For given positive integers $a$ and $b$, let us consider the equation$$a + \gcd(b, x) = b + \gcd(a, x).$$ [list=a] [*]For $a = 20$ and $b = 25$, find the least positive integer $x$ satisfying this equation. [*]Prove that for any positive integers $a$ and $b$, there exist infinitely many positive integers $x$ satisfying this equation. [/list] [i](Here, $\gcd(m, n)$ denotes the greatest common divisor of positive integers $m$ and $n$.)[/i]

Let $k\geqslant 2$ be a natural number. Prove that the natural numbers with an even sum of digits give all the possible residues when divided by $k{}$. [i]Proposed by P. Kozlov and I. Bogdanov[/i]

Does there exist a finite sequence of integers $c_1,c_2,\ldots ,c_n$ such that all the numbers $a+c_1,a+c_2,\ldots ,a+c_n$ are primes for more than one but not infinitely many different integers $a$?

How many ways are there to insert plus signs $+$ between the digits of number $111111 ...111$ which includes thirty of digits $1$ so that the result will be a multiple of $30$?

Let $1=d_1<d_2<\ldots<d_{2m}=n$ be the divisors of a positive integer $n$, where $n$ is not a perfect square. Consider the determinant $$D=\begin{vmatrix}n+d_1&n&\ldots&n\\n&n+d_2&\ldots&n\\\ldots&\ldots&&\ldots\\n&n&\ldots&n+d_{2m}\end{vmatrix}.$$ (a) Prove that $n^m$ divides $D$. (b) Prove that $1+d_1+d_2+\ldots+d_{2m}$ divides $D$.

In the decimal expression of a $ 2009$-digit natural number there are only the digits $ 5$ and $ 8$. Prove that we can get a $ 2008$-digit number divisible by $ 11$ if we remove just one digit from the number.

A set of positive integers is called [i]fragrant[/i] if it contains at least two elements and each of its elements has a prime factor in common with at least one of the other elements. Let $P(n)=n^2+n+1$. What is the least possible positive integer value of $b$ such that there exists a non-negative integer $a$ for which the set $$\{P(a+1),P(a+2),\ldots,P(a+b)\}$$ is fragrant?

Find all positive integers $k,m$ and $n$ such that $k!+3^m=3^n$

What is the last two digits of the number $(11^2 + 15^2 + 19^2 +
...

+ 2007^2)^
2$?

Is there a six-digit number where every two consecutive digits make up a certain number two-digit number that is the square of an integer? Justify your answer.

Find four different three-digit decimal numbers starting with the same digit, such that their sum is divisible by three of them.

[b]p1.[/b] Find the greatest integer $n$ such that $n \log_{10} 4$ does not exceed $\log_{10} 1998$. [b]p2.[/b] Rectangle $ABCD$ has sides $AB = CD = 12/5$, $BC = DA = 5$. Point $P$ is on $AD$ with $\angle BPC = 90^o$. Compute $BP + PC$. [b]p3.[/b] Compute the number of sequences of four decimal digits $(a, b, c, d)$ (each between $0$ and $9$ inclusive) containing no adjacent repeated digits. (That is, each digit is distinct from the digits directly before and directly after it.) [b]p4.[/b] Solve for $t$, $-\pi/4 \le t \le \pi/4 $: $$\sin^3 t + \sin^2 t \cos t + \sin t \cos^2 t + \cos^3 t =\frac{\sqrt6}{2}$$ [b]p5.[/b] Find all integers $n$ such that $n - 3$ divides $n^2 + 2$. [b]p6.[/b] Find the maximum number of bishops that can occupy an $8 \times 8$ chessboard so that no two of the bishops attack each other. (Bishops can attack an arbitrary number of squares in any diagonal direction.) [b]p7.[/b] Points $A, B, C$, and $D$ are on a Cartesian coordinate system with $A = (0, 1)$, $B = (1, 1)$, $C = (1,-1)$, and $D = (-1, 0)$. Compute the minimum possible value of $PA + PB + PC + PD$ over all points $P$. [b]p8.[/b] Find the number of distinct real values of $x$ which satisfy $$(x-1)(x-2)(x-3)(x-4)(x-5)(x-6)(x-7)(x-8)(x-9)(x-10)+(1^2 \cdot 3^2\cdot 5^2\cdot 7^2\cdot 9^2)/2^{10} = 0.$$ PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Consider the set $S= \{(a + b)^7 - a^7 - b^7 : a,b \in Z\}$. Find the greatest common divisor of all members in $S$.

Find all the pairs of positive numbers such that the last digit of their sum is 3, their difference is a primer number and their product is a perfect square.

Determine all integers $m \geq 2$ such that every $n$ with $\frac{m}{3} \leq n \leq \frac{m}{2}$ divides the binomial coefficient $\binom{n}{m-2n}$.

For $\pi\leq\theta<2\pi$, let \[ P=\dfrac12\cos\theta-\dfrac14\sin2\theta-\dfrac18\cos3\theta+\dfrac1{16}\sin4\theta+\dfrac1{32}\cos5\theta-\dfrac1{64}\sin6\theta-\dfrac1{128}\cos7\theta+\ldots \] and \[ Q=1-\dfrac12\sin\theta-\dfrac14\cos2\theta+\dfrac1{8}\sin3\theta+\dfrac1{16}\cos4\theta-\dfrac1{32}\sin5\theta-\dfrac1{64}\cos6\theta+\dfrac1{128}\sin7\theta +\ldots \] so that $\tfrac PQ = \tfrac{2\sqrt2}7$. Then $\sin\theta = -\tfrac mn$ where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

A positive integer is called [i]Tiranian[/i] if it can be written as $x^2+6xy+y^2$, where $x$ and $y$ are (not necessarily distinct) positive integers. The integer $36^{2023}$ is written as the sum of $k$ Tiranian integers. What is the smallest possible value of $k$? [i]Proposed by Miroslav Marinov, Bulgaria[/i]

Mark writes the expression $\sqrt{\underline{abcd}}$ on the board, where $\underline{abcd}$ is a four-digit number and $a \neq 0.$ Derek, a toddler, decides to move the $a,$ changing Mark's expression to $a\sqrt{\underline{bcd}}.$ Surprisingly, these two expressions are equal. Compute the only possible four-digit number $\underline{abcd}.$

[u]Round 9[/u] [b]p25.[/b] Let $S$ be the set of the first $2017$ positive integers. Find the number of elements $n \in S$ such that $\sum^n_{i=1} \left\lfloor \frac{n}{i} \right\rfloor$ is even. [b]p26.[/b] Let $\{x_n\}_{n \ge 0}$ be a sequence with $x_0 = 0$,$x_1 = \frac{1}{20}$ ,$x_2 = \frac{1}{17}$ ,$x_3 = \frac{1}{10}$ , and $x_n = \frac12 ((x_{n-2} +x_{n-4})$ for $n\ge 4$. Compute $$ \left\lfloor \frac{1}{x_{2017!} -x_{2017!-1}} \right\rfloor.$$ [b]p27.[/b] Let $ABCDE$ be be a cyclic pentagon. Given that $\angle CEB = 17^o$, find $\angle CDE + \angle EAB$, in degrees. [u]Round 10[/u] [b]p28.[/b] Let $S = \{1,2,4, ... ,2^{2016},2^{2017}\}$. For each $0 \le i \le 2017$, let $x_i$ be chosen uniformly at random from the subset of $S$ consisting of the divisors of $2^i$ . What is the expected number of distinct values in the set $\{x_0,x_1,x_2,... ,x_{2016},x_{2017}\}$? [b]p29.[/b] For positive real numbers $a$ and $b$, the points $(a, 0)$, $(20,17)$ and $(0,b)$ are collinear. Find the minimum possible value of $a+b$. [b]p30.[/b] Find the sum of the distinct prime factors of $2^{36}-1$. [u]Round 11[/u] [b]p31.[/b] There exist two angle bisectors of the lines $y = 20x$ and $y = 17x$ with slopes $m_1$ and $m_2$. Find the unordered pair $(m_1,m_2)$. [b]p32.[/b] Triangle 4ABC has sidelengths $AB = 13$, $BC = 14$, $C A =15$ and orthocenter $H$. Let $\Omega_1$ be the circle through $B$ and $H$, tangent to $BC$, and let $\Omega_2$ be the circle through $C$ and $H$, tangent to $BC$. Finally, let $R \ne H$ denote the second intersection of $\Omega_1$ and $\Omega_2$. Find the length $AR$. [b]p33.[/b] For a positive integer $n$, let $S_n = \{1,2,3, ...,n\}$ be the set of positive integers less than or equal to $n$. Additionally, let $$f (n) = |\{x \in S_n : x^{2017}\equiv x \,\, (mod \,\, n)\}|.$$ Find $f (2016)- f (2015)+ f (2014)- f (2013)$. [u]Round 12[/u] [b]p34. [/b] Estimate the value of $\sum^{2017}_{n=1} \phi (n)$, where $\phi (n)$ is the number of numbers less than or equal $n$ that are relatively prime to n. If your estimate is $E$ and the correct answer is $A$, your score for this problem will be max $\max \left(0,\lfloor 15 - 75 \frac{|A-E|}{A} \rceil \right).$ [b]p35.[/b] An up-down permutation of order $n$ is a permutation $\sigma$ of $(1,2,3, ..., n)$ such that $\sigma(i ) <\sigma (i +1)$ if and only if $i$ is odd. Denote by $P_n$ the number of up-down permutations of order $n$. Estimate the value of $P_{20} +P_{17}$. If your estimate is $E$ and the correct answer is $A$, your score for this problem will be $\max \left(0, 16 -\lceil \max \left(\frac{E}{A}, 2- \frac{E}{A}\right) \rceil \right).$ [b]p36.[/b] For positive integers $n$, superfactorial of $n$, denoted $n\$ $, is defined as the product of the first $n$ factorials. In other words, we have $n\$ = \prod^n_{i=1}(i !)$. Estimate the number of digits in the product $(20\$)\cdot (17\$)$. If your estimate is $E$ and the correct answer is $A$, your score for this problem will be $\max \left(0, \lfloor 15 -\frac12 |A-E| \rfloor \right).$ PS. You should use hide for answers. Rounds 1-4 have been posted [url=https://artofproblemsolving.com/community/c3h3158491p28715220]here [/url] and 5-8 [url=https://artofproblemsolving.com/community/c3h3158514p28715373]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Find all prime numbers $p$ and $q$ such that $3p^2q+2pq^2=483$

For a positive integer $a$, define $F_1 ^{(a)}=1$, $F_2 ^{(a)}=a$ and for $n>2$, $F_n ^{(a)}=F_{n-1} ^{(a)}+F_{n-2} ^{(a)}$. A positive integer is fibonatic when it is equal to $F_n ^{(a)}$ for a positive integer $a$ and $n>3$. Prove that there are infintely many not fibonatic integers.

Find all integer solutions $(m,n)$ to $m^{n}=n^{m}.$

a. Does there exist 4 distinct positive integers such that the sum of any 3 of them is prime? b. Does there exist 5 distinct positive integers such that the sum of any 3 of them is prime?

[u]Round 1[/u] [b]p1.[/b] Five children, Aisha, Baesha, Cosha, Dasha, and Erisha, competed in running, jumping, and throwing. In each event, first place was won by someone from Renton, second place by someone from Seattle, and third place by someone from Tacoma. Aisha was last in running, Cosha was last in jumping, and Erisha was last in throwing. Could Baesha and Dasha be from the same city? [b]p2.[/b] Fifty-five Brits and Italians met in a coffee shop, and each of them ordered either coffee or tea. Brits tell the truth when they drink tea and lie when they drink coffee; Italians do it the other way around. A reporter ran a quick survey: Forty-four people answered “yes” to the question, “Are you drinking coffee?” Thirty-three people answered “yes” to the question, “Are you Italian?” Twenty-two people agreed with the statement, “It is raining outside.” How many Brits in the coffee shop are drinking tea? [b]p3.[/b] Doctor Strange is lost in a strange house with a large number of identical rooms, connected to each other in a loop. Each room has a light and a switch that could be turned on and off. The lights might initially be on in some rooms and off in others. How can Dr. Strange determine the number of rooms in the house if he is only allowed to switch lights on and off? [b]p4.[/b] Fifty street artists are scheduled to give solo shows with three consecutive acts: juggling, drumming, and gymnastics, in that order. Each artist will spend equal time on each of the three activities, but the lengths may be different for different artists. At least one artist will be drumming at every moment from dawn to dusk. A new law was just passed that says two artists may not drum at the same time. Show that it is possible to cancel some of the artists' complete shows, without rescheduling the rest, so that at least one show is going on at every moment from dawn to dusk, and the schedule complies with the new law. [b]p5.[/b] Alice and Bob split the numbers from $1$ to $12$ into two piles with six numbers in each pile. Alice lists the numbers in the first pile in increasing order as $a_1 < a_2 < a_3 < a_4 < a_5 < a_6$ and Bob lists the numbers in the second pile in decreasing order $b_1 > b_1 > b_3 > b_4 > b_5 > b_6$. Show that no matter how they split the numbers, $$|a_1 -b_1| + |a_2 -b_2| + |a_3 -b_3| + |a_4 -b_4| + |a_5 -b_5| + |a_6 -b_6| = 36.$$ [u]Round 2[/u] [b]p6.[/b] The Martian alphabet has ? letters. Marvin writes down a word and notices that within every sub-word (a contiguous stretch of letters) at least one letter occurs an odd number of times. What is the length of the longest possible word he could have written? [b]p7.[/b] For a long space journey, two astronauts with compatible personalities are to be selected from $24$ candidates. To find a good fit, each candidate was asked $24$ questions that required a simple yes or no answer. Two astronauts are compatible if exactly $12$ of their answers matched (that is, both answered yes or both answered no). Miraculously, every pair of these $24$ astronauts was compatible! Show that there were exactly $12$ astronauts whose answer to the question “Can you repair a flux capacitor?” was exactly the same as their answer to the question “Are you afraid of heights?” (that is, yes to both or no to both). PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

[b]p1.[/b] Two players play the following game. On the lowest left square of an $8\times 8$ chessboard there is a rook. The first player is allowed to move the rook up or to the right by an arbitrary number of squares. The second player is also allowed to move the rook up or to the right by an arbitrary number of squares. Then the first player is allowed to do this again, and so on. The one who moves the rook to the upper right square wins. Who has a winning strategy? [b]p2.[/b] In Crocodile Country there are banknotes of $1$ dollar, $10$ dollars, $100$ dollars, and $1,000$ dollars. Is it possible to get 1,000,000 dollars by using $250,000$ banknotes? [b]p3.[/b] Fifteen positive numbers (not necessarily whole numbers) are placed around the circle. It is known that the sum of every four consecutive numbers is $30$. Prove that each number is less than $15$. [b]p4.[/b] Donald Duck has $100$ sticks, each of which has length $1$ cm or $3$ cm. Prove that he can break into $2$ pieces no more than one stick, after which he can compose a rectangle using all sticks. [b]p5.[/b] Three consecutive $2$ digit numbers are written next to each other. It turns out that the resulting $6$ digit number is divisible by $17$. Find all such numbers. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].