Determine all positive integers $M$ such that the sequence $a_0, a_1, a_2, \cdots$ defined by \[ a_0 = M + \frac{1}{2} \qquad \textrm{and} \qquad a_{k+1} = a_k\lfloor a_k \rfloor \quad \textrm{for} \, k = 0, 1, 2, \cdots \] contains at least one integer term.

$ k$ is a given natural number. Find all functions $ f: \mathbb{N}\rightarrow\mathbb{N}$ such that for each $ m,n\in\mathbb{N}$ the following holds: \[ f(m)\plus{}f(n)\mid (m\plus{}n)^k\]

Given sets $A = \{1, 4, 5, 6, 7, 9, 11, 16, 17\}$, $B = \{2, 3, 8, 10, 12, 13, 14, 15, 18\}$, if a positive integer leaves a remainder (the smallest non-negative remainder) that belongs to $A$ when divided by 19, then that positive integer is called an $\alpha$ number. If a positive integer leaves a remainder that belongs to $B$ when divided by 19, then that positive integer is called a $\beta$ number. (1) For what positive integer $n$, among all its positive divisors, are the numbers of $\alpha$ divisors and $\beta$ divisors equal? (2) For which positive integers $k$, are the numbers of $\alpha$ divisors less than the numbers of $\beta$ divisors? For which positive integers $l$, are the numbers of $\alpha$ divisors greater than the numbers of $\beta$ divisors?

[b]7.1 . [/b] The area of the quadrilateral is $3$ cm$^2$ , and the lengths of its diagonals are $6$ cm and $2$ cm. Find the angle between the diagonals. [b]7.2[/b] Prove that the number $1 + 2^{3456789}$ is composite. [b]7.3[/b] $20$ people took part in the chess tournament. The participant who took clear (undivided) $19$th place scored $9.5$ points. How could they distribute points among other participants? [b]7.4[/b] The sum of the distances between the midpoints of opposite sides of a quadrilateral is equal to its semi-perimeter. Prove that this quadrilateral is a parallelogram. [b]7.5[/b] $40$ people travel on a bus without a conductor passengers carrying only coins in denominations of $10$, $15$ and $20$ kopecks. Total passengers have $ 49$ coins. Prove that passengers will not be able to pay the required amount of money to the ticket office and pay each other correctly. (Cost of a bus ticket in 1963 was 5 kopecks.) [b]7.6[/b] Some natural number $a$ is divided with a remainder by all natural numbers less than $a$. The sum of all the different (!) remainders turned out to be equal to $a$. Find $a$. [b]7.7[/b] Two squares were cut out of a chessboard. In what case is it possible and in what case not to cover the remaining squares of the board with dominoes (i.e., figures of the form $2\times 1$) without overlapping? PS. You should use hide for answers.Collected [url=https://artofproblemsolving.com/community/c3983460_1963_leningrad_math_olympiad]here[/url].

$m, n $ are positive integers and $p$ is a prime number. Find all triples $(m, n, p)$ satisfying $(m^3+n)(n^3+m)=p^3$

Find all positive integers $n$ such that the following holds. $$\tau(n)|2^{\sigma(n)}-1$$

Find all positive integers $ (x,y)$ such that $ \frac{y^2x}{x\plus{}y}$ is a prime number

Given positive integers $a,b,c$ which are pairwise relatively prime, show that \[2abc-ab-bc-ac \] is the biggest number that can't be expressed in the form $xbc+yca+zab$ with $x,y,z$ being natural numbers.

Find the number of integers $k$ in the set $\{0, 1, 2, \dots, 2012\}$ such that $\binom{2012}{k}$ is a multiple of $2012$.

Let $b$ be a positive integer such that $\gcd(b,6)=1$. Show that there are positive integers $x$ and $y$ such that $\frac1x+\frac1y=\frac3b$ if and only if $b$ is divisible by some prime number of form $6k-1$.

A number with $2016$ zeros that is written as $101010 \dots 0101$ is given, in which the zeros and ones alternate. Prove that this number is not prime.

The function is given $f(x) = \frac{2x^3 -6x^2 + 13x + 10}{2x^2 - 9x}$. Determine all positive integers $x$ for which $f(x)$ is an integer

[b]p1.[/b] One hundred hobbits sit in a circle. The hobbits realize that whenever a hobbit and his two neighbors add up their total rubles, the sum is always $2007$. Prove that each hobbit has $669$ rubles. [b]p2.[/b] There was a young lady named Chris, Who, when asked her age, answered this: "Two thirds of its square Is a cube, I declare." Now what was the age of the miss? (a) Find the smallest possible age for Chris. You must justify your answer. (Note: ages are positive integers; "cube" means the cube of a positive integer.) (b) Find the second smallest possible age for Chris. You must justify your answer. (Ignore the word "young.") [b]p3.[/b] Show that $$\sum_{n=1}^{2007}\frac{1}{n^3+3n^2+2n}<\frac14$$ [b]p4.[/b] (a) Show that a triangle $ABC$ is isosceles if and only if there are two distinct points $P_1$ and $P_2$ on side $BC$ such that the sum of the distances from $P_1$ to the sides $AB$ and $AC$ equals the sum of the distances from $P_2$ to the sides $AB$ and $AC$. (b) A convex quadrilateral is such that the sum of the distances of any interior point to its four sides is constant. Prove that the quadrilateral is a parallelogram. (Note: "distance to a side" means the shortest distance to the line obtained by extending the side.) [b]p5.[/b] Each point in the plane is colored either red or green. Let $ABC$ be a fixed triangle. Prove that there is a triangle $DEF$ in the plane such that $DEF$ is similar to $ABC$ and the vertices of $DEF$ all have the same color. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

What is the largest prime factor of 8091?

For $a,n\in\mathbb{Z}^+$, consider the following equation: \[ a^2x+6ay+36z=n\quad (1) \] where $x,y,z\in\mathbb{N}$. a) Find all $a$ such that for all $n\geq 250$, $(1)$ always has natural roots $(x,y,z)$. b) Given that $a>1$ and $\gcd (a,6)=1$. Find the greatest value of $n$ in terms of $a$ such that $(1)$ doesn't have natural root $(x,y,z)$.

For positive integers $a$ and $b$, let $M(a,b)$ denote their greatest common divisor. Determine all pairs of positive integers $(m,n)$ such that for any two positive integers $x$ and $y$ such that $x\mid m$ and $y\mid n$, $$M(x+y,mn)>1.$$ [i]Proposed by Ivan Novak[/i]

Let $x, y$ be distinct positive integers. Show that the number $$\frac{(x + y)^2}{x^3 + xy^2- x^2y -y^3}$$ is not an integer.

All six-digit natural numbers from $100000$ to $999999$ are written on the page in ascending order without spaces. What is the largest value of$ k$ for which the same $k$-digit number can be found in at least two different places in this string?

Let $a, b, c$ be pairwise coprime natural numbers. A positive integer $n$ is said to be [i]stubborn[/i] if it cannot be written in the form $n = bcx+cay+abz$, for some $x, y, z \in\mathbb{ N}.$ Determine the number of stubborn numbers.

A positive integer $m$ is perfect if the sum of all its positive divisors, $1$ and $m$ inclusive, is equal to $2m$. Determine the positive integers $n$ such that $n^n + 1$ is a perfect number.

If $n$ is a positive integer and $a, b$ are relatively prime positive integers, calculate $(a + b,a^n + b^n)$.

[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]The longest diagonal of a regular hexagon is 12 inches long. What is the area of the hexagon, in square inches? [b]p2.[/b] When Al and Bob play a game, either Al wins, Bob wins, or they tie. The probability that Al does not win is $\frac23$ , and the probability that Bob does not win is $\frac34$ . What is the probability that they tie? [b]p3.[/b] Find the sum of the $a + b$ values over all pairs of integers $(a, b)$ such that $1 \le a < b \le 10$. That is, compute the sum $$(1 + 2) + (1 + 3) + (1 + 4) + ...+ (2 + 3) + (2 + 4) + ...+ (9 + 10).$$ [b]p4.[/b] A $3 \times 11$ cm rectangular box has one vertex at the origin, and the other vertices are above the $x$-axis. Its sides lie on the lines $y = x$ and $y = -x$. What is the $y$-coordinate of the highest point on the box, in centimeters? [b]p5.[/b] Six blocks are stacked on top of each other to create a pyramid, as shown below. Carl removes blocks one at a time from the pyramid, until all the blocks have been removed. He never removes a block until all the blocks that rest on top of it have been removed. In how many different orders can Carl remove the blocks? [img]https://cdn.artofproblemsolving.com/attachments/b/e/9694d92eeb70b4066b1717fedfbfc601631ced.png[/img] [b]p6.[/b] Suppose that a right triangle has sides of lengths $\sqrt{a + b\sqrt{3}}$,$\sqrt{3 + 2\sqrt{3}}$, and $\sqrt{4 + 5\sqrt{3}}$, where $a, b$ are positive integers. Find all possible ordered pairs $(a, b)$. [b]p7.[/b] Farmer Chong Gu glues together $4$ equilateral triangles of side length $ 1$ such that their edges coincide. He then drives in a stake at each vertex of the original triangles and puts a rubber band around all the stakes. Find the minimum possible length of the rubber band. [b]p8.[/b] Compute the number of ordered pairs $(a, b)$ of positive integers less than or equal to $100$, such that a $b -1$ is a multiple of $4$. [b]p9.[/b] In triangle $ABC$, $\angle C = 90^o$. Point $P$ lies on segment $BC$ and is not $B$ or $C$. Point $I$ lies on segment $AP$. If $\angle BIP = \angle PBI = \angle CAB = m^o$ for some positive integer $m$, find the sum of all possible values of $m$. [b]p10.[/b] Bob has $2$ identical red coins and $2$ identical blue coins, as well as $4$ distinguishable buckets. He places some, but not necessarily all, of the coins into the buckets such that no bucket contains two coins of the same color, and at least one bucket is not empty. In how many ways can he do this? [b]p11.[/b] Albert takes a $4 \times 4$ checkerboard and paints all the squares white. Afterward, he wants to paint some of the square black, such that each square shares an edge with an odd number of black squares. Help him out by drawing one possible configuration in which this holds. (Note: the answer sheet contains a $4 \times 4$ grid.) [b]p12.[/b] Let $S$ be the set of points $(x, y)$ with $0 \le x \le 5$, $0 \le y \le 5$ where $x$ and $y$ are integers. Let $T$ be the set of all points in the plane that are the midpoints of two distinct points in $S$. Let $U$ be the set of all points in the plane that are the midpoints of two distinct points in $T$. How many distinct points are in $U$? (Note: The points in $T$ and $U$ do not necessarily have integer coordinates.) [b]p13.[/b] In how many ways can one express $6036$ as the sum of at least two (not necessarily positive) consecutive integers? [b]p14.[/b] Let $a, b, c, d, e, f$ be integers (not necessarily distinct) between $-100$ and $100$, inclusive, such that $a + b + c + d + e + f = 100$. Let $M$ and $m$ be the maximum and minimum possible values, respectively, of $$abc + bcd + cde + def + ef a + f ab + ace + bdf.$$ Find $\frac{M}{m}$. [b]p15.[/b] In quadrilateral $ABCD$, diagonal $AC$ bisects diagonal $BD$. Given that $AB = 20$, $BC = 15$, $CD = 13$, $AC = 25$, find $DA$. PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

How many different integer solutions to the inequality $|x| + |y| < 100$ are there?

Find a four-digit number that is perfect square and such that the first two digits are the same and the last two as well.