For any nonnegative integer $ n$, let $ f(n)$ be the greatest integer such that $ 2^{f(n)} | n \plus{} 1$. A pair $ (n, p)$ of nonnegative integers is called nice if $ 2^{f(n)} > p$. Find all triples $ (n, p, q)$ of nonnegative integers such that the pairs $ (n, p)$, $ (p, q)$ and $ (n \plus{} p \plus{} q, n)$ are all nice.

Morteza wishes to take two real numbers $S$ and $P$, and then to arrange six pairwise distinct real numbers on a circle so that for each three consecutive numbers at least one of the two following conditions holds: 1) their sum equals $S$ 2) their product equals $P$. Determine if Morteza’s wish could be fulfilled.

Let $\phi(n)$ denote the number of positive integers less than or equal to $n$ that are relatively prime to $n$, and let $d(n)$ denote the number of positive integer divisors of $n$. For example, $\phi(6) = 2$ and $d(6) = 4$. Find the sum of all odd integers $n \le 5000$ such that $n \mid \phi(n) d(n)$. [i]Alex Zhu[/i]

Suppose $n$ and $r$ are nonnegative integers such that no number of the form $n^2+r-k(k+1) \text{ }(k\in\mathbb{N})$ equals to $-1$ or a positive composite number. Show that $4n^2+4r+1$ is $1$, $9$, or prime.

Integers $1 \le n \le 200$ are written on a blackboard just one by one. We surrounded just $100$ integers with circle. We call a square of the sum of surrounded integers minus the sum of not surrounded integers $score$ of this situation. Calculate the average score in all ways.

Does there exist an integer containing only digits $2$ and $0$ which is a $k$-th power of a positive integer ($k \ge2$)?

Find all triplets $ (x,y,z) $ of positive integers such that \[ x^y + y^x = z^y \]\[ x^y + 2012 = y^{z+1} \]

Prove that there are no positive integers $a,b$ such that $(15a +b)(a +15b)$ is a power of $3.$

$a,b>1$ - are naturals, and $a^2+b,a+b^2$ are primes. Prove $(ab+1,a+b)=1$

Find, with proof, the maximal length of a non-constant arithmetic progression with all the terms squares of positive integers.

Let $f(x)$ is such function, that $f(x)=1$ for integer $x$ and $f(x)=0$ for non integer $x$. Build such function using only variable $x$, integer numbers, and operations $+,-,*,/,[.]$(plus, minus, multiply,divide and integer part)

Set $S = \{1, 2, 3, ..., 2005\}$. If among any $n$ pairwise coprime numbers in $S$ there exists at least a prime number, find the minimum of $n$.

A positive integer $n$ is called $special$ if there exist integers $a > 1$ and $b > 1$ such that $n=a^b + b$. Is there a set of $2014$ consecutive positive integers that contains exactly $2012$ $special$ numbers?

Alice and Bob live on the same road. At time $t$, they both decide to walk to each other's houses at constant speed. However, they were busy thinking about math so that they didn't realize passing each other. Alice arrived at Bob's house at $3:19\text{pm}$, and Bob arrived at Alice's house at $3:29\text{pm}$. Charlie, who was driving by, noted that Alice and Bob passed each other at $3:11\text{pm}$. Find the difference in minutes between the time Alice and Bob left their own houses and noon on that day. [i]Proposed by Kevin You[/i]

Given a positive integer $N$, determine all positive integers $n$, satisfying the following condition: for any list $d_1,d_2,\ldots,d_k$ of (not necessarily distinct) divisors of $n$ such that $\frac{1}{d_1} + \frac{1}{d_2} + \ldots + \frac{1}{d_k} > N$, some of the fractions $\frac{1}{d_1}, \frac{1}{d_2}, \ldots, \frac{1}{d_k}$ add up to exactly $N$.

Find the maximum integer $m$ such that $m! \cdot 2022!$ is a factorial of an integer.

Let $s(n)$ the sum of digits of $n$. Find the greatest 3-digits number $m$ such that $3s(m)=s(3m)$.

Prove that there is no $1989$-digit natural number at least three of whose digits are equal to $5$ and such that the product of its digits equals their sum.

Let $a_{1}, a_{2}, \cdots, a_{k}$ be relatively prime positive integers. Determine the largest integer which cannot be expressed in the form \[x_{1}a_{2}a_{3}\cdots a_{k}+x_{2}a_{1}a_{3}\cdots a_{k}+\cdots+x_{k}a_{1}a_{2}\cdots a_{k-1}\] for some nonnegative integers $x_{1}, x_{2}, \cdots, x_{k}$.

Let $a, b, c$ be real numbers, and define $S_n = a^n + b^n + c^n$ for positive integers $n$. Suppose that $S_1, S_2, S_3$ are integers satisfying $6 | 5S_1 - 3S_2 - 2S_3$. Show that $S_n$ is an integer for all positive integers $n$.

[u]Round 6[/u] [b]p16.[/b] Let $a_1, a_2, ... , a_{2011}$ be a sequence of numbers such that $a_1 = 2011$ and $a_1+a_2+...+a_n = n^2 \cdot a_n$ for $n = 1, 2, ... 2011$. (That is, $a_1 = 1^2\cdot a_1$, $a_1 + a_2 = 2^2 \cdot a_2$, $...$) Compute $a_{2011}$. [b]p17.[/b] Three rectangles, with dimensions $3 \times 5$, $4 \times 2$, and $6 \times 4$, are each divided into unit squares which are alternately colored black and white like a checkerboard. Each rectangle is cut along one of its diagonals into two triangles. For each triangle, let m be the total black area and n the total white area. Find the maximum value of $|m - n|$ for the $6$ triangles. [b]p18.[/b] In triangle $ABC$, $\angle BAC = 90^o$, and the length of segment $AB$ is $2011$. Let $M$ be the midpoint of $BC$ and $D$ the midpoint of $AM$. Let $E$ be the point on segment $AB$ such that $EM \parallel CD$. What is the length of segment $BE$? [u]Round 7[/u] [b]p19.[/b] How many integers from $1$ to $100$, inclusive, can be expressed as the difference of two perfect squares? (For example, $3 = 2^2 - 1^2$). [b]p20.[/b] In triangle $ABC$, $\angle ABC = 45$ and $\angle ACB = 60^o$. Let $P$ and $Q$ be points on segment $BC$, $F$ a point on segment $AB$, and $E$ a point on segment $AC$ such that $F Q \parallel AC$ and $EP \parallel AB$. Let $D$ be the foot of the altitude from $A$ to $BC$. The lines $AD$, $F Q$, and $P E$ form a triangle. Find the positive difference, in degrees, between the largest and smallest angles of this triangle. [b]p21.[/b] For real number $x$, $\lceil x \rceil$ is equal to the smallest integer larger than or equal to $x$. For example, $\lceil 3 \rceil = 3$ and $\lceil 2.5 \rceil = 3$. Let $f(n)$ be a function such that $f(n) = \left\lceil \frac{n}{2}\right\rceil + f\left( \left\lceil \frac{n}{2}\right\rceil\right)$ for every integer $n$ greater than $1$. If $f(1) = 1$, find the maximum value of $f(k) - k$, where $k$ is a positive integer less than or equal to $2011$. [u]Round 8[/u] The answer to each of the three questions in this round depends on the answer to one of the other questions. There is only one set of correct answers to these problems; however, each question will be scored independently, regardless of whether the answers to the other questions are correct. [b]p22.[/b] Let $W$ be the answer to problem 24 in this guts round. Let $f(a) = \frac{1}{1 -\frac{1}{1- \frac{1}{a}}}$. Determine$|f(2) + ... + f(W)|$. [b]p23.[/b] Let $X$ be the answer to problem $22$ in this guts round. How many odd perfect squares are less than $8X$? [b]p24.[/b] Let $Y$ be the answer to problem $23$ in this guts round. What is the maximum number of points of intersections of two regular $(Y - 5)$-sided polygons, if no side of the first polygon is parallel to any side of the second polygon? [u]Round 9[/u] [b]p25.[/b] Cross country skiers $s_1, s_2, s_3, ..., s_7$ start a race one by one in that order. While each skier skis at a constant pace, the skiers do not all ski at the same rate. In the course of the race, each skier either overtakes another skier or is overtaken by another skier exactly two times. Find all the possible orders in which they can finish. Write each possible finish as an ordered septuplet $(a, b, c, d, e, f, g)$ where $a, b, c, d, e, f, g$ are the numbers $1-7$ in some order. (So a finishes first, b finishes second, etc.) [b]p26.[/b] Archie the Alchemist is making a list of all the elements in the world, and the proportion of earth, air, fire, and water needed to produce each. He writes the proportions in the form E:A:F:W. If each of the letters represents a whole number from $0$ to $4$, inclusive, how many different elements can Archie list? Note that if Archie lists wood as $2:0:1:2$, then $4:0:2:4$ would also produce wood. In addition, $0:0:0:0$ does not produce an element. [b]p27.[/b] Let $ABCD$ be a rectangle with $AB = 10$ and $BC = 12$. Let $M$ be the midpoint of $CD$, and $P$ be the point on $BM$ such that $DP = DA$. Find the area of quadrilateral $ABPD$. [u]Round 10[/u] [b]p28.[/b] David the farmer has an infinitely large grass-covered field which contains a straight wall. He ties his cow to the wall with a rope of integer length. The point where David ties his rope to the wall divides the wall into two parts of length $a$ and $b$, where $a > b$ and both are integers. The rope is shorter than the wall but is longer than $a$. Suppose that the cow can reach grass covering an area of $\frac{165\pi}{2}$. Find the ratio $\frac{a}{b}$ . You may assume that the wall has $0$ width. [b]p29.[/b] Let $S$ be the number of ordered quintuples $(a, b, x, y, n)$ of positive integers such that $$\frac{a}{x}+\frac{b}{y}=\frac{1}{n}$$ $$abn = 2011^{2011}$$ Compute the remainder when $S$ is divided by $2012$. [b]p30.[/b] Let $n$ be a positive integer. An $n \times n$ square grid is formed by $n^2$ unit squares. Each unit square is then colored either red or blue such that each row or column has exactly $10$ blue squares. A move consists of choosing a row or a column, and recolor each unit square in the chosen row or column – if it is red, we recolor it blue, and if it is blue, we recolor it red. Suppose that it is possible to obtain fewer than $10n$ blue squares after a sequence of finite number of moves. Find the maximum possible value of $n$. PS. You should use hide for answers. First rounds have been posted [url=https://artofproblemsolving.com/community/c4h2786905p24497746]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

We write pairs of integers on a blackboard. Initially, the pair $(1,2)$ is written. On a move, if $(a, b)$ is on the blackboard, we can add $(-a, -b)$ or $(-b, a+b)$. In addition, if $(a, b)$ and $(c, d)$ are written on the blackboard, we can add $(a+c, b+d)$. Can we reach $(2022, 2023)$?

A wishing well is located at the point $(11,11)$ in the $xy$-plane. Rachelle randomly selects an integer $y$ from the set $\left\{ 0, 1, \dots, 10 \right\}$. Then she randomly selects, with replacement, two integers $a,b$ from the set $\left\{ 1,2,\dots,10 \right\}$. The probability the line through $(0,y)$ and $(a,b)$ passes through the well can be expressed as $\frac mn$, where $m$ and $n$ are relatively prime positive integers. Compute $m+n$. [i]Proposed by Evan Chen[/i]

On the blackboard, there are numbers $1, 2, \dots , 100$. At each move, Bob erases arbitrary two numbers $a$ and $b$, where $a \ge b > 0$, and writes the single number $\lfloor{a/b}\rfloor$. After $99$ such moves the blackboard will contain a single number. What is its maximum possible value? (Reminder that $\lfloor{x}\rfloor$ is the maximum integer not exceeding $x$.)

The numbers from $1$ to $2015$ are written on sheets so that if if $n-m$ is a prime, then $n$ and $m$ are on different sheets. What is the minimum number of sheets required?