Let $m,n$ be positive integers. Find the minimum positive integer $N$ which satisfies the following condition. If there exists a set $S$ of integers that contains a complete residue system module $m$ such that $| S | = N$, then there exists a nonempty set $A \subseteq S$ so that $n\mid {\sum\limits_{x \in A} x }$.

Suppose $x$ is a real number such that $x^2=10x+7$. Find the unique ordered pair of integers $(m,n)$ such that $x^3=mx+n$.

Ankit wants to create a pseudo-random number generator using modular arithmetic. To do so he starts with a seed $x_0$ and a function $f(x) = 2x + 25$ (mod $31$). To compute the $k$-th pseudo random number, he calls $g(k)$ dened as follows: $$g(k) = \begin{cases} x_0 \,\,\, \text{if} \,\,\, k = 0 \\ f(g(k- 1)) \,\,\, \text{if} \,\,\, k > 0 \end{cases}$$ If $x_0$ is $2017$, compute $\sum^{2017}_{j=0} g(j)$ (mod $31$).

Let $p$ be a prime such that $2^{p-1}\equiv 1 \pmod{p^2}$. Show that $(p-1)(p!+2^n)$ has at least three distinct prime divisors for each $n\in \mathbb{N}$ .

Let $p \geq 5$ be a prime number. Prove that there exists an integer $a$ with $1 \leq a \leq p-2$ such that neither $a^{p-1}-1$ nor $(a+1)^{p-1}-1$ is divisible by $p^2$.

If $a_1, a_2, \cdots, a_7$ are not necessarily distinct real numbers such that $1 < a_i < 13$ for all $i$, then show that we can choose three of them such that they are the lengths of the sides of a triangle.

[b]p1.[/b] In the following $3$ by $3$ grid, $a, b, c$ are numbers such that the sum of each row is listed at the right and the sum of each column is written below it: [center][img]https://cdn.artofproblemsolving.com/attachments/d/9/4f6fd2bc959c25e49add58e6e09a7b7eed9346.png[/img][/center] What is $n$? [b]p2.[/b] Suppose in your sock drawer of $14$ socks there are 5 different colors and $3$ different lengths present. One day, you decide you want to wear two socks that have both different colors and different lengths. Given only this information, what is the maximum number of choices you might have? [b]p3.[/b] The population of Arveymuddica is $2014$, which is divided into some number of equal groups. During an election, each person votes for one of two candidates, and the person who was voted for by $2/3$ or more of the group wins. When neither candidate gets $2/3$ of the vote, no one wins the group. The person who wins the most groups wins the election. What should the size of the groups be if we want to minimize the minimum total number of votes required to win an election? [b]p4.[/b] A farmer learns that he will die at the end of the year (day $365$, where today is day $0$) and that he has a number of sheep. He decides that his utility is given by ab where a is the money he makes by selling his sheep (which always have a fixed price) and $b$ is the number of days he has left to enjoy the profit; i.e., $365-k$ where $k$ is the day. If every day his sheep breed and multiply their numbers by $103/101$ (yes, there are small, fractional sheep), on which day should he sell them all? [b]p5.[/b] Line segments $\overline{AB}$ and $\overline{AC}$ are tangent to a convex arc $BC$ and $\angle BAC = \frac{\pi}{3}$ . If $\overline{AB} = \overline{AC} = 3\sqrt3$, find the length of arc $BC$. [b]p6.[/b] Suppose that you start with the number $8$ and always have two legal moves: $\bullet$ Square the number $\bullet$ Add one if the number is divisible by $8$ or multiply by $4$ otherwise How many sequences of $4$ moves are there that return to a multiple of $8$? [b]p7.[/b] A robot is shuffling a $9$ card deck. Being very well machined, it does every shuffle in exactly the same way: it splits the deck into two piles, one containing the $5$ cards from the bottom of the deck and the other with the $4$ cards from the top. It then interleaves the cards from the two piles, starting with a card from the bottom of the larger pile at the bottom of the new deck, and then alternating cards from the two piles while maintaining the relative order of each pile. The top card of the new deck will be the top card of the bottom pile. The robot repeats this shuffling procedure a total of n times, and notices that the cards are in the same order as they were when it started shuffling. What is the smallest possible value of $n$? [b]p8.[/b] A secant line incident to a circle at points $A$ and $C$ intersects the circle's diameter at point $B$ with a $45^o$ angle. If the length of $AB$ is $1$ and the length of $BC$ is $7$, then what is the circle's radius? [b]p9.[/b] If a complex number $z$ satisfies $z + 1/z = 1$, then what is $z^{96} + 1/z^{96}$? [b]p10.[/b] Let $a, b$ be two acute angles where $\tan a = 5 \tan b$. Find the maximum possible value of $\sin (a - b)$. [b]p11.[/b] A pyramid, represented by $SABCD$ has parallelogram $ABCD$ as base ($A$ is across from $C$) and vertex $S$. Let the midpoint of edge $SC$ be $P$. Consider plane $AMPN$ where$ M$ is on edge $SB$ and $N$ is on edge $SD$. Find the minimum value $r_1$ and maximum value $r_2$ of $\frac{V_1}{V_2}$ where $V_1$ is the volume of pyramid $SAMPN$ and $V_2$ is the volume of pyramid $SABCD$. Express your answer as an ordered pair $(r_1, r_2)$. [b]p12.[/b] A $5 \times 5$ grid is missing one of its main diagonals. In how many ways can we place $5$ pieces on the grid such that no two pieces share a row or column? [b]p13.[/b] There are $20$ cities in a country, some of which have highways connecting them. Each highway goes from one city to another, both ways. There is no way to start in a city, drive along the highways of the country such that you travel through each city exactly once, and return to the same city you started in. What is the maximum number of roads this country could have? [b]p14.[/b] Find the area of the cyclic quadrilateral with side lengths given by the solutions to $$x^4-10x^3+34x^2- 45x + 19 = 0.$$ [b]p15.[/b] Suppose that we know $u_{0,m} = m^2 + m$ and $u_{1,m} = m^2 + 3m$ for all integers $m$, and that $$u_{n-1,m} + u_{n+1,m} = u_{n,m-1} + u_{n,m+1}$$ Find $u_{30,-5}$. PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Find all integers $n \ge 2$ for which there exists $n$ integers $a_1,a_2,\dots,a_n \ge 2$ such that for all indices $i \ne j$, we have $a_i \mid a_j^2+1$.

Determine all triples $(m , n , p)$ satisfying : \[n^{2p}=m^2+n^2+p+1\] where $m$ and $n$ are integers and $p$ is a prime number.

Let $a_0$ and $a_n$ be different divisors of a natural number $m$, and $a_0, a_1, \ldots, a_n$ be a sequence of natural numbers such that it satisfies \[a_{i+1} = |a_i\pm a_{i-1}|\text{ for }0 < i < n\] If $gcd(a_0,a_1,\ldots, a_n) = 1$, show that there exists a term of the sequence that is smaller than $\sqrt{m}$ . [i]Proposed by Dusan Djukic[/i]

There exists a unique pair of positive integers $k,n$ such that $k$ is divisible by $6$, and $\sum_{i=1}^ki^2=n^2$. Find $(k,n)$.

Let $n,k$ be positive integers such that $n$ is not divisible by $3$ and $k\ge n$. Prove that there is an integer $m$ divisible by $n$ whose sum of digits in base $10$ equals $k$.

Let all positive integers $n$ satisfy the following condition: for each non-negative integers $k, m$ with $k + m \le n$, the numbers $\binom{n-k}{m}$ and $\binom{n-m}{k}$ leave the same remainder when divided by $2$. (Poland) PS. The translation was done using Google translate and in case it is not right, there is the original text in Slovak

We say that a positive integer $n$ is [i]fantastic[/i] if there exist positive rational numbers $a$ and $b$ such that $$ n = a + \frac 1a + b + \frac 1b.$$ [b](a)[/b] Prove that there exist infinitely many prime numbers $p$ such that no multiple of $p$ is fantastic. [b](b)[/b] Prove that there exist infinitely many prime numbers $p$ such that some multiple of $p$ is fantastic. [i]Proposed by Walther Janous, Austria[/i]

The sequence of positive integers $a_1,a_2,a_3,\dots$ is [i]brazilian[/i] if $a_1=1$ and $a_n$ is the least integer greater than $a_{n-1}$ and $a_n$ is [b]coprime[/b] with at least half elements of the set $\{a_1,a_2,\dots, a_{n-1}\}$. Is there any odd number which does [b]not[/b] belong to the brazilian sequence?

[i]20 problems for 25 minutes.[/i] [b]p1.[/b] Evaluate $20 \times 21 + 2021$. [b]p2.[/b] Let points $A$, $B$, $C$, and $D$ lie on a line in that order. Given that $AB = 5CD$ and $BD = 2BC$, compute $\frac{AC}{BD}$. [b]p3.[/b] There are $18$ students in Vincent the Bug's math class. Given that $11$ of the students take U.S. History, $15$ of the students take English, and $2$ of the students take neither, how many students take both U.S. History and English? [b]p4.[/b] Among all pairs of positive integers $(x, y)$ such that $xy = 12$, what is the least possible value of $x + y$? [b]p5.[/b] What is the smallest positive integer $n$ such that $n! + 1$ is composite? [b]p6.[/b] How many ordered triples of positive integers $(a, b,c)$ are there such that $a + b + c = 6$? [b]p7.[/b] Thomas orders some pizzas and splits each into $8$ slices. Hungry Yunseo eats one slice and then finds that she is able to distribute all the remaining slices equally among the $29$ other math club students. What is the fewest number of pizzas that Thomas could have ordered? [b]p8.[/b] Stephanie has two distinct prime numbers $a$ and $b$ such that $a^2-9b^2$ is also a prime. Compute $a + b$. [b]p9.[/b] Let $ABCD$ be a unit square and $E$ be a point on diagonal $AC$ such that $AE = 1$. Compute $\angle BED$, in degrees. [b]p10.[/b] Sheldon wants to trace each edge of a cube exactly once with a pen. What is the fewest number of continuous strokes that he needs to make? A continuous stroke is one that goes along the edges and does not leave the surface of the cube. [b]p11.[/b] In base $b$, $130_b$ is equal to $3n$ in base ten, and $1300_b$ is equal to $n^2$ in base ten. What is the value of $n$, expressed in base ten? [b]p12.[/b] Lin is writing a book with $n$ pages, numbered $1,2,..., n$. Given that $n > 20$, what is the least value of $n$ such that the average number of digits of the page numbers is an integer? [b]p13.[/b] Max is playing bingo on a $5\times 5$ board. He needs to fill in four of the twelve rows, columns, and main diagonals of his bingo board to win. What is the minimum number of boxes he needs to fill in to win? [b]p14.[/b] Given that $x$ and $y$ are distinct real numbers such that $x^2 + y = y^2 + x = 211$, compute the value of $|x - y|$. [b]p15.[/b] How many ways are there to place 8 indistinguishable pieces on a $4\times 4$ checkerboard such that there are two pieces in each row and two pieces in each column? [b]p16.[/b] The Manhattan distance between two points $(a, b)$ and $(c, d)$ in the plane is defined to be $|a - c| + |b - d|$. Suppose Neil, Neel, and Nail are at the points $(5, 3)$, $(-2,-2)$ and $(6, 0)$, respectively, and wish to meet at a point $(x, y)$ such that their Manhattan distances to$ (x, y)$ are equal. Find $10x + y$. [b]p17.[/b] How many positive integers that have a composite number of divisors are there between $1$ and $100$, inclusive? [b]p18.[/b] Find the number of distinct roots of the polynomial $$(x - 1)(x - 2) ... (x - 90)(x^2 - 1)(x^2 - 2) ... (x^2 - 90)(x^4 - 1)(x^4 - 2)...(x^4 - 90)$$. [b]p19.[/b] In triangle $ABC$, let $D$ be the foot of the altitude from $ A$ to $BC$. Let $P,Q$ be points on $AB$, $AC$, respectively, such that $PQ$ is parallel to $BC$ and $\angle PDQ = 90^o$. Given that $AD = 25$, $BD = 9$, and $CD = 16$, compute $111 \times PQ$. [b]p20.[/b] The simplified fraction with numerator less than $1000$ that is closest but not equal to $\frac{47}{18}$ is $\frac{p}{q}$ , where $p$ and $q$ are relatively prime positive integers. Compute $p$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

How many seven-digit numbers are multiples of $388$ and end in $388$?

Find all nonnegative integers $k, n$ which satisfy $2^{2k+1} + 9\cdot 2^k + 5 = n^2.$

Anna is out shopping for fruit. She observes that four oranges, three bananas and one lemon costs exactly the same as three oranges and two lemons (all prices are in whole kroner). Just then her friend Bengt calls, and Anna tells this to him. Bengt complains, that ''information is not enough for me to know how much each fruit costs''. ''No'', says Anna,' 'but three oranges and two lemons cost as many kroner as your mother is old''. Unfortunately, it's not enough either, but if she had been younger then the information would have been sufficient for you to be able to figure out what the fruits costs. How old is Bengt's mother?

In a sequence ,first term is $2$ and after $2.$ term all terms is equal to sum of the previous number's digits' $5.$ power. (Like this $2.$term is $2^5=32$ , $3.$term is $3^5+2^5=243+32=275\dotsm$) Prove that, this infinite sequence has at least $2$ two numbers are equal.

For any integer $n+1,\ldots, 2n$ ($n$ is a natural number) consider its greatest odd divisor. Prove that the sum of all these divisors equals $n^2.$

Find all pair of positive integers $(m,n)$ such that $$mn(m^2+6mn+n^2)$$is a perfect square. [i]Proposed by Li4 and Untro368[/i]

Call a number [i]feared [/i] if it contains the digits $13$ as a contiguous substring and [i]fearless [/i] otherwise. (For example, $132$ is feared, while $123$ is fearless.) Compute the smallest positive integer $n$ such that there exists a positive integer $a < 100$ such that $n$ and $n + 10a$ are fearless while $n +a$, $n + 2a$, $. . . $, $n + 9a$ are all feared.

Let be two natural numbers $ m,n\ge 2, $ two increasing finite sequences of real numbers $ \left( a_i \right)_{1\le i\le n} ,\left( b_j \right)_{1\le j\le m} , $ and the set $$ \left\{ a_i+b_j| 1\le i\le n,1\le j\le m \right\} . $$ Show that the set above has $ n+m-1 $ elements if and only if the two sequences above are arithmetic progressions and these have the same ratio.

Define an $n$-digit pair cycle to be a number with $n^2 + 1$ digits between $1$ and $n$ with every possible pair of consecutive digits. For instance, $11221$ is a 2-digit pair cycle since it contains the consecutive digits $11$, $12$, $22$, and $21$. How many $3$-digit pair cycles exist?