Let $p$ be a prime number. Call a positive integer $n$ interesting if \[x^n-1=(x^p-x+1)f(x)+pg(x)\] for some polynomials $f$ and $g$ with integer coefficients. a) Prove that the number $p^p-1$ is interesting. b) For which $p$ is $p^p-1$ the minimal interesting number?

Consider the function $f : R^+ \to R^+$ and satisfying $$f(x + 2y + f(x + y)) = f(2x) + f(3y), \,\, \forall \,\, x, y > 0.$$ 1. Find all functions $f(x)$ that satisfy the given condition. 2. Suppose that $f(4\sin^4x)f(4\cos^4x) \ge f^2(1)$ for all $x \in \left(0\frac{\pi}{2}\right) $. Find the minimum value of $f(2022)$.

Is there a number $s$ in the set $\{\pi,2\pi,3\pi,...,\} $ such that the first three digits after the decimal point of $s$ are $.001$? Fully justify your answer.

Let $M$ be an empty set of real numbers. For any $x \in M$ the functions $f: M\to M$ and $g: M\to M$ satisfy the relations $f (g (x)) = g (f (x)) = x$ and $f (x) + g (x) = x$. Show that $- x \in M$ ¸ and $f (-x) = -f (x)$ whatever $x \in M$.

Anna and Boris move simultaneously towards each other, from points A and B respectively. Their speeds are constant, but not necessarily equal. Had Anna started 30 minutes earlier, they would have met 2 kilometers nearer to B. Had Boris started 30 minutes earlier instead, they would have met some distance nearer to A. Can this distance be uniquely determined? [i](3 points)[/i]

Let $ {{a} _ {1}}, {{a} _ {2}}, \ldots, {{a} _ {n}} $ be permutation of numbers $ 1,2, \ldots, n $, where $ n \geq 2 $. Find the maximum value of the sum $$ S (n) = | {{a} _ {1}} - {{a} _ {2}} | + | {{a} _ {2}} - {{a} _ {3}} | + \cdots + | {{a} _ {n-1}} - {{a} _ {n}} |. $$

Given a sequence $\{ a_n\}_{n\in \mathbb{Z}^+}$ defined by $a_1=1$ and $a_{2k}=a_{2k-1}+a_k,a_{2k+1}=a_{2k}$ for all positive integer $k$. Prove that, for any positive integer $n$, $a_{2^n}>2^{\frac{n^2}{4}}$.

Let $n$ be a positive integer and $x$ positive real number such that none of numbers $x,2x,\dots,nx$ and none of $\frac{1}{x},\frac{2}{x},\dots,\frac{\left\lfloor nx\right\rfloor }{x}$ is an integer. Prove that \[ \left\lfloor x\right\rfloor +\left\lfloor 2x\right\rfloor +\dots+\left\lfloor nx\right\rfloor +\left\lfloor \frac{1}{x}\right\rfloor +\left\lfloor \frac{2}{x}\right\rfloor +\dots+\left\lfloor \frac{\left\lfloor nx\right\rfloor }{x}\right\rfloor =n\left\lfloor nx\right\rfloor \]

[u]Round 1[/u] [b]p1.[/b] Alex is on a week-long mining quest. Each morning, she mines at least $1$ and at most $10$ diamonds and adds them to her treasure chest (which already contains some diamonds). Every night she counts the total number of diamonds in her collection and finds that it is divisible by either $22$ or $25$. Show that she miscounted. [b]p2.[/b] Hermione set out a row of $11$ Bertie Bott’s Every Flavor Beans for Ron to try. There are $5$ chocolateflavored beans that Ron likes and $6$ beans flavored like earwax, which he finds disgusting. All beans look the same, and Hermione tells Ron that a chocolate bean always has another chocolate bean next to it. What is the smallest number of beans that Ron must taste to guarantee he finds a chocolate one? [b]p3.[/b] There are $101$ pirates on a pirate ship: the captain and $100$ crew. Each pirate, including the captain, starts with $1$ gold coin. The captain makes proposals for redistributing the coins, and the crew vote on these proposals. The captain does not vote. For every proposal, each crew member greedily votes “yes” if he gains coins as a result of the proposal, “no” if he loses coins, and passes otherwise. If strictly more crew members vote “yes” than “no,” the proposal takes effect. The captain can make any number of proposals, one after the other. What is the largest number of coins the captain can accumulate? [b]p4.[/b] There are $100$ food trucks in a circle and $10$ gnomes who sample their menus. For the first course, all the gnomes eat at different trucks. For each course after the first, gnome #$1$ moves $1$ truck left or right and eats there; gnome #$2$ moves $2$ trucks left or right and eats there; ... gnome #$10$ moves $10$ trucks left or right and eats there. All gnomes move at the same time. After some number of courses, each food truck had served at least one gnome. Show that at least one gnome ate at some food truck twice. [b]p5.[/b] The town of Lumenville has $100$ houses and is preparing for the math festival. The Tesla wiring company lays lengths of power wire in straight lines between the houses so that power flows between any two houses, possibly by passing through other houses.The Edison lighting company hangs strings of lights in straight lines between pairs of houses so that each house is connected by a string to exactly one other. Show that however the houses are arranged, the Edison company can always hang their strings of lights so that the total length of the strings is no more than the total length of the power wires the Tesla company used. [img]https://cdn.artofproblemsolving.com/attachments/9/2/763de9f4138b4dc552247e9316175036c649b6.png[/img] [u]Round 2[/u] [b]p6.[/b] What is the largest number of zeros that could appear at the end of $1^n + 2^n + 3^n + 4^n$, where n can be any positive integer? [b]p7.[/b] A tennis academy has $2023$ members. For every group of 1011 people, there is a person outside of the group who played a match against everyone in it. Show there is someone who has played against all $2022$ other members. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

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]

A set $S$ of rational numbers is ordered in a tree-diagram in such a way that each rational number $\frac{a}{b}$ (where $a$ and $b$ are coprime integers) has exactly two successors: $\frac{a}{a+b}$ and $\frac{b}{a+b}$. How should the initial element be selected such that this tree contains the set of all rationals $r$ with $0 < r < 1$? Give a procedure for determining the level of a rational number $\frac{p}{q}$ in this tree.

Let $a_1, a_2, a_3, \ldots$ be a sequence of positive real numbers, and $s$ be a positive integer, such that \[a_n = \max \{ a_k + a_{n-k} \mid 1 \leq k \leq n-1 \} \ \textrm{ for all } \ n > s.\] Prove there exist positive integers $\ell \leq s$ and $N$, such that \[a_n = a_{\ell} + a_{n - \ell} \ \textrm{ for all } \ n \geq N.\] [i]Proposed by Morteza Saghafiyan, Iran[/i]

The real numbers $a_1,a_2,a_3$ and $b{}$ are given. The equation \[(x-a_1)(x-a_2)(x-a_3)=b\]has three distinct real roots, $c_1,c_2,c_3.$ Determine the roots of the equation \[(x+c_1)(x+c_2)(x+c_3)=b.\][i]Proposed by A. Antropov and K. Sukhov[/i]

Let $c$ be a positive integer. Consider the sequence $x_1,x_2,\ldots$ which satisfies $x_1=c$ and, for $n\ge 2$, \[x_n=x_{n-1}+\left\lfloor\frac{2x_{n-1}-(n+2)}{n}\right\rfloor+1\] where $\lfloor x\rfloor$ denotes the largest integer not greater than $x$. Determine an expression for $x_n$ in terms of $n$ and $c$. [i]Huang Yumin[/i]

Find all real solutions to $ x^3 \minus{} 3x^2 \minus{} 8x \plus{} 40 \minus{} 8\sqrt[4]{4x \plus{} 4} \equal{} 0$

Let $n$ be a positive integer. Prove that all roots of the equation $$x(x + 2) (x + 4 )... (x + 2n) + (x +1) (x + 3 )... (x + 2n - 1) = 0$$ are real and irrational.

Find the smallest odd integer $ k$ such that: for every $ 3\minus{}$degree polynomials $ f$ with integer coefficients, if there exist $ k$ integer $ n$ such that $ |f(n)|$ is a prime number, then $ f$ is irreducible in $ \mathbb{Z}[n]$.

Find all functions $f:\mathbb N\to\mathbb N_0$ such that for all $m,n\in\mathbb N$, \begin{align*} f(mn)&=f(m)f(n)\\ f(m+n)&=\min(f(m),f(n))\qquad\text{if }f(m)\ne f(n)\end{align*}

In a magic square $n \times n$ composed from the numbers $1,2,\cdots,n^2$, the centers of any two squares are joined by a vector going from the smaller number to the bigger one. Prove that the sum of all these vectors is zero. (A magic square is a square matrix such that the sums of entries in all its rows and columns are equal.)

Find all real polynomials $P(x)$ of degree $5$ such that $(x-1)^3| P(x)+1$ and $(x+1)^3| P(x)-1$.

Let $ f(x)$ be a polynomial with rational coefficients and $ \alpha$ be a real number such that \[ \alpha^3 \minus{} \alpha \equal{} [f(\alpha)]^3 \minus{} f(\alpha) \equal{} 33^{1992}.\] Prove that for each $ n \geq 1,$ \[ \left [ f^{n}(\alpha) \right]^3 \minus{} f^{n}(\alpha) \equal{} 33^{1992},\] where $ f^{n}(x) \equal{} f(f(\cdots f(x))),$ and $ n$ is a positive integer.

Find all functions $g:R\rightarrow R$ for which there exists a strictly increasing function $ f:R\rightarrow R $ such that $f(x+y)=f(x)g(y)+f(y)$ $\forall x,y \in R$.

Let $a$, $b$ be two distinct real numbers and let $c$ be a positive real numbers such that $a^4 - 2019a = b^4 - 2019b = c$. Prove that $- \sqrt{c} < ab < 0$.

The measures, in degrees, of the angles , $\alpha, \beta$ and $\theta$ are greater than $0$ less than $60$. Find the value of $\theta$ knowing, also, that $\alpha + \beta = 2\theta$ and that $$\sin \alpha \sin \beta \sin \theta = \sin(60 - \alpha ) \sin(60 - \beta) \sin(60 - \theta ).$$

For positive real numbers $a, b, c$ with sum $\frac{3}{2}$, find the smallest possible value of the following expression: $$\frac{a^3}{bc} + \frac{b^3}{ca} + \frac{c^3}{ab} + \frac{1}{abc}$$ [i]Proposed by Serhii Torba[/i]