Let $n>1$ be a given positive integer. Petro and Vasyl play the following game. They take turns making moves and Petro goes first. In one turn, a player chooses one of the numbers from $1$ to $n$ that wasn't selected before and writes it on the board. The first player after whose turn the product of the numbers on the board will be divisible by $n$ loses. Who wins if every player wants to win? Find answer for each $n>1$. [i]Proposed by Mykhailo Shtandenko, Anton Trygub[/i]

Tanya chose a natural number $X\le100$, and Sasha is trying to guess this number. He can select two natural numbers $M$ and $N$ less than $100$ and ask about $\gcd(X+M,N)$. Show that Sasha can determine Tanya's number with at most seven questions.

A positive integer is called [i]cool[/i] if it is divisible by the square of each of its prime divisors. Prove that $n$ and $n+1$ are simultaneously cool for infinitely many $n$.

Let $ n \ge 2009$ be an integer and define the set: \[ S \equal{} \{2^x|7 \le x \le n, x \in \mathbb{N}\}. \] Let $ A$ be a subset of $ S$ and the sum of last three digits of each element of $ A$ is $ 8$. Let $ n(X)$ be the number of elements of $ X$. Prove that \[ \frac {28}{2009} < \frac {n(A)}{n(S)} < \frac {82}{2009}. \]

Prove that for all integers $n\ge3$ there exists a set $A_n=\{a_1,a_2,\ldots,a_n\}$ of $n$ distinct natural numbers such that, for each $i=1,2,\ldots,n$, $$\prod_{\small{\begin{matrix}1\le k\le n\\k\ne i\end{matrix}}}a_k\equiv1\pmod{a_i}.$$

1) Cells of $8 \times 8$ table contain pairwise distinct positive integers. Each integer is prime or a product of two primes. It is known that for any integer $a$ from the table there exists integer written in the same row or in the same column such that it is not relatively prime with $a$. Find maximum possible number of prime integers in the table. 2) Cells of $2n \times 2n$ table, $n \ge 2,$ contain pairwise distinct positive integers. Each integer is prime or a product of two primes. It is known that for any integer $a$ from the table there exist integers written in the same row and in the same column such that they are not relatively prime with $a$. Find maximum possible number of prime integers in the table.

Prove that there is no pair of relatively prime positive integers $(a, b)$ that satisfy the equation $$a^3 + 2017a = b^3 -2017b.$$

Let $a_n$ be a series of positive integers with $a_1=1$ and for any arbitrary prime number $p$, the set $\{a_1,a_2,\cdots,a_p\}$ is a complete remainder system modulo $p$. Prove that $\lim_{n\rightarrow \infty} \cfrac{a_n}{n}=1$.

Determine if there exists a (three-variable) polynomial $P(x,y,z)$ with integer coefficients satisfying the following property: a positive integer $n$ is [i]not[/i] a perfect square if and only if there is a triple $(x,y,z)$ of positive integers such that $P(x,y,z) = n$.

Let $b,n > 1$ be integers. Suppose that for each $k > 1$ there exists an integer $a_k$ such that $b - a^n_k$ is divisible by $k$. Prove that $b = A^n$ for some integer $A$. [i]Author: Dan Brown, Canada[/i]

Prove that the number \[ 3 \underbrace{99\ldots9}_{2025} \underbrace{60\ldots01}_{2025} \] is a square of a positive integer.

For positive integers $x$, $y$ and $z$ holds $\frac{1}{x^2}+\frac{1}{y^2}=\frac{1}{z^2}$. Prove that $xyz\geq 3600$

[i]a)[/i] Call a four-digit number $(xyzt)_B$ in the number system with base $B$ stable if $(xyzt)_B = (dcba)_B - (abcd)_B$, where $a \leq b \leq c \leq d$ are the digits of $(xyzt)_B$ in ascending order. Determine all stable numbers in the number system with base $B.$ [i]b)[/i] With assumptions as in [i]a[/i], determine the number of bases $B \leq 1985$ such that there exists a stable number with base $B.$

[u]Round 1[/u] [b]p1.[/b] A box contains $1$ ball labelledW, $1$ ball labelled $E$, $1$ ball labelled $L$, $1$ ball labelled $C$, $1$ ball labelled $O$, $8$ balls labelled $M$, and $1$ last ball labelled $E$. One ball is randomly drawn from the box. The probability that the ball is labelled $E$ is $\frac{1}{a}$ . Find $a$. [b]p2.[/b] Let $$G +E +N = 7$$ $$G +E +O = 15$$ $$N +T = 22.$$ Find the value of $T +O$. [b]p3.[/b] The area of $\vartriangle LMT$ is $22$. Given that $MT = 4$ and that there is a right angle at $M$, find the length of $LM$. [u]Round 2[/u] [b]p4.[/b] Kevin chooses a positive $2$-digit integer, then adds $6$ times its unit digit and subtracts $3$ times its tens digit from itself. Find the greatest common factor of all possible resulting numbers. [b]p5.[/b] Find the maximum possible number of times circle $D$ can intersect pentagon $GRASS'$ over all possible choices of points $G$, $R$, $A$, $S$, and $S'$. [b]p6.[/b] Find the sum of the digits of the integer solution to $(\log_2 x) \cdot (\log_4 \sqrt{x}) = 36$. [u]Round 3[/u] [b]p7.[/b] Given that $x$ and $y$ are positive real numbers such that $x^2 + y = 20$, the maximum possible value of $x + y$ can be written as $\frac{a}{b}$ where $a$ and $b$ are relatively prime positive integers. Find $a +b$. [b]p8.[/b] In $\vartriangle DRK$, $DR = 13$, $DK = 14$, and $RK = 15$. Let $E$ be the point such that $ED = ER = EK$. Find the value of $\lfloor DE +RE +KE \rfloor$. [b]p9.[/b] Subaru the frog lives on lily pad $1$. There is a line of lily pads, numbered $2$, $3$, $4$, $5$, $6$, and $7$. Every minute, Subaru jumps from his current lily pad to a lily pad whose number is either $1$ or $2$ greater, chosen at random from valid possibilities. There are alligators on lily pads $2$ and $5$. If Subaru lands on an alligator, he dies and time rewinds back to when he was on lily pad number $1$. Find how many times Subaru is expected to die before he reaches pad $7$. [u]Round 4[/u] [b]p10.[/b] Find the sum of the following series: $$\sum^{\infty}_{i=1} = \frac{\sum^i_{j=1} j}{2^i}=\frac{1}{2^1}+\frac{1+2}{2^2}+\frac{1+2+3}{2^3}+\frac{1+2+3+4}{2^4}+... $$ [b]p11.[/b] Let $\phi (x)$ be the number of positive integers less than or equal to $x$ that are relatively prime to $x$. Find the sum of all $x$ such that $\phi (\phi(x)) = x -3$. Note that $1$ is relatively prime to every positive integer. [b]p12.[/b] On a piece of paper, Kevin draws a circle. Then, he draws two perpendicular lines. Finally, he draws two perpendicular rays originating from the same point (an $L$ shape). What is the maximum number of sections into which the lines and rays can split the circle? [u]Round 5 [/u] [b]p13.[/b] In quadrilateral $ABCD$, $\angle A = 90^o$, $\angle C = 60^o$, $\angle ABD = 25^o$, and $\angle BDC = 5^o$. Given that $AB = 4\sqrt3$, the area of quadrilateral $ABCD$ can be written as $a\sqrt{b}$. Find $10a +b$. [b]p14.[/b] The value of $$\sum^6_{n=2} \left( \frac{n^4 +1}{n^4 -1}\right) -2 \sum^6_{n=2}\left(\frac{n^3 -n^2+n}{n^4 -1}\right)$$ can be written as $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $100m+n$. [b]p15.[/b] Positive real numbers $x$ and $y$ satisfy the following $2$ equations. $$x^{1+x^{1+x^{1+...}}}= 8$$ $$\sqrt[24]{y +\sqrt[24]{y + \sqrt[24]{y +...}}} = x$$ Find the value of $\lfloor y \rfloor$. PS. You should use hide for answers. Rounds 6-9 have been posted [url=https://artofproblemsolving.com/community/c3h3167130p28823260]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

For positive integers $n$ and $k \geq 2$, define $E_k(n)$ as the greatest exponent $r$ such that $k^r$ divides $n!$. Prove that there are infinitely many $n$ such that $E_{10}(n) > E_9(n)$ and infinitely many $m$ such that $E_{10}(m) < E_9(m)$.

Call a pair of integers $a$ and $b$ square makers , if $ab+1$ is a perfect square. Determine for which $n$ is it possible to divide the set $\{1,2, \dots , 2n\}$ into $n$ pairs of square makers.

[b]p1. [/b] A clock has a long hand for minutes and a short hand for hours. A placement of those hands is [i]natural [/i] if you will see it in a correctly functioning clock. So, having both hands pointing straight up toward $12$ is natural and so is having the long hand pointing toward $6$ and the short hand half-way between $2$ and $3$. A natural placement of the hands is symmetric if you get another natural placement by interchanging the long and short hands. One kind of symmetric natural placement is when the hands are pointed in exactly the same direction. Are there symmetric natural placements of the hands in which the two hands are not pointed in exactly the same direction? If so, describe one such placement. If not, explain why none are possible. [b]p2.[/b] Let $\frac{m}{n}$ be a fraction such that when you write out the decimal expansion of $\frac{m}{n}$ , it eventually ends up with the four digits $2001$ repeated over and over and over. Prove that $101$ divides $n$. [b]p3.[/b] Consider the following two questions: Question $1$: I am thinking of a number between $0$ and $15$. You get to ask me seven yes-or-no questions, and I am allowed to lie at most once in answering your questions. What seven questions can you ask that will always allow you to determine the number? Note: You need to come up with seven questions that are independent of the answers that are received. In other words, you are not allowed to say, "If the answer to question $1$ is yes, then question $2$ is XXX; but if the answer to question $1$ is no, then question $2$ is YYY." Question $2$: Consider the set $S$ of all seven-tuples of zeros and ones. What sixteen elements of $S$ can you choose so that every pair of your chosen seven-tuples differ in at least three coordinates? a. These two questions are closely related. Show that an answer to Question $1$ gives an answer to Question $2$. b. Answer either Question $1$ or Question $2$. [b]p4.[/b] You may wish to use the angle addition formulas for the sine and cosine functions: $\sin (\alpha + \beta) = \sin \alpha \cos \beta + \cos \alpha \sin \beta$ $\cos (\alpha + \beta) = \cos \alpha \cos \beta - \sin \alpha \sin \beta$ a) Prove the identity $(\sin x)(1 + 2 \cos 2x) = \sin (3x)$. b) For any positive integer $n$, prove the identity $$(sin x)(1 + 2 \cos 2x + 2\cos 4x + ... +2\cos 2nx) = \sin ((2n +1)x)$$ [b]p5.[/b] Define the set $\Omega$ in the $xy$-plane as the union of the regions bounded by the three geometric figures: triangle $A$ with vertices $(0.5, 1.5)$, $(1.5, 0.5)$ and $(0.5,-0.5)$, triangle $B$ with vertices $(-0.5,1.5)$, $(-1.5,-0.5)$ and $(-0.5, 0.5)$, and rectangle $C$ with corners $(0.5, 1.0)$, $(-0.5, 1.0)$, $(-0.5,-1.0)$, and $(0.5,-1.0)$. a. Explain how copies of $\Omega$ can be used to cover the $xy$-plane. The copies are obtained by translating $\Omega$ in the $xy$-plane, and copies can intersect only along their edges. b. We can define a transformation of the plane as follows: map any point $(x, y)$ to $(x + G, x + y + G)$, where $G = 1$ if $y < -2x$, $G = -1$ if $y > -2x$, and $G = 0$ if $y = -2x$. Prove that every point in $\Omega$ is transformed into another point in $\Omega$, and that there are at least two points in $\Omega$ that are transformed into the same point. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Solve in the natural numbers the equation $ \log_{6n-19} (n!+1) =2. $ [i]Dragoș Crișan[/i]

p1. A regular $6$-face dice is thrown three times. Calculate the probability that the number of dice points on all three throws is $ 12$? p2. Given two positive real numbers $x$ and $y$ with $xy = 1$. Determine the minimum value of $\frac{1}{x^4}+\frac{1}{4y^4}.$ p3. Known a square network which is continuous and arranged in forming corners as in the following picture. Determine the value of the angle marked with the letter $x$. [img]https://cdn.artofproblemsolving.com/attachments/6/3/aee36501b00c4aaeacd398f184574bd66ac899.png[/img] p4. Find the smallest natural number $n$ such that the sum of the measures of the angles of the $n$-gon, with $n > 6$ is less than $n^2$ degrees. p5. There are a few magic cards. By stating on which card a number is there, without looking at the card at all, someone can precisely guess the number. If the number is on Card $A$ and $B$, then the number in question is $1 + 2$ (sum of corner number top left) cards $A$ and $B$. If the numbers are in $A$, $B$, and $C$, the number what is meant is $1 + 2 + 4$ or equal to $7$ (which is obtained by adding the numbers in the upper left corner of each card $A$,$B$, and $C$). [img]https://cdn.artofproblemsolving.com/attachments/e/5/9e80d4f3bba36a999c819c28c417792fbeff18.png[/img] a. How can this be explained? b. Suppose we are going to make cards containing numbers from $1$ to with $15$ based on the rules above. Try making the cards. [hide=original wording for p5, as the wording isn't that clear]Ada suatu kartu ajaib. Dengan menyebutkan di kartu yang mana suatu bilan gan berada, tanpa melihat kartu sama sekali, seseorang dengan tepat bisa menebak bilangan yang dimaksud. Kalau bilangan tersebut ada pada Kartu A dan B, maka bilangan yang dimaksud adalah 1 + 2 (jumlah bilangan pojok kiri atas) kartu A dan B. Kalau bilangan tersebut ada di A, B, dan C, bilangan yang dimaksud adalah 1 + 2 + 4 atau sama dengan 7 (yang diperoleh dengan menambahkan bilangan-bilangan di pojok kiri atas masing-masing kartu A, B, dan C) a. Bagaimana hal ini bisa dijelaskan? b. Andai kita akan membuat kartu-kartu yang memuat bilangan dari 1 sampai dengan 15 berdasarkan aturan di atas. Coba buatkan kartu-kartunya[/hide]

$p(x)$ is an irreducible polynomial with integer coefficients, and $q$ is a fixed prime number. Let $a_n$ be a number of solutions of the equation $p(x)\equiv 0\mod q^n$. Prove that we can find $M$ such that $\{a_n\}_{n\ge M}$ is constant.

Let $Q(n)$ denote the sum of the digits of $n$ in its decimal representation. Prove that for every positive integer $k$, there exists a multiple $n$ of $k$ such that $Q(n)=Q(n^2)$.

[u]Round 1[/u] [b]p1.[/b] Alice has a pizza with eight slices. On each slice, she either adds only salt, only pepper, or leaves it plain. Determine how many ways there are for Alice to season her entire pizza. [b]p2.[/b] Call a number almost prime if it has exactly three divisors. Find the number of almost prime numbers less than $100$. [b]p3.[/b] Determine the maximum number of points of intersection between a circle and a regular pentagon. [u]Round 2[/u] [b]p4.[/b] Let $d(n)$ denote the number of positive integer divisors of $n$. Find $d(d(20^{18}))$. [b]p5.[/b] $20$ chubbles are equal to $19$ flubbles. $20$ flubbles are equal to $18$ bubbles. How many bubbles are $1000$ chubbles worth? [b]p6.[/b] Square $ABCD$ and equilateral triangle $EFG$ have equal area. Compute $\frac{AB}{EF}$ . [u]Round 3[/u] [b]p7.[/b] Find the minimumvalue of $y$ such that $y = x^2 -6x -9$ where x is a real number. [b]p8.[/b] I have $2$ pairs of red socks, $5$ pairs of white socks, and $7$ pairs of blue socks. If I randomly pull out one sock at a time without replacement, how many socks do I need to draw to guarantee that I have drawn $3$ pairs of socks of the same color? [b]p9. [/b]There are $23$ paths from my house to the school, $29$ paths from the school to the library, and $3$ paths fromthe library to town center. Additionally, there are $6$ paths directly from my house to the library. If I have to pass through the library to get to town center, howmany ways are there to travel from my house all the way to the town center? [u]Round 4[/u] [b]p10.[/b] A circle of radius $25$ and a circle of radius $4$ are externally tangent. A line is tangent to the circle of radius $25$ at $A$ and the circle of radius $4$ at $B$, where $A \ne B$. Compute the length of $AB$. [b]p11.[/b] A gambler spins two wheels, one numbered $1$ to $4$ and another numbered $1$ to $5$, and the amount of money he wins is the sum of the two numbers he spins in dollars. Determine the expected amount of money he will win. [b]p12.[/b] Find the remainder when $20^{19}$ is divided by $18$. PS. You should use hide for answers. Rounds 5-8 have been posted [url=https://artofproblemsolving.com/community/c3h3166012p28809547]here [/url] and 9-12 [url=https://artofproblemsolving.com/community/c3h3166099p28810427]here[/url].Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

$p$ is an odd prime number. Prove that there exists a natural number $x$ such that $x$ and $4x$ are both primitive roots modulo $p$. [i]Proposed by Mohammad Gharakhani[/i]

Find all integers $n$ for which $n(n + 2010)$ is a perfect square.

Find all natural numbers $n$ for which the number of all positive divisors of the number lcm $(1,2,..., n)$ is equal to $2^k$ for some non-negative integer $k$.