Let $r$ and $s$ be two rational numbers. Find all functions $f: \mathbb Q \to \mathbb Q$ such that for all $x,y\in\mathbb Q$ we have \[ f(x+f(y)) = f(x+r)+y+s. \]

Let $S^1 = \{ z \in \mathbb{C} \mid |z| =1 \}$ be the unit circle in the complex plane. Let $f \colon S^1 \longrightarrow S^2$ be the map given by $f(z) = z^2$. We define $f^{(1)} \colon = f$ and $f^{(k+1)} \colon = f \circ f^{(k)}$ for $k \geq 1$. The smallest positive integer $n$ such that $f^{(n)}(z) = z$ is called the [i]period[/i] of $z$. Determine the total number of points in $S^1$ of period $2025$. (Hint : $2025 = 3^4 \times 5^2$)

Let $P$ be a polynomial of degree greater than or equal to $4$ with integer coefficients. An integer $x$ is called $P$-[i]representable[/i] if there exists integer numbers $a$ and $b$ such that $x = P(a) - P(b)$. Prove that, if for all $N \geq 0$, more than half of the integers of the set $\{0,1,\dots,N\}$ are $P$-[i]representable[/i], then all the even integers are $P$-[i]representable[/i] or all the odd integers are $P$-[i]representable[/i].

Find all triples of non-negative integers $(a,b,c)$ which simultaneously satisfy the conditions: [list] [*] $1\leq a<b<c\leq 100$, [*] $b$ is the geometric mean of $a$ and $c$, [*] $\{\sqrt{b}\}$ is the arithmetic mean of $\{\sqrt{a}\}$ and $\{\sqrt{c}\}$.

Professor Oak is feeding his $100$ Pokémon. Each Pokémon has a bowl whose capacity is a positive real number of kilograms. These capacities are known to Professor Oak. The total capacity of all the bowls is $100$ kilograms. Professor Oak distributes $100$ kilograms of food in such a way that each Pokémon receives a non-negative integer number of kilograms of food (which may be larger than the capacity of the bowl). The [i]dissatisfaction level[/i] of a Pokémon who received $N$ kilograms of food and whose bowl has a capacity of $C$ kilograms is equal to $\lvert N-C\rvert$. Find the smallest real number $D$ such that, regardless of the capacities of the bowls, Professor Oak can distribute food in a way that the sum of the dissatisfaction levels over all the $100$ Pokémon is at most $D$. [i]Oleksii Masalitin, Ukraine[/i]

[b]p1.[/b] Let $f(x) = \frac{3x^3+7x^2-12x+2}{x^2+2x-3}$ . Find all integers $n$ such that $f(n)$ is an integer. [b]p2.[/b] How many ways are there to arrange $10$ trees in a line where every tree is either a yew or an oak and no two oak trees are adjacent? [b]p3.[/b] $20$ students sit in a circle in a math class. The teacher randomly selects three students to give a presentation. What is the probability that none of these three students sit next to each other? [b]p4.[/b] Let $f_0(x) = x + |x - 10| - |x + 10|$, and for $n \ge 1$, let $f_n(x) = |f_{n-1}(x)| - 1$. For how many values of $x$ is $f_{10}(x) = 0$? [b]p5.[/b] $2$ red balls, $2$ blue balls, and $6$ yellow balls are in a jar. Zion picks $4$ balls from the jar at random. What is the probability that Zion picks at least $1$ red ball and$ 1$ blue ball? [b]p6.[/b] Let $\vartriangle ABC$ be a right-angled triangle with $\angle ABC = 90^o$ and $AB = 4$. Let $D$ on $AB$ such that $AD = 3DB$ and $\sin \angle ACD = \frac35$ . What is the length of $BC$? [b]p7.[/b] Find the value of of $$\dfrac{1}{1 +\dfrac{1}{2+ \dfrac{1}{1+ \dfrac{1}{2+ \dfrac{1}{1+ ...}}}}}$$ [b]p8.[/b] Consider all possible quadrilaterals $ABCD$ that have the following properties; $ABCD$ has integer side lengths with $AB\parallel CD$, the distance between $\overline{AB}$ and $\overline{CD}$ is $20$, and $AB = 18$. What is the maximum area among all these quadrilaterals, minus the minimum area? [b]p9.[/b] How many perfect cubes exist in the set $\{1^{2018},2^{2017}, 3^{2016},.., 2017^2, 2018^1\}$? [b]p10.[/b] Let $n$ be the number of ways you can fill a $2018\times 2018$ array with the digits $1$ through $9$ such that for every $11\times 3$ rectangle (not necessarily for every $3 \times 11$ rectangle), the sum of the $33$ integers in the rectangle is divisible by $9$. Compute $\log_3 n$. PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

The remainder $R$ obtained by dividing $x^{100}$ by $x^2-3x+2$ is a polynomial of degree less than $2$. Then $R$ may be written as: $\textbf{(A) }2^{100}-1\qquad \textbf{(B) }2^{100}(x-1)-(x-2)\qquad \textbf{(C) }2^{100}(x-3)\qquad$ $\textbf{(D) }x(2^{100}-1)+2(2^{99}-1)\qquad \textbf{(E) }2^{100}(x+1)-(x+2)$

Find all functions $f : \mathbb{R} \rightarrow \mathbb{R}$ such that $$(y + 1)f(x) + f(xf(y) + f(x + y))= y$$ for all $x, y \in \mathbb{R}$.

Let $\{a_k\}_{k\geq 1}$ be a sequence of non-negative integers, such that $a_k \geq a_{2k} + a_{2k+1}$, for all $k\geq 1$. a) Prove that for all positive integers $n\geq 1$ there exist $n$ consecutive terms equal with 0 in the sequence $\{a_k\}_k$; b) State an example of sequence with the property in the hypothesis which contains an infinite number of non-zero terms.

Prove that there exists a convex 1990-gon with the following two properties : [b]a.)[/b] All angles are equal. [b]b.)[/b] The lengths of the 1990 sides are the numbers $ 1^2$, $ 2^2$, $ 3^2$, $ \cdots$, $ 1990^2$ in some order.

[u]Round 1[/u] [b]p1.[/b] Alice has a pizza with eight slices. On each slice, she either adds only salt, only pepper, or leaves it plain. Determine how many ways there are for Alice to season her entire pizza. [b]p2.[/b] Call a number almost prime if it has exactly three divisors. Find the number of almost prime numbers less than $100$. [b]p3.[/b] Determine the maximum number of points of intersection between a circle and a regular pentagon. [u]Round 2[/u] [b]p4.[/b] Let $d(n)$ denote the number of positive integer divisors of $n$. Find $d(d(20^{18}))$. [b]p5.[/b] $20$ chubbles are equal to $19$ flubbles. $20$ flubbles are equal to $18$ bubbles. How many bubbles are $1000$ chubbles worth? [b]p6.[/b] Square $ABCD$ and equilateral triangle $EFG$ have equal area. Compute $\frac{AB}{EF}$ . [u]Round 3[/u] [b]p7.[/b] Find the minimumvalue of $y$ such that $y = x^2 -6x -9$ where x is a real number. [b]p8.[/b] I have $2$ pairs of red socks, $5$ pairs of white socks, and $7$ pairs of blue socks. If I randomly pull out one sock at a time without replacement, how many socks do I need to draw to guarantee that I have drawn $3$ pairs of socks of the same color? [b]p9. [/b]There are $23$ paths from my house to the school, $29$ paths from the school to the library, and $3$ paths fromthe library to town center. Additionally, there are $6$ paths directly from my house to the library. If I have to pass through the library to get to town center, howmany ways are there to travel from my house all the way to the town center? [u]Round 4[/u] [b]p10.[/b] A circle of radius $25$ and a circle of radius $4$ are externally tangent. A line is tangent to the circle of radius $25$ at $A$ and the circle of radius $4$ at $B$, where $A \ne B$. Compute the length of $AB$. [b]p11.[/b] A gambler spins two wheels, one numbered $1$ to $4$ and another numbered $1$ to $5$, and the amount of money he wins is the sum of the two numbers he spins in dollars. Determine the expected amount of money he will win. [b]p12.[/b] Find the remainder when $20^{19}$ is divided by $18$. PS. You should use hide for answers. Rounds 5-8 have been posted [url=https://artofproblemsolving.com/community/c3h3166012p28809547]here [/url] and 9-12 [url=https://artofproblemsolving.com/community/c3h3166099p28810427]here[/url].Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

$f$ and $g$ are real-valued functions defined on the real line. For all $x$ and $y, f(x+y)+f(x-y)=2f(x)g(y)$. $f$ is not identically zero and $|f(x)|\le1$ for all $x$. Prove that $|g(x)|\le1$ for all $x$.

Two graphs $G_1$ and $G_2$ of quadratic polynomials intersect at points $A$ and $B$. Let $O$ be the vertex of $G_1$. Lines $OA$ and $OB$ intersect $G_2$ again at points $C$ and $D$. Prove that $CD$ is parallel to the $x$-axis.

Let $k$ be any fixed positive integer. Let's look at integer pairs $(a, b)$, for which the quadratic equations $x^2 - 2ax + b = 0$ and $y^2 + 2ay + b = 0$ are real solutions (not necessarily different), which can be denoted by $x_1, x_2$ and $y_1, y_2$, respectively, in such an order that the equation $x_1 y_1 - x_2 y_2 = 4k$. a) Find the largest possible value of the second component $b$ of such a pair of numbers ($a, b)$. b) Find the sum of the other components of all such pairs of numbers.

Find all functions $ f: \mathbb{R} \rightarrow \mathbb{R}$ for which the equality \[f(f(x)\minus{}y)\equal{}f(x)\plus{}f(f(y)\minus{}f(\minus{}x))\plus{}x\] holds for all real $x,y$.

[i]Introductory part [/i] We call an $n$-tuple $x = (x_1, x_2, ... , x_n)$, with $x_k \in R$ (or respectively with all $x_k \in Z$) a real vector (or respectively an integer vector). The set of all real vectors (respectively all integer vectors) is usually denoted by $R^n$ (respectively $Z^n$). A vector $x$ is null if and only if $x_k = 0$, for all $k \in \{1, 2,... , n\}$. Also let $U_n$ be the set of all real vectors $x = (x_1, x_2, ... , x_n)$, such that $x^2_1 + x^2_2 + ...+ x^2_n = 1$. For two vectors $x = (x_1, ... , x_n), y = (y_1, ..., y_n)$ we define the scalar product as the real number $x\cdot y = x_1y_1 + x_2y_2 +...+ x_ny_n$. We define the norm of the vector $x$ as $||x|| =\sqrt{x^2_1 + x^2_2 + ...+ x^2_n}$ [i]The problem[/i] Let $A(k, r) = \{x \in U_n |$ for all $z \in Z^n$ we have either $|x \cdot z| \ge \frac{k}{||z||^r}$ or $z$ is null $\}$. Prove that if $r > n - 1$ the we can find a positive number $k$ such that $A(k, r)$ is not empty, and if $r < n - 1$ we cannot find such a positive number $k$.

The points $(1,1),(2,3),(4,5)$ and $(999,111)$ are marked in the coordinate system. We continue to mark points in the following way : [list] [*]If points $(a,b)$ are marked then $(b,a)$ and $(a-b,a+b)$ can be marked [*]If points $(a,b)$ and $(c,d)$ are marked then so can be $(ad+bc, 4ac-4bd)$. [/list] Can we, after some finite number of these steps, mark a point belonging to the line $y=2x$.

The sequence $a_1,a_2,a_3,...$, is constructed according to the rule $$a_1=1, a_2=a_1+1/a_1, ... , a_{n+1}=a_n+1/a_n, ...$$ Prove that $a_{100} > 14$.

A parabola $p$ is drawn on the coordinate plane — the graph of the equation $y =-x^2$, and a point $A$ is marked that does not lie on the parabola $p$. All possible parabolas $q$ of the form $y = x^2+ax+b$ are drawn through point $A$, intersecting $p$ at two points $X$ and $Y$ . Prove that all possible $XY$ lines pass through a fixed point in the plane. [i]P.A.Kozhevnikov[/i]

Find all monotone functions $ f: \mathbb{R} \to \mathbb{R} $ satisfying the equation $$ f(4x)-f(3x) = 2x \ \ \text{ for } \ \ x \in \mathbb{R}.$$

Solve $2|3-2x|-|x-2|=x$ , $x\in\mathbb{R}$ .

Is there an integer coefficients polynomial $P(x)$ satisfying \[ \begin{cases} P(1+\sqrt[3]{2})=1+\sqrt[3]{2}\\ P(1+\sqrt{5})=2+3\sqrt{5}\end{cases} \]

[b]p1.[/b] Consider a triangular grid: nodes of the grid are painted black and white. At a single step you are allowed to change colors of all nodes situated on any straight line (with the slope $0^o$ ,$60^o$, or $120^o$ ) going through the nodes of the grid. Can you transform the combination in the left picture into the one in the right picture in a finite number of steps? [img]https://cdn.artofproblemsolving.com/attachments/3/a/957b199149269ce1d0f66b91a1ac0737cf3f89.png[/img] [b]p2.[/b] Find $x$ satisfying $\sqrt{x\sqrt{x \sqrt{x ...}}} = \sqrt{2022}$ where it is an infinite expression on the left side. [b]p3.[/b] $179$ glasses are placed upside down on a table. You are allowed to do the following moves. An integer number $k$ is fixed. In one move you are allowed to turn any $k$ glasses . (a) Is it possible in a finite number of moves to turn all $179$ glasses into “bottom-down” positions if $k=3$? (b) Is it possible to do it if $k=4$? [b]p4.[/b] An interval of length $1$ is drawn on a paper. Using a compass and a simple ruler construct an interval of length $\sqrt{93}$. [b]p5.[/b] Show that $5^{2n+1} + 3^{n+2} 2^{n-1} $ is divisible by $19$ for any positive integer $n$. [b]p6.[/b] Solve the system $$\begin{cases} \dfrac{xy}{x+y}=1-z \\ \dfrac{yz}{y+z}=2-x \\ \dfrac{xz}{x+z}=2-y \end{cases}$$ PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

[b]p1.[/b] There are $2024$ apples in a very large basket. First, Julie takes away half of the apples in the basket; then, Diane takes away $202$ apples from the remaining bunch. How many apples remain in the basket? [b]p2.[/b] The set of all permutations (different arrangements) of the letters in ”ABMC” are listed in alphabetical order. The first item on the list is numbered $1$, the second item is numbered $2$, and in general, the kth item on the list is numbered $k$. What number is given to ”ABMC”? [b]p3.[/b] Daniel has a water bottle that is three-quarters full. After drinking $3$ ounces of water, the water bottle is three-fifths full. The density of water is $1$ gram per milliliter, and there are around $28$ grams per ounce. How many milliliters of water could the bottle fit at full capacity? [b]p4.[/b] How many ways can four distinct $2$-by-$1$ rectangles fit on a $2$-by-$4$ board such that each rectangle is fully on the board? [b]p5.[/b] Iris and Ivy start reading a $240$ page textbook with $120$ left-hand pages and $120$ right-hand pages. Iris takes $4$ minutes to read each page, while Ivy takes $5$ minutes to read a left-hand page and $3$ minutes to read a right-hand page. Iris and Ivy move onto the next page only when both sisters have completed reading. If a sister finishes reading a page first, the other sister will start reading three times as fast until she completes the page. How many minutes after they start reading will both sisters finish the textbook? [b]p6.[/b] Let $\vartriangle ABC$ be an equilateral triangle with side length $24$. Then, let $M$ be the midpoint of $BC$. Define $P$ to be the set of all points $P$ such that $2PM = BC$. The minimum value of $AP$ can be expressed as $\sqrt{a}- b$, where $a$ and $b$ are positive integers. Find $a + b$. [b]p7.[/b] Jonathan has $10$ songs in his playlist: $4$ rap songs and $6$ pop songs. He will select three unique songs to listen to while he studies. Let $p$ be the probability that at least two songs are rap, and let $q$ be the probability that none of them are rap. Find $\frac{p}{q}$ . [b]p8.[/b] A number $K$ is called $6,8$-similar if $K$ written in base $6$ and $K$ written in base $8$ have the same number of digits. Find the number of $6,8$-similar values between $1$ and $1000$, inclusive. [b]p9.[/b] Quadrilateral $ABCD$ has $\angle ABC = 90^o$, $\angle ADC = 120^o$, $AB = 5$, $BC = 18$, and $CD = 3$. Find $AD^2$. [b]p10.[/b] Bob, Eric, and Raymond are playing a game. Each player rolls a fair $6$-sided die, and whoever has the highest roll wins. If players are tied for the highest roll, the ones that are tied reroll until one wins. At the start, Bob rolls a $4$. The probability that Eric wins the game can be expressed as $\frac{p}{q}$ where $p$ and $q$ are relatively prime positive integers. Find $p + q$. [b]p11.[/b] Define the following infinite sequence $s$: $$s = \left\{\frac92,\frac{99}{2^2},\frac{999}{2^3} , ... , \overbrace{\frac{999...999}{2^k}}^{k\,\,nines}, ...\right\}$$ The sum of the first $2024$ terms in $s$, denoted $S$, can be expressed as $$S =\frac{5^a - b}{4}+\frac{1}{2^c},$$ where $a, b$, and $c$ are positive integers. Find $a + b + c$. [b]p12.[/b] Andy is adding numbers in base $5$. However, he accidentally forgets to write the units digit of each number. If he writes all the consecutive integers starting at $0$ and ending at $50$ (base $10$) and adds them together, what is the difference between Andy’s sum and the correct sum? (Express your answer in base-$10$.) [b]p13.[/b] Let $n$ be the positive real number such that the system of equations $$y =\frac{1}{\sqrt{2024 - x^2}}$$ $$y =\sqrt{x^2 - n}$$ has exactly two real solutions for $(x, y)$: $(a, b)$ and $(-a, b)$. Then, $|a|$ can be expressed as $j\sqrt{k}$, where $j$ and $k$ are integers such that $k$ is not divisible by any perfect square other than $1$. Find $j · k$. [b]p14.[/b] Nakio is playing a game with three fair $4$-sided dice. But being the cheater he is, he has secretly replaced one of the three die with his own $4$-sided die, such that there is a $1/2$ chance of rolling a $4$, and a $1/6$ chance to roll each number from $1$ to $3$. To play, a random die is chosen with equal probability and rolled. If Nakio guesses the number that is on the die, he wins. Unfortunately for him, Nakio’s friends have an anti-cheating mechanism in place: when the die is picked, they will roll it three times. If each roll lands on the same number, that die is thrown out and one of the two unused dice is chosen instead with equal probability. If Nakio always guesses $4$, the probability that he wins the game can be expressed as $\frac{m}{n}$ , where $m$ and $n$ are relatively prime. Find $m + n$. [b]p15.[/b] A particle starts in the center of a $2$m-by-$2$m square. It moves in a random direction such that the angle between its direction and a side of the square is a multiple of $30^o$. It travels in that direction at $1$ m/s, bouncing off of the walls of the square. After a minute, the position of the particle is recorded. The expected distance from this point to the start point can be written as $$\frac{1}{a}\left(b - c\sqrt{d}\right),$$ where $a$ and $b$ are relatively prime, and d is not divisible by any perfect square. Find $a + b + c + d$. PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Find all positive integers $n$ with $n \ge 2$ such that the polynomial \[ P(a_1, a_2, ..., a_n) = a_1^n+a_2^n + ... + a_n^n - n a_1 a_2 ... a_n \] in the $n$ variables $a_1$, $a_2$, $\dots$, $a_n$ is irreducible over the real numbers, i.e. it cannot be factored as the product of two nonconstant polynomials with real coefficients. [i]Proposed by Yang Liu[/i]