Let $a_1,a_2,a_3,...$ be a strictly increasing sequence of positive integers. A number $a_n$ in the sequence is said to be [i]lucky [/i] if it is the sum of several (not necessarily distinct) smaller terms of the sequence, and [i]unlucky [/i]otherwise. (For example, in the sequence $4,6,14,15,25,...$ numbers $4,6,15$ are [i]unlucky[/i], while $14 = 4+4+6$ and $25 = 4+6+15$ are [i]lucky[/i].) Prove that there are only finitely many [i]unlucky [/i]numbers in the sequence.

How many rational solutions are there to $5x^2+2y^2=1$?

An integer number $m\geq 1$ is [i]mexica[/i] if it's of the form $n^{d(n)}$, where $n$ is a positive integer and $d(n)$ is the number of positive integers which divide $n$. Find all mexica numbers less than $2019$. Note. The divisors of $n$ include $1$ and $n$; for example, $d(12)=6$, since $1, 2, 3, 4, 6, 12$ are all the positive divisors of $12$. [i]Proposed by Cuauhtémoc Gómez[/i]

Let $P_1, \ldots , P_s$ be arithmetic progressions of integers, the following conditions being satisfied: [b](i)[/b] each integer belongs to at least one of them; [b](ii)[/b] each progression contains a number which does not belong to other progressions. Denote by $n$ the least common multiple of the ratios of these progressions; let $n=p_1^{\alpha_1} \cdots p_k^{\alpha_k}$ its prime factorization. Prove that \[s \geq 1 + \sum^k_{i=1} \alpha_i (p_i - 1).\] [i]Proposed by Dierk Schleicher, Germany[/i]

Prove that there are infinitely many positive integers $n$ with the following property: For any $n$ integers $a_{1},a_{2},...,a_{n}$ which form in arithmetic progression, both the mean and the standard deviation of the set $\{a_{1},a_{2},...,a_{n}\}$ are integers. [i]Remark[/i]. The mean and standard deviation of the set $\{x_{1},x_{2},...,x_{n}\}$ are defined by $\overline{x}=\frac{x_{1}+x_{2}+...+x_{n}}{n}$ and $\sqrt{\frac{\sum (x_{i}-\overline{x})^{2}}{n}}$, respectively.

Determine the minimum possible amount of distinct prime divisors of $19^{4n}+4$, for a positive integer $n$.

[b]p1.[/b] Steven has just learned about polynomials and he is struggling with the following problem: expand $(1-2x)^7$ as $a_0 +a_1x+...+a_7x^7$ . Help Steven solve this problem by telling him what $a_1 +a_2 +...+a_7$ is. [b]p2.[/b] Each element of the set ${2, 3, 4, ..., 100}$ is colored. A number has the same color as any divisor of it. What is the maximum number of colors? [b]p3.[/b] Fuchsia is selecting $24$ balls out of $3$ boxes. One box contains blue balls, one red balls and one yellow balls. They each have a hundred balls. It is required that she takes at least one ball from each box and that the numbers of balls selected from each box are distinct. In how many ways can she select the $24$ balls? [b]p4.[/b] Find the perfect square that can be written in the form $\overline{abcd} - \overline{dcba}$ where $a, b, c, d$ are non zero digits and $b < c$. $\overline{abcd}$ is the number in base $10$ with digits $a, b, c, d$ written in this order. [b]p5.[/b] Steven has $100$ boxes labeled from $ 1$ to $100$. Every box contains at most $10$ balls. The number of balls in boxes labeled with consecutive numbers differ by $ 1$. The boxes labeled $1,4,7,10,...,100$ have a total of $301$ balls. What is the maximum number of balls Steven can have? [b]p6.[/b] In acute $\vartriangle ABC$, $AB=4$. Let $D$ be the point on $BC$ such that $\angle BAD = \angle CAD$. Let $AD$ intersect the circumcircle of $\vartriangle ABC$ at $X$. Let $\Gamma$ be the circle through $D$ and $X$ that is tangent to $AB$ at $P$. If $AP = 6$, compute $AC$. [b]p7.[/b] Consider a $15\times 15$ square decomposed into unit squares. Consider a coloring of the vertices of the unit squares into two colors, red and blue such that there are $133$ red vertices. Out of these $133$, two vertices are vertices of the big square and $32$ of them are located on the sides of the big square. The sides of the unit squares are colored into three colors. If both endpoints of a side are colored red then the side is colored red. If both endpoints of a side are colored blue then the side is colored blue. Otherwise the side is colored green. If we have $196$ green sides, how many blue sides do we have? [b]p8.[/b] Carl has $10$ piles of rocks, each pile with a different number of rocks. He notices that he can redistribute the rocks in any pile to the other $9$ piles to make the other $9$ piles have the same number of rocks. What is the minimum number of rocks in the biggest pile? [b]p9.[/b] Suppose that Tony picks a random integer between $1$ and $6$ inclusive such that the probability that he picks a number is directly proportional to the the number itself. Danny picks a number between $1$ and $7$ inclusive using the same rule as Tony. What is the probability that Tony’s number is greater than Danny’s number? [b]p10.[/b] Mike wrote on the board the numbers $1, 2, ..., n$. At every step, he chooses two of these numbers, deletes them and replaces them with the least prime factor of their sum. He does this until he is left with the number $101$ on the board. What is the minimum value of $n$ for which this is possible? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

(a) Let $\gcd(m, k) = 1$. Prove that there exist integers $a_1, a_2, . . . , a_m$ and $b_1, b_2, . . . , b_k$ such that each product $a_ib_j$ ($i = 1, 2, \cdots ,m; \ j = 1, 2, \cdots, k$) gives a different residue when divided by $mk.$ (b) Let $\gcd(m, k) > 1$. Prove that for any integers $a_1, a_2, . . . , a_m$ and $b_1, b_2, . . . , b_k$ there must be two products $a_ib_j$ and $a_sb_t$ ($(i, j) \neq (s, t)$) that give the same residue when divided by $mk.$ [i]Proposed by Hungary.[/i]

Define $p(n)$ to be th product of all non-zero digits of $n$. For instance $p(5)=5$, $p(27)=14$, $p(101)=1$ and so on. Find the greatest prime divisor of the following expression: \[p(1)+p(2)+p(3)+...+p(999).\]

We will say that a natural number is [i]acceptable[/i] if it has at most $9$ distinct prime divisors. There is a stack of $100!=1\times2\times\cdots\times100$ stones. A [i]legal move[/i] consists in removing $k$ stones from the stack, where $k$ is an acceptable number. Two players, Lucas and Nicolas, take turns making legal moves; Lucas starts the game. The one who removes the last stone wins. Determine which of the players has a winning strategy and describe this strategy.

a) Prove that for every positive integer $y$ the equality ${\rm lcm}(x,y+1)\cdot {\rm lcm}(x+1,y)=x(x+1)$ holds for infinitely many positive integers $x.$ b) Prove that there exists positive integer $y$ such that the equality ${\rm lcm}(x,y+1)\cdot {\rm lcm}(x+1,y)=y(y+1)$ holds for at least 2014 positive integers $x.$

Find all positive integers $n$ for which the largest prime divisor of $n^2+3$ is equal to the least prime divisor of $n^4+6.$

$x,y,z,a,b,c$ are positive integers that satisfy $xy \equiv a \pmod z$, $yz \equiv b \pmod x$, $zx \equiv c \pmod y$. Prove that $\min{\{x,y,z\}} \le ab+bc+ca$.

For the all $(m,n,k)$ positive integer triples such that $|m^k-n!| \le n$ find the maximum value of $\frac{n}{m}$ [i]Proposed by Melih Üçer[/i]

Let $A_n$ be the number of arranged n-tuples of natural numbers $(a_1,a_2…a_n)$, such that $\frac{1}{a_1} +\frac{1}{a_2} +...+\frac{1}{a_n} =1$. Find the parity of $A_{68}$.

Find all triples $(p, x, y)$ consisting of a prime number $p$ and two positive integers $x$ and $y$ such that $x^{p -1} + y$ and $x + y^ {p -1}$ are both powers of $p$. [i]Proposed by Belgium[/i]

A rectangle has side lengths $6$ and $8$. There are relatively prime positive integers $m$ and $n$ so that $\tfrac{m}{n}$ is the probability that a point randomly selected from the inside of the rectangle is closer to a side of the rectangle than to either diagonal of the rectangle. Find $m + n$.

A natural number $n$ is called breakable if there exist positive integers $a,b,x,y$ such that $a+b = n$ and $\frac{x}{a}+\frac{y}{b}= 1$. Find all breakable numbers.

Prove that for all $n\in \mathbb{N}$ there exist natural numbers $a_1,a_2,...,a_n$ such that: $(i)a_1>a_2>...>a_n$ $(ii)a_i|a^2_{i+1},\forall i\in\{1,2,...,n-1\}$ $(iii)a_i\nmid a_j,\forall i,j\in \{1,2,...,n\},i\neq j$

There are a buch of 2000 stones. Two players play alternatively, following the next rules: ($a$)On each turn, the player can take 1, 2, 3, 4 or 5 stones [b]of[/b] the bunch. ($b$) On each turn, the player has forbidden to take the exact same amount of stones that the other player took just before of him in the last play. The loser is the player who can't make a valid play. Determine which player has winning strategy and give such strategy.

Determine all digits $z$ such that for each integer $k \ge 1$ there exists an integer $n\ge 1$ with the property that the decimal representation of $n^9$ ends with at least $k$ digits $z$. [i](Proposed by Walther Janous)[/i]

On the board written numbers from 1 to 25 . Bob can pick any three of them say $a,b,c$ and replace by $a^3+b^3+c^3$ . Prove that last number on the board can not be $2013^3$.

a) Show that there are five integers $A, B, C, D$, and $E$ such that $2018 = A^5 + B^5 + C^5 + D^5 + E^5$ b) Show that there are no four integers $A, B, C$ and $D$ such that $2018 = A^5 + B^5 + C^5 + D^5$

[u]Round 1[/u] [b]p1.[/b] Tom and Jerry stole a chain of $7$ sausages and are now trying to divide the bounty. They take turns biting the sausages at one of the connections. When one of them breaks a connection, he may eat any single sausages that may fall out. Tom takes the first bite. Each of them is trying his best to eat more sausages than his opponent. Who will succeed? [b]p2. [/b]The King of the Mountain Dwarves wants to light his underground throne room by placing several torches so that the whole room is lit. The king, being very miserly, wants to use as few torches as possible. What is the least number of torches he could use? (You should show why he can't do it with a smaller number of torches.) This is the shape of the throne room: [img]https://cdn.artofproblemsolving.com/attachments/b/2/719daafd91fc9a11b8e147bb24cb66b7a684e9.png[/img] Also, the walls in all rooms are lined with velvet and do not reflect the light. For example, the picture on the right shows how another room in the castle is partially lit. [img]https://cdn.artofproblemsolving.com/attachments/5/1/0f6971274e8c2ff3f2d0fa484b567ff3d631fb.png[/img] [b]p3.[/b] In the Hundred Acre Wood, all the animals are either knights or liars. Knights always tell the truth and liars always lie. One day in the Wood, Winnie-the-Pooh, a knight, decides to visit his friend Rabbit, also a noble knight. Upon arrival, Pooh finds his friend sitting at a round table with $5$ other guests. One-by-one, Pooh asks each person at the table how many of his two neighbors are knights. Surprisingly, he gets the same answer from everybody! "Oh bother!" proclaims Pooh. "I still don't have enough information to figure out how many knights are at this table." "But it's my birthday," adds one of the guests. "Yes, it's his birthday!" agrees his neighbor. Now Pooh can tell how many knights are at the table. Can you? [b]p4.[/b] Several girls participate in a tennis tournament in which each player plays each other player exactly once. At the end of the tournament, it turns out that each player has lost at least one of her games. Prove that it is possible to find three players $A$, $B$, and $C$ such that $A$ defeated $B$, $B$ defeated $C$, and $C$ defeated $A$. [b]p5.[/b] There are $40$ piles of stones with an equal number of stones in each. Two players, Ann and Bob, can select any two piles of stones and combine them into one bigger pile, as long as this pile would not contain more than half of all the stones on the table. A player who can’t make a move loses. Ann goes first. Who wins? [u]Round 2[/u] [b]p6.[/b] In a galaxy far, far away, there is a United Galactic Senate with $100$ Senators. Each Senator has no more than three enemies. Tired of their arguments, the Senators want to split into two parties so that each Senator has no more than one enemy in his own party. Prove that they can do this. (Note: If $A$ is an enemy of $B$, then $B$ is an enemy of $A$.) [b]p7.[/b] Harry has a $2012$ by $2012$ chessboard and checkers numbered from $1$ to $2012 \times 2012$. Can he place all the checkers on the chessboard in such a way that whatever row and column Professor Snape picks, Harry will be able to choose three checkers from this row and this column such that the product of the numbers on two of the checkers will be equal to the number on the third? [img]https://cdn.artofproblemsolving.com/attachments/b/3/a87d559b340ceefee485f41c8fe44ae9a59113.png[/img] PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

The number "3" is written on a board. Ana and Bernardo take turns, starting with Ana, to play the following game. If the number written on the board is $n$, the player in his/her turn must replace it by an integer $m$ coprime with $n$ and such that $n<m<n^2$. The first player that reaches a number greater or equal than 2016 loses. Determine which of the players has a winning strategy and describe it.