 1}^n r^{2k - 1}$. Prove that $F(n,1)$ divides $F(n,k)$.

A positive integer is called [i]nice[/i] if the sum of its digits in the number system with base $ 3$ is divisible by $ 3$. Calculate the sum of the first $ 2005$ nice positive integers.

Determine all pairs $(p, q)$ of primes for which $p^{q+1}+q^{p+1}$ is a perfect square.

Find all odd prime $p$ such that $1+k(p-1)$ is prime for all integer $k$ where $1 \le k \le \dfrac{p-1}{2}$.

A rectangular floor that is $10$ feet wide and $17$ feet long is tiled with $170$ one-foot square tiles. A bug walks from one corner to the opposite corner in a straight line. Including the first and the last tile, how many tiles does the bug visit? $\textbf{(A) } 17 \qquad\textbf{(B) } 25 \qquad\textbf{(C) } 26 \qquad\textbf{(D) } 27 \qquad\textbf{(E) } 28$

Given a prime number $p$, find all integer solutions of $p(x+y) = xy$.

The sets $A = \{z : z^{18} = 1\}$ and $B = \{w : w^{48} = 1\}$ are both sets of complex roots of unity. The set $C = \{zw : z \in A \ \text{and} \ w \in B\}$ is also a set of complex roots of unity. How many distinct elements are in $C$?

Prove that for all $a, b$ and $x_0$ positive integers, in the sequence $x_1, x_2, x_3, \cdots$ defined by $$x_{n+1}=ax_n+b, n\geq 0$$ Exist an $x_i$ that is not prime for some $i\geq 1$

Prove that if the length and breadth of a rectangle are both odd integers, then there does not exist a point $P$ inside the rectangle such that each of the distances from $P$ to the 4 corners of the rectangle is an integer.

For the integer $n>1$, define $D(n)=\{ a-b\mid ab=n, a>b>0, a,b\in\mathbb{N} \}$. Prove that for any integer $k>1$, there exists pairwise distinct positive integers $n_1,n_2,\ldots,n_k$ such that $n_1,\ldots,n_k>1$ and $|D(n_1)\cap D(n_2)\cap\cdots\cap D(n_k)|\geq 2$.

Let $(a_n)_{n\geq0}$ and $(b_n)_{n \geq 0}$ be two sequences of natural numbers. Determine whether there exists a pair $(p, q)$ of natural numbers that satisfy \[p < q \quad \text{ and } \quad a_p \leq a_q, b_p \leq b_q.\]

Find all triples $(x,y,z)$ of positive integers such that $x \leq y \leq z$ and \[x^3(y^3+z^3)=2012(xyz+2).\]

Find the least prime factor of the number $\frac{2016^{2016}-3}{3}$.