$\mathbb{N}$ is the set of positive integers. Determine all functions $f:\mathbb{N}\to\mathbb{N}$ such that for every pair $(m,n)\in\mathbb{N}^2$ we have that: \[f(m)+f(n) \ | \ m+n .\]

[b]p1.[/b] If $$ \begin{cases} a^2 + b^2 + c^2 = 1000 \\ (a + b + c)^2 = 100 \\ ab + bc = 10 \end{cases}$$ what is $ac$? [b]p2.[/b] If a and b are real numbers such that $a \ne 0$ and the numbers $1$, $a + b$, and $a$ are, in some order, the numbers $0$, $\frac{b}{a}$ , and $b$, what is $b - a$? [b]p3.[/b] Of the first $120$ natural numbers, how many are divisible by at least one of $3$, $4$, $5$, $12$, $15$, $20$, and $60$? [b]p4.[/b] For positive real numbers $a$, let $p_a$ and $q_a$ be the maximum and minimum values, respectively, of $\log_a(x)$ for $a \le x \le 2a$. If $p_a - q_a = \frac12$ , what is $a$? [b]p5.[/b] Let $ABC$ be an acute triangle and let $a$, $b$, and $c$ be the sides opposite the vertices $A$, $B$, and $C$, respectively. If $a = 2b \sin A$, what is the measure of angle $B$? [b]p6.[/b] How many ordered triples $(x, y, z)$ of positive integers satisfy the equation $$x^3 + 2y^3 + 4z^3 = 9?$$ [b]p7.[/b] Joe has invented a robot that travels along the sides of a regular octagon. The robot starts at a vertex of the octagon and every minute chooses one of two directions (clockwise or counterclockwise) with equal probability and moves to the next vertex in that direction. What is the probability that after $8$ minutes the robot is directly opposite the vertex it started from? [b]p8.[/b] Find the nonnegative integer $n$ such that when $$\left(x^2 -\frac{1}{x}\right)^n$$ is completely expanded the constant coefficient is $15$. [b]p9.[/b] For each positive integer $k$, let $$f_k(x) = \frac{kx + 9}{x + 3}.$$ Compute $$f_1 \circ f_2\circ ... \circ f_{13}(2).$$ [b]p10.[/b] Exactly one of the following five integers cannot be written in the form $x^2 + y^2 + 5z^2$, where $x$, $y$, and $z$ are integers. Which one is it? $$2003, 2004, 2005, 2006, 2007$$ [b]p11.[/b] Suppose that two circles $C_1$ and $C_2$ intersect at two distinct points $M$ and $N$. Suppose that $P$ is a point on the line $MN$ that is outside of both $C_1$ and $C_2$. Let $A$ and $B$ be the two distinct points on $C_1$ such that AP and BP are each tangent to $C_1$ and $B$ is inside $C_2$. Similarly, let $D$ and $E$ be the two distinct points on $C_2$ such that $DP$ and $EP$ are each tangent to $C_2$ and $D$ is inside $C_1$. If $AB = \frac{5\sqrt2}{2}$ , $AD = 2$, $BD = 2$, $EB = 1$, and $ED =\sqrt2$, find $AE$. [b]p12.[/b] How many ordered pairs $(x, y)$ of positive integers satisfy the following equation? $$\sqrt{x} +\sqrt{y} =\sqrt{2007}.$$ [b]p13.[/b] The sides $BC$, $CA$, and $CB$ of triangle $ABC$ have midpoints $K$, $L$, and $M$, respectively. If $$AB^2 + BC^2 + CA^2 = 200,$$ what is $AK^2 + BL^2 + CM^2$? [b]p14.[/b] Let $x$ and $y$ be real numbers that satisfy: $$x + \frac{4}{x}= y +\frac{4}{y}=\frac{20}{xy}.$$ Compute the maximum value of $|x - y|$. [b]p15.[/b] $30$ math meet teams receive different scores which are then shuffled around to lend an aura of mystery to the grading. What is the probability that no team receives their own score? Express your answer as a decimal accurate to the nearest hundredth. PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Does the system of equation \begin{align*} \begin{cases} x_1 + x_2 &= y_1 + y_2 + y_3 + y_4 \\ x_1^2 + x_2^2 &= y_1^2 + y_2^2 + y_3^2 + y_4^2 \\ x_1^3 + x_2^3 &= y_1^3 + y_2^3 + y_3^3 + y_4^3 \end{cases} \end{align*} admit a solution in integers such that the absolute value of each of these integers is greater than $2020$?

Is it possible to colour all integers greater than $1{}$ in three colours (each integer in one colour, all three colours must be used) so that the colour of the product of any two differently coloured numbers is different from the colour of each of the factors?

$(\sqrt6 + \sqrt7)^{1000}$ in base ten has a tens digit of $a$ and a ones digit of $b$. Determine $10a + b$.

Let n be an odd positive integer.Let M be a set of $n$ positive integers ,which are 2x2 different. A set $T$ $\in$ $M$ is called "good" if the product of its elements is divisible by the sum of the elements in $M$, but is not divisible by the square of the same sum. Given that $M$ is "good",how many "good" subsets of $M$ can there be?

Let $a, b, c$ and $d$ be non-negative integers. Prove that the numbers $2^a7^b$ and $2^c7^d$ give the same remainder when divided by $15$ iff the numbers $3^a5^b$ and $3^c5^d$ give the same remainder when divided by $16$.

What is the smallest positive integer $t$ such that there exist integers $x_1,x_2,\ldots,x_t$ with \[x^3_1+x^3_2+\,\ldots\,+x^3_t=2002^{2002}\,?\]

Determine all even numbers $n$, $n \in \mathbb N$ such that \[{ \frac{1}{d_{1}}+\frac{1}{d_{2}}+ \cdots +\frac{1}{d_{k}}=\frac{1620}{1003}}, \] where $d_1, d_2, \ldots, d_k$ are all different divisors of $n$.

What is the largest integer $n < 2018$ such that for all integers $b > 1$, $n$ has at least as many $1$'s in its base-$4$ representation as it has in its base-$b$ representation?

Determine the positive integers expressible in the form $\frac{x^2+y}{xy+1}$, for at least $2$ pairs $(x,y)$ of positive integers

[b]p1.[/b] Thirty players participate in a chess tournament. Every player plays one game with every other player. What maximal number of players can get exactly $5$ points? (any game adds $1$ point to the winner’s score, $0$ points to a loser’s score, in the case of a draw each player obtains $1/2$ point.) [b]p2.[/b] A father and his son returned from a fishing trip. To make their catches equal the father gave to his son some of his fish. If, instead, the son had given his father the same number of fish, then father would have had twice as many fish as his son. What percent more is the father's catch more than his son's? [b]p3.[/b] What is the maximal number of pieces of two shapes, [img]https://cdn.artofproblemsolving.com/attachments/a/5/6c567cf6a04b0aa9e998dbae3803b6eeb24a35.png[/img] and [img]https://cdn.artofproblemsolving.com/attachments/8/a/7a7754d0f2517c93c5bb931fb7b5ae8f5e3217.png[/img], that can be used to tile a $7\times 7$ square? [b]p4.[/b] Six shooters participate in a shooting competition. Every participant has $5$ shots. Each shot adds from 1 to $10$ points to shooter’s score. Every person can score totally for all five shots from $5$ to $50$ points. Each participant gets $7$ points for at least one of his shots. The scores of all participants are different. We enumerate the shooters $1$ to $6$ according to their scores, the person with maximal score obtains number $1$, the next one obtains number $2$, the person with minimal score obtains number $6$. What score does obtain the participant number 3? The total number of all obtained points is $264$. [b]p5.[/b] There are $2014$ stones in a pile. Two players play the following game. First, player $A$ takes some number of stones (from $1$ to $30$) from the pile, then player B takes $1$ or $2$ stones, then player $A$ takes $2$ or $3$ stones, then player $B$ takes $3$ or $4$ stones, then player A takes $4$ or $5$ stones, etc. The player who gets the last stone is the winner. If no player gets the last stone (there is at least one stone in the pile but the next move is not allowed) then the game results in a draw. Who wins the game using the right strategy? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $X$ be a non empty subset of $\mathbb{N} = \{1,2,\ldots \}$. Suppose that for all $x \in X$, $4x \in X$ and $\lfloor \sqrt{x} \rfloor \in X$. Prove that $X=\mathbb{N}$.

[u]Round 9[/u] [b]p25.[/b] Maisy the Bear is at the origin of the Cartesian Plane. WhenMaisy is on the point $(m,n)$ then it can jump to either $(m,n +1)$ or $(m+1,n)$. Let $L(x, y)$ be the number of pathsMaisy can take to reach the point $(x, y)$. The sum of $L(x, y)$ over all lattice points $(x, y)$ with both coordinates between $0$ and $2020$, inclusive, can be written as ${2k \choose k} - j$ for a minimum positive integer k and corresponding positive integer $j$ . Find $k + j$ . [b]p26.[/b] A circle with center $O$ and radius $2$ and a circle with center $P$ and radius $3$ are externally tangent at $A$. Points $B$ and $C$ are on the circle with center $O$ such that $\vartriangle ABC$ is equilateral. Segment $AB$ extends past B to point $D$ and $AC$ extends past $C$ to point $E$ such that $BD = CE =\sqrt3$. A line through $D$ is tangent to circle $P$ at $F$. The value of $EF^2$ can be expressed as $\frac{a+b\sqrt{c}}{d}$ where $a$, $b$, $c$, and $d$ are integers, c is squarefree, and $gcd(a,b,d) = 1$. Find $a +b +c +d$. [b]p27.[/b] Find the number of trailing zeroes at the end of $$\sum^{2021}_{i=1}(2021^i -1) = (2021^1 -1)...(2021^{2021}-1).$$ [u]Round 10[/u] [b]p28.[/b] Points $A, B, C, P$, and $D$ lie on circle ω in that order. Let $AC$ and $BD$ intersect at $I$ . Given that $PI = PC = PD$, $\angle DAB = 137^o$, and $\angle ABC = 109^o$, find the measure of $\angle BIC$ in degrees. [b]p29.[/b] Find the sum of all positive integers $n < 2021$ such that when ${d_1,d_2,... ,d_k}$ are the positive integer factors of $n$, then $$\left( \sum^{k}_{i=1}d_i \right) \left( \sum^{k}_{i=1} \frac{1}{d_i} \right)= r^2$$ for some rational number $r$ . [b]p30.[/b] Let $a, b, c, d$ and $e$ be positive real numbers. Define the function $f (x, y) = \frac{x}{y}+\frac{y}{x}$ for all positive real numbers. Given that $f (a,b) = 7$, $f (b,c) = 5$, $f (c,d) = 3$, and $f (d,e) = 2$, find the sum of all possible values of $f (e,a)$. [u]Round 11[/u] [b]p31.[/b] There exist $100$ (not necessarily distinct) complex numbers $r_1, r_2,..., r_{100}$ such that for any positive integer $1 \le k \le 100$, we have that $P(r_k ) = 0$ where the polynomial $P$ is defined as $$P(x) = \sum^{101}_{i=1}i \cdot x^{101-i} = x^{100} +2x^{99} +3x^{98} +...+99x^2 +100x +101.$$ Find the value of $$\prod^{100}_{j=1} (r^2_j+1) = (r^2_1 +1)(r^2_2 +1)...(r^2_{100} +1).$$ [b]p32.[/b] Let $BT$ be the diameter of a circle $\omega_1$, and $AT$ be a tangent of $\omega_1$. Line $AB$ intersects $\omega_1$ at $C$, and $\vartriangle ACT$ has circumcircle $\omega_2$. Points $P$ and $S$ exist such that $PA$ and $PC$ are tangent to $\omega_2$ and $SB = BT = 20$. Given that $AT = 15$, the length of $PS$ can be written as $\frac{a\sqrt{b}}{c}$ , where $a$, $b$, and $c$ are positive integers, $b$ is squarefree, and $gcd(a,b) = 1$. Find $a +b +c$. [b]p33.[/b] There are a hundred students in math team. Each pair of students are either mutually friends or mutually enemies. It is given that if any three students are chosen, then they are not all mutually friends. The maximum possible number of ways to choose four students such that it is possible to label them $A, B, C$, and $D$ such that $A$ and $B$ are friends, $B$ and $C$ are friends, $C$ and $D$ are friends, and D and A are friends can be expressed as $n^4$. Find $n$. [u]Round 12[/u] [b]p34.[/b] Let $\{p_i\}$ be the prime numbers, such that $p_1 = 2, p_2 = 3, p_3 = 5, ...$ For each $i$ , let $q_i$ be the nearest perfect square to $p_i$ . Estimate $\sum^{2021}_{i=1}|p_i=q_i |$. If the correct answer is $A$ and your answer is $E$, your score will be $\left \lfloor 30 \cdot \max - \left(0,1-5 \cdot \left| \log_{10} \frac{A}{E} \right| \right)\right \rfloor.$ [b]p35.[/b] Estimate the number of digits of $(2021!)^{2021}$. If the correct answer is $A$ and your answer is $E$, your score will be $\left \lfloor 15 \cdot \max \left(0,2- \cdot \left| \log_{10} \frac{A}{E} \right| \right) \right \rfloor.$ [b]p36.[/b] Pick a positive integer between$ 1$ and $1000$, inclusive. If your answer is $E$ and a quarter of the mean of all the responses to this problem is $A$, your score will be $$ \lfloor \max \left(0,30- |A-E|, 2-|E-1000| \right) \rfloor.$$ Note that if you pick $1000$, you will automatically get $2$ points. PS. You should use hide for answers. Rounds 1-4 have been posted [url=https://artofproblemsolving.com/community/c3h3166489p28814241]here [/url] and 5-8 [url=https://artofproblemsolving.com/community/c3h3166494p28814284]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

[b]4.[/b] Find all positive integers $\alpha , \beta (\alpha >1)$ and all prime numbers $p, q, r$ which satisfy the equation $p^{\alpha}= q^{\beta}+r^{\alpha}$ ($\alpha , \beta , p, q, r$ need not necessarily be different). [b](N. 12)[/b]

Find the largest constant $c>0$ such that for every positive integer $n\ge 2$, there always exist a positive divisor $d$ of $n$ such that $$d\le \sqrt{n}\hspace{0.5cm} \text{and} \hspace{0.5cm} \tau(d)\ge c\sqrt{\tau(n)}$$ where $\tau(n)$ is the number of divisors of $n$. [i]Proposed by Mohd. Suhaimi Ramly[/i]

Pedro and Cecilia play the following game: Pedro chooses a positive integer number $a$ and Cecilia wins if she finds a positive integrer number $b$, prime with $a$, such that, in the factorization of $a^3+b^3$ will appear three different prime numbers. Prove that Cecilia can always win.

[b]p1.[/b] Consider a right triangle with legs of lengths $a$ and $b$ and hypotenuse of length $c$ such that the perimeter of the right triangle is numerically (ignoring units) equal to its area. Prove that there is only one possible value of $a + b - c$, and determine that value. [b]p2.[/b] Last August, Jennifer McLoud-Mann, along with her husband Casey Mann and an undergraduate David Von Derau at the University of Washington, Bothell, discovered a new tiling pattern of the plane with a pentagon. This is the fifteenth pattern of using a pentagon to cover the plane with no gaps or overlaps. It is unknown whether other pentagons tile the plane, or even if the number of patterns is finite. Below is a portion of this new tiling pattern. [img]https://services.artofproblemsolving.com/download.php?id=YXR0YWNobWVudHMvOS8xLzM4M2RjZDEzZTliYTlhYTJkZDU4YTA4ZGMwMTA0MzA5ODk1NjI0LnBuZw==&rn=bW1wYyAyMDE1LnBuZw==[/img] Determine the five angles (in degrees) of the pentagon $ABCDE$ used in this tiling. Explain your reasoning, and give the values you determine for the angles at the bottom. [b]p3.[/b] Let $f(x) =\sqrt{2019 + 4\sqrt{2015}} +\sqrt{2015} x$. Find all rational numbers $x$ such that $f(x)$ is a rational number. [b]p4.[/b] Alice has a whiteboard and a blackboard. The whiteboard has two positive integers on it, and the blackboard is initially blank. Alice repeats the following process. $\bullet$ Let the numbers on the whiteboard be $a$ and $b$, with $a \le b$. $\bullet$ Write $a^2$ on the blackboard. $\bullet$ Erase $b$ from the whiteboard and replace it with $b - a$. For example, if the whiteboard began with 5 and 8, Alice first writes $25$ on the blackboard and changes the whiteboard to $5$ and $3$. Her next move is to write $9$ on the blackboard and change the whiteboard to $2$ and $3$. Alice stops when one of the numbers on the whiteboard is 0. At this point the sum of the numbers on the blackboard is $2015$. a. If one of the starting numbers is $1$, what is the other? b. What are all possible starting pairs of numbers? [b]p5.[/b] Professor Beatrix Quirky has many multi-volume sets of books on her shelves. When she places a numbered set of $n$ books on her shelves, she doesn’t necessarily place them in order with book $1$ on the left and book $n$ on the right. Any volume can be placed at the far left. The only rule is that, except the leftmost volume, each volume must have a volume somewhere to its left numbered either one more or one less. For example, with a series of six volumes, Professor Quirky could place them in the order $123456$, or $324561$, or $564321$, but not $321564$ (because neither $4$ nor $6$ is to the left of $5$). Let’s call a sequence of numbers a [i]quirky [/i] sequence of length $n$ if: 1. the sequence contains each of the numbers from $1$ to $n$, once each, and 2. if $k$ is not the first term of the sequence, then either $k + 1$ or $k - 1$ occurs somewhere before $k$ in the sequence. Let $q_n$ be the number of quirky sequences of length $n$. For example, $q_3 = 4$ since the quirky sequences of length $3$ are $123$, $213$, $231$, and $321$. a. List all quirky sequences of length $4$. b. Find an explicit formula for $q_n$. Prove that your formula is correct. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Positive integers $a, b, c$ satisfying the equality $a^2 + b^2 = c^2$. Show that the number $\frac12(c - a) (c - b)$ is square of an integer.

Find the sum $$\sum^{2020}_{n=1} \gcd (n^3 -2n^2 +2021,n^2 -3n +3).$$

$2000$ numbers are considered: $11, 101, 1001, . . $. Prove that at least $99\%$ of these numbers are composite.

Find all the positive integers $n$ such that: $(1)$ $n$ has at least $4$ positive divisors. $(2)$ if all positive divisors of $n$ are $d_1,d_2,\cdots ,d_k,$ then $d_2-d_1,d_3-d_2,\cdots ,d_k-d_{k-1}$ form a geometric sequence.

Determine all functions $f$ defined on the set of all positive integers and taking non-negative integer values, satisfying the three conditions: [list] [*] $(i)$ $f(n) \neq 0$ for at least one $n$; [*] $(ii)$ $f(x y)=f(x)+f(y)$ for every positive integers $x$ and $y$; [*] $(iii)$ there are infinitely many positive integers $n$ such that $f(k)=f(n-k)$ for all $k<n$. [/list]

What is the remainder when $\sum_{k=1}^{2005}k^{2005\cdot 2^{2005}}$ is divided by $2^{2005}$?

Determine the number of positive integers $N = \overline{abcd}$, with $a, b, c, d$ nonzero digits, which satisfy $(2a -1) (2b -1) (2c- 1) (2d - 1) = 2abcd -1$.