Let $p$ be a prime number, and let $a_1, a_2, \dots , a_p$ and $b_1, b_2, \dots , b_p$ be $2p$ (not necessarily distinct) integers chosen from the set $\{1, 2, \dots , p - 1\}$. Prove that there exist integers $i$ and $j$ such that $1 \le i < j \le p$ and $p$ divides $a_ib_j-a_jb_i$.

Find the unique real number $c$ such that the polynomial $x^3+cx+c$ has exactly two real roots.

(a) Let $a,x,y$ be positive integers. Prove: if $x\ne y$, the also \[ax+\gcd(a,x)+\text{lcm}(a,x)\ne ay+\gcd(a,y)+\text{lcm}(a,y).\] (b) Show that there are no two positive integers $a$ and $b$ such that \[ab+\gcd(a,b)+\text{lcm}(a,b)=2014.\]

Does there exist on the Cartesian plane a convex $2023$-gon with vertices at integer points, such that the lengths of all its sides are equal? [i]Proposed by Anton Trygub[/i]

Odd numbers $x,y,z$ such that $gcd(x,y,z)=1$ are given. It turned out that $x^2+y^2+z^2 \vdots x+y+z$ Prove that $x+y+z-2$ is not divisible by $3$

Let $S$ be a set of $49$-digit numbers $n$, with the property that each of the digits $1, 2, 3, \dots, 7$ appears in the decimal expansion of $n$ seven times (and $8, 9$ and $0$ do not appear). Show that no two distinct elements of $S$ divide each other.

Let $\gamma=\alpha \times \beta$ where \[\alpha=999 \cdots 9\] (2010 '9') and \[\beta=444 \cdots 4\] (2010 '4') Find the sum of digits of $\gamma$.

Find all pairs of positive prime numbers $(p,q)$ such that $p^5+p^3+2=q^2-q$

Let $n$ be an odd positive integer. Given are $n$ balls - black and white, placed on a circle. For a integer $1\leq k \leq n-1$, call $f(k)$ the number of balls, such that after shifting them with $k$ positions clockwise, their color doesn't change. a) Prove that for all $n$, there is a $k$ with $f(k) \geq \frac{n-1}{2}$. b) Prove that there are infinitely many $n$ (and corresponding colorings for them) such that $f(k)\leq \frac{n-1}{2}$ for all $k$.

Let $a,b$ be nonnegative integers. Prove that $5a>7b$ if and only if there exist nonnegative integers $x,y,z,t$ such that \begin{align*} x+2y+3z+7t&=a,\\ y+2z+5t&=b. \end{align*}

Determine the values of $n \in N$ such that a square of side $n$ can be split into a square of side $1$ and five rectangles whose side measures are $10$ distinct natural numbers and all greater than $1$.

1. For [i]n[/i] a positive integer, [i]n[/i]! = 1 $\cdot$ 2 $\cdot$ 3 $\dots$ ([i]n[/i] - 1) $\cdot$ [i]n[/i] is the product of the positive integers from 1 to [i]n[/i]. Determine, with proof, all positive integers [i]n[/i] for which [i]n[/i]! + 3 is a power of 3.

Find all positive integers $n\geq1$ such that there exists a pair $(a,b)$ of positive integers, such that $a^2+b+3$ is not divisible by the cube of any prime, and $$n=\frac{ab+3b+8}{a^2+b+3}.$$

Let $a$ and $n$ be positive integers such that the greatest common divisor of $a$ and $n!$ is $1$. Prove that $n!$ divides $a^{n!}-1$.

Prove that if $7$ divides $a^2 + b^2 + 1$, then $7$ does not divide $a + b$.

Determine all positive integers whose squares end in $196$.

Find all natural numbers $n > 1$ for which there exists such natural numbers $a_1,a_2,...,a_n$ for which the numbers $\{a_i +a_j | 1 \le i \le j \le n \}$ form a full system modulo $\frac{n(n+1)}{2}$.

Consider a sequence of non-negative integers g$_1,g_2,g_3,...$ each consisting of three digits (numbers smaller than $100$ are also written with three digits; the number $27$, for example, is written as $027$). Each number consists of the preceding by taking the product of the three digits that make up the preceding. The resulting sequence is of course dependent on the choice of $g_1$ (e.g. $g_1 = 359$ leads to $g_2= 135$, $g_3= 015$, $g_4 = 000$).Prove that independent of the choice of $g_1$: (a) $g_{n+1}\le g_n$ (b) $g_{10}= 000$.

Let $w (n)$ denote the number of different prime numbers by which $n$ is divisible. Prove that there are infinitely many natural numbers $n$ such that $w(n) < w(n + 1) < w(n + 2)$.

[u]Round 1 [/u] [b]p1.[/b] Pirate Jim had $8$ boxes with gun powder weighing $1, 2, 3, 4, 5, 6, 7$, and $8$ pounds (the weight is printed on top of every box). Pirate Bob hid a $1$-pound gold bar in one of these boxes. Pirate Jim has a balance scale that he can use, but he cannot open any of the boxes. Help him find the box with the gold bar using two weighings on the balance scale. [b]p2.[/b] James Bond will spend one day at Dr. Evil's mansion to try to determine the answers to two questions: a) Is Dr. Evil at home? b) Does Dr. Evil have an army of ninjas? The parlor in Dr. Evil's mansion has three windows. At noon, Mr. Bond will sneak into the parlor and use open or closed windows to signal his answers. When he enters the parlor, some windows may already be opened, and Mr. Bond will only have time to open or close one window (or leave them all as they are). Help Mr. Bond and Moneypenny design a code that will tell Moneypenny the answers to both questions when she drives by later that night and looks at the windows. Note that Moneypenny will not have any way to know which window Mr. Bond opened or closed. [b]p3.[/b] Suppose that you have a triangle in which all three side lengths and all three heights are integers. Prove that if these six lengths are all different, there cannot be four prime numbers among them. p4. Fred and George have designed the Amazing Maze, a $5\times 5$ grid of rooms, with Adorable Doors in each wall between rooms. If you pass through a door in one direction, you gain a gold coin. If you pass through the same door in the opposite direction, you lose a gold coin. The brothers designed the maze so that if you ever come back to the room in which you started, you will find that your money has not changed. Ron entered the northwest corner of the maze with no money. After walking through the maze for a while, he had $8$ shiny gold coins in his pocket, at which point he magically teleported himself out of the maze. Knowing this, can you determine whether you will gain or lose a coin when you leave the central room through the north door? [b]p5.[/b] Bill and Charlie are playing a game on an infinite strip of graph paper. On Bill’s turn, he marks two empty squares of his choice (not necessarily adjacent) with crosses. Charlie, on his turn, can erase any number of crosses, as long as they are all adjacent to each other. Bill wants to create a line of $2013$ crosses in a row. Can Charlie stop him? [u]Round 2 [/u] [b]p6.[/b] $1000$ non-zero numbers are written around a circle and every other number is underlined. It happens that each underlined number is equal to the sum of its two neighbors and that each non-underlined number is equal to the product of its two neighbors. What could the sum of all the numbers written on the circle be? [b]p7.[/b] A grasshopper is sitting at the edge of a circle of radius $3$ inches. He can hop exactly $4$ inches in any direction, as long as he stays within the circle. Which points inside the circle can the grasshopper reach if he can make as many jumps as he likes? [img]https://cdn.artofproblemsolving.com/attachments/1/d/39b34b2b4afe607c1232f4ce9dec040a34b0c8.png[/img] PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Prove that there are infinitely many pairs $(a, b)$ of positive integers with the following properties: $\bullet$ $a+b$ divides $ab+1$, $\bullet$ $a-b$ divides $ab -1$, $\bullet$ $b > 2$ and $a > b\sqrt3 - 1$.

[b]p1.[/b] There are $5$ weights of masses $1,2,3,5$, and $10$ grams. One of the weights is counterfeit (its weight is different from what is written, it is unknown if the weight is heavier or lighter). How to find the counterfeit weight using simple balance scales only twice? [b]p2.[/b] There are $998$ candies and chocolate bars and $499$ bags. Each bag may contain two items (either two candies, or two chocolate bars, or one candy and one chocolate bar). Ann distributed candies and chocolate bars in such a way that half of the candies share a bag with a chocolate bar. Helen wants to redistribute items in the same bags in such a way that half of the chocolate bars would share a bag with a candy. Is it possible to achieve that? [b]p3.[/b] Insert in sequence $2222222222$ arithmetic operations and brackets to get the number $999$ (For instance, from the sequence $22222$ one can get the number $45$: $22*2+2/2 = 45$). [b]p4.[/b] Put numbers from $15$ to $23$ in a $ 3\times 3$ table in such a way to make all sums of numbers in two neighboring cells distinct (neighboring cells share one common side). [b]p5.[/b] All integers from $1$ to $200$ are colored in white and black colors. Integers $1$ and $200$ are black, $11$ and $20$ are white. Prove that there are two black and two white numbers whose sums are equal. [b]p6.[/b] Show that $38$ is the sum of few positive integers (not necessarily, distinct), the sum of whose reciprocals is equal to $1$. (For instance, $11=6+3+2$, $1/16+1/13+1/12=1$.) PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Show that the following implication holds for any two complex numbers $x$ and $y$: if $x+y$, $x^2+y^2$, $x^3+y^3$, $x^4+y^4\in\mathbb Z$, then $x^n+y^n\in\mathbb Z$ for all natural n.

Suppose that we have two natural numbers $x , y \le 100!$ with undetermined values. Prove that there exist natural numbers $m , n$ such that values of $x , y$ get uniquely determined according to value of $\varphi(d(my))+d(\varphi(nx))$. ( for each natural number $n$ , $d(n)$ is number of its positive divisors and $\varphi(n)$ is the number of the numbers less that $n$ which are relatively prime to $n$. ) [i]Proposed by Mehran Talaei[/i]

The $n$-tuple $(a_1,a_2,\ldots,a_n)$ of integers satisfies the following: [list](i) $1\le a_1<a_2<\cdots < a_n\le 50$ (ii) for each $n$-tuple $(b_1,b_2,\ldots,b_n)$ of positive integers, there exist a positive integer $m$ and an $n$-tuple $(c_1,c_2,\ldots,c_n)$ of positive integers such that \[mb_i=c_i^{a_i}\qquad\text{for } i=1,2,\ldots,n. \] [/list]Prove that $n\le 16$ and determine the number of $n$-tuples $(a_1,a_2,\ldots,a_n$) satisfying these conditions for $n=16$.