Let $k$ be a positive integer congruent to $1$ modulo $4$ which is not a perfect square and let $a=\frac{1+\sqrt{k}}{2}$. Show that $\{\left \lfloor{a^2n}\right \rfloor-\left \lfloor{a\left \lfloor{an}\right \rfloor}\right \rfloor : n \in \mathbb{N}_{>0}\}=\{1 , 2 , \ldots ,\left \lfloor{a}\right \rfloor\}$.

Let $a$ and $b$ be integers. Is it possible to find integers $p$ and $q$ such that the integers $p+na$ and $q +nb$ have no common prime factor no matter how the integer $n$ is chosen ?

Determine all pairs $(n, k)$ of distinct positive integers such that there exists a positive integer $s$ for which the number of divisors of $sn$ and of $sk$ are equal.

[b]p1.[/b] Find the smallest whole number $n \ge 2$ such that the product $(2^2 - 1)(3^2 - 1) ... (n^2 - 1)$ is the square of a whole number. [b]p2.[/b] The figure below shows a $ 10 \times 10$ square with small $2 \times 2$ squares removed from the corners. What is the area of the shaded region? [img]https://cdn.artofproblemsolving.com/attachments/7/5/a829487cc5d937060e8965f6da3f4744ba5588.png[/img] [b]p3.[/b] Three cars are racing: a Ford $[F]$, a Toyota $[T]$, and a Honda $[H]$. They began the race with $F$ first, then $T$, and $H$ last. During the race, $F$ was passed a total of $3$ times, $T$ was passed $5$ times, and $H$ was passed $8$ times. In what order did the cars finish? [b]p4.[/b] There are $11$ big boxes. Each one is either empty or contains $8$ medium-sized boxes inside. Each medium box is either empty or contains $8$ small boxes inside. All small boxes are empty. Among all the boxes, there are a total of $102$ empty boxes. How many boxes are there altogether? [b]p5.[/b] Ann, Mary, Pete, and finally Vlad eat ice cream from a tub, in order, one after another. Each eats at a constant rate, each at his or her own rate. Each eats for exactly the period of time that it would take the three remaining people, eating together, to consume half of the tub. After Vlad eats his portion there is no more ice cream in the tube. How many times faster would it take them to consume the tub if they all ate together? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

A list of natural numbers is written on a blackboard. The following operation is performed and repeated: choose any two numbers $a, b$, wipe them out and instead write gcd$(a, b)$ and lcm$(a, b)$. Show that the content of the list no longer changed after a certain point in time.

[u]Round 5[/u] [b]p13.[/b] Determine the number of different circular bracelets can be made with $7$ beads, all either colored red or black. [b]p14.[/b] The product of $260$ and $n$ is a perfect square. The $2020$th least possible positive integer value of $n$ can be written as$ p^{e_1}_1 \cdot p^{e_2}_2\cdot p^{e_3}_3\cdot p^{e_4}_4$ . Find the sum $p_1 +p_2 +p_3 +p_4 +e_1 +e_2 +_e3 +e_4$. [b]p15.[/b] Let $B$ and $C$ be points along the circumference of circle $\omega$. Let $A$ be the intersection of the tangents at $B$ and $C$ and let $D \ne A$ be on $\overrightarrow{AC}$ such that $AC =CD = 6$. Given $\angle BAC = 60^o$, find the distance from point $D$ to the center of $\omega$. [u]Round 6[/u] [b]p16.[/b] Evaluate $\sqrt{2+\sqrt{2+\sqrt{2+...}}}$. [b]p17.[/b] Let $n(A)$ be the number of elements of set $A$ and $||A||$ be the number of subsets of set $A$. Given that $||A||+2||B|| = 2^{2020}$, find the value of $n(B)$. [b]p18.[/b] $a$ and $b$ are positive integers and $8^a9^b$ has $578$ factors. Find $ab$. [u]Round 7[/u] [b]p19.[/b] Determine the probability that a randomly chosen positive integer is relatively prime to $2019$. [b]p20.[/b] A $3$-by-$3$ grid of squares is to be numbered with the digits $1$ through $9$ such that each number is used once and no two even-numbered squares are adjacent. Determine the number of ways to number the grid. [b]p21.[/b] In $\vartriangle ABC$, point $D$ is on $AC$ so that $\frac{AD}{DC}= \frac{1}{13}$ . Let point $E$ be on $BC$, and let $F$ be the intersection of $AE$ and $BD$. If $\frac{DF}{FB}=\frac{2}{7}$ and the area of $\vartriangle DBC$ is $26$, compute the area of $\vartriangle F AB$. [u]Round 8[/u] [b]p22.[/b] A quarter circle with radius $1$ is located on a line with its horizontal base on the line and to the left of the vertical side. It is then rolled to the right until it reaches its original orientation. Determine the distance traveled by the center of the quarter circle. [b]p23.[/b] In $1734$, mathematician Leonhard Euler proved that $\frac{\pi^2}{6}=\frac11+\frac14+\frac19+\frac{1}{16}+...$. With this in mind, calculate the value of $\frac11-\frac14+\frac19-\frac{1}{16}+...$ (the series obtained by negating every other term of the original series). [b]p24.[/b] Billy the biker is competing in a bike show where he can do a variety of tricks. He knows that one trick is worth $2$ points, $1$ trick is worth $3$ points, and 1 is worth $5$ points, but he doesn’t remember which trick is worth what amount. When it’s Billy’s turn to perform, he does $6$ tricks, randomly choosing which trick to do. Compute the sum of all the possible values of points that Billy could receive in total. PS. You should use hide for answers. Rounds 1-4 have been posted [url=https://artofproblemsolving.com/community/c3h3166016p28809598]here [/url] and 9-12 [url=https://artofproblemsolving.com/community/c3h3166115p28810631]here[/url].Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Find all pairs of natural numbers $ (a, b)$ such that $ 7^a \minus{} 3^b$ divides $ a^4 \plus{} b^2$. [i]Author: Stephan Wagner, Austria[/i]

Let $n$ be a positive integer less than $1000$. The remainders obtained when dividing $n$ by $2, 2^2, 2^3, ... , 2^8$, and $2^9$ , are calculated. If the sum of all these remainders is $137$, what are all the possible values ​​of $n$?

How many different ways are there to write 2004 as a sum of one or more positive integers which are all "aproximately equal" to each other? Two numbers are called aproximately equal if their difference is at most 1. The order of terms does not matter: two ways which only differ in the order of terms are not considered different.

Let $ a_0$ be a natural number. The sequence $ (a_n)$ is defined by $ a_{n\plus{}1}\equal{}\frac{a_n}{5}$ if $ a_n$ is divisible by $ 5$ and $ a_{n\plus{}1}\equal{}[a_n \sqrt{5}]$ otherwise . Show that the sequence $ a_n$ is increasing starting from some term.

For a positive integer $n$ denote $d(n)$ its greatest odd divisor. Find the value of the sum $$d(1008)+d(1009)+...+d(2015)$$

Prove that there exist infinitely many composite positive integers $n$ such that $n$ divides $3^{n-1}-2^{n-1}$.

Prove that from every set of $200$ integers you can choose a subset of $100$ with the total sum divisible by $100$.

Let $ \mathbb{N}_0$ denote the set of nonnegative integers. Find all functions $ f$ from $ \mathbb{N}_0$ to itself such that \[ f(m \plus{} f(n)) \equal{} f(f(m)) \plus{} f(n)\qquad \text{for all} \; m, n \in \mathbb{N}_0. \]

Let $a$, $b$ be such integers that $gcd(a + n,b + n) > 1$ for every integer $n \geq 1$. Prove that $a = b$.

Prove that for each positive integer $ n$, there are pairwise relatively prime integers $ k_0,k_1,\ldots,k_n$, all strictly greater than $ 1$, such that $ k_0k_1\ldots k_n\minus{}1$ is the product of two consecutive integers.

Call a positive integer $x$ $n$-[i]cube-invariant[/i] if the last $n$ digits of $x$ are equal to the last $n$ digits of $x^3$. For example, $1$ is $n$-cube invariant for any integer $n$. How many $2015$-cube-invariant numbers $x$ are there such that $x < 10^{2015}$?

Ana and Celia sell various objects and obtain for each object as many euros as objects they sold. The money obtained is made up of some $10$ euro bills and less than $10$ coins of $1$ euro . They decide to distribute the money as follows: Ana takes a $10$ euro bill and then Celia, and so on successively until Ana takes the last $10$ euro note, and Celia takes all the $1$ euro coins . How many euros more than Celia did Ana take? Give all the possibilities. [hide=original wording]Ana y Celia venden varios objetos y obtienen por cada objeto tantos euros como objetos vendieron. El dinero obtenido está constituido por algunos billetes de 10 euros y menos de 10 monedas de 1 euro. Deciden repartir el dinero del siguiente modo: Ana toma un billete de 10 euros y después Celia, y así sucesivamente hasta que Ana toma el último billete de 10 euros, y Celia se lleva todas las monedas de 1 euro. ¿Cuántos euros más que Celia se llevó Ana? Dar todas las posibilidades.[/hide]

Let $n$ be a positive integer. Find the greatest common divisor of the numbers $\binom{2n}{1},\binom{2n}{3},\binom{2n}{5},...,\binom{2n}{2n-1}$.

[u]Set 6[/u] [b]p16.[/b] Let $x$ and $y$ be non-negative integers. We say point $(x, y)$ is square if $x^2 + y$ is a perfect square. Find the sum of the coordinates of all distinct square points which also satisfy $x^2 + y \le 64$. [b]p17.[/b] Two integers $a$ and $b$ are randomly chosen from the set $\{1, 2, 13, 17, 19, 87, 115, 121\}$, with $a > b$. What is the expected value of the number of factors of $ab$? [b]p18.[/b] Marnie the Magical Cello is jumping on nonnegative integers on number line. She starts at $0$ and jumps following two specific rules. For each jump she can either jump forward by $1$ or jump to the next multiple of $4$ (the next multiple must be strictly greater than the number she is currently on). How many ways are there for her to jump to $2022$? (Two ways are considered distinct only if the sequence of numbers she lands on is different.) PS. You should use hide for answers.

[b]p1.[/b] A teenager coining home after midnight heard the hall clock striking the hour. At some moment between $15$ and $20$ minutes later, the minute hand hid the hour hand. To the nearest second, what time was it then? [b]p2.[/b] The ratio of two positive integers $a$ and $b$ is $2/7$, and their sum is a four digit number which is a perfect cube. Find all such integer pairs. [b]p3.[/b] Given the integers $1, 2 , ..., n$ , how many distinct numbers are of the form $\sum_{k=1}^n( \pm k) $ , where the sign ($\pm$) may be chosen as desired? Express answer as a function of $n$. For example, if $n = 5$ , then we may form numbers: $ 1 + 2 + 3- 4 + 5 = 7$ $-1 + 2 - 3- 4 + 5 = -1$ $1 + 2 + 3 + 4 + 5 = 15$ , etc. [b]p4.[/b] $\overline{DE}$ is a common external tangent to two intersecting circles with centers at $O$ and $O'$. Prove that the lines $AD$ and $BE$ are perpendicular. [img]https://cdn.artofproblemsolving.com/attachments/1/f/40ffc1bdf63638cd9947319734b9600ebad961.png[/img] [b]p5.[/b] Find all polynomials $f(x)$ such that $(x-2) f(x+1) - (x+1) f(x) = 0$ for all $x$ . PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

$n$ is an odd positive integer and $x,y$ are two rational numbers satisfying $$x^n+2y=y^n+2x.$$Prove that $x=y$.

Let $n = 6901$. There are $6732$ positive integers less than or equal to $n$ that are also relatively prime to $n$. Find the sum of the distinct prime factors of $n$.

There are two types of items in Alik's collection: badges and bracelets and there are more badges than bracelets. Alik noticed that if he increases the number of bracelets some (not necessarily integer) number of times without changing the number of icons, then in its collection will be $100$ items. And if, on the contrary, he increases the initial number of badges by the same number of times, leaving the same number of bracelets, then he will have $101$ items. How many badges and how many bracelets could there be in Alik's collection?

Let $M$ be a subset of $\mathbb{R}$ such that the following conditions are satisfied: a) For any $x \in M, n \in \mathbb{Z}$, one has that $x+n \in \mathbb{M}$. b) For any $x \in M$, one has that $-x \in M$. c) Both $M$ and $\mathbb{R}$ \ $M$ contain an interval of length larger than $0$. For any real $x$, let $M(x) = \{ n \in \mathbb{Z}^{+} | nx \in M \}$. Show that if $\alpha,\beta$ are reals such that $M(\alpha) = M(\beta)$, then we must have one of $\alpha + \beta$ and $\alpha - \beta$ to be rational.