There is a broken computer such that only three primitive data $c$, $1$ and $-1$ are reserved. Only allowed operation may take $u$ and $v$ and output $u \cdot v + v.$ At the beginning, $u,v \in \{c, 1, -1\}.$ After then, it can also take the value of the previous step (only one step back) besides $\{c, 1, -1\}$. Prove that for any polynomial $P_{n}(x) = a_0 \cdot x^n + a_1 \cdot x^{n-1} + \ldots + a_n$ with integer coefficients, the value of $P_n(c)$ can be computed using this computer after only finite operation.

Find all functions $f:\mathbb{R} \rightarrow \mathbb{R}$ with the property that $$f(x)f(y) = (xy - 1)^2f\left(\frac{x + y - 1}{xy - 1}\right)$$ for all real numbers $x, y$ with $xy \neq 1$.

Let $a_0, a_1, . . . , a_N$ be real numbers satisfying $a_0 = a_N = 0$ and \[a_{i+1} - 2a_i + a_{i-1} = a^2_i\] for $i = 1, 2, . . . , N - 1.$ Prove that $a_i\leq 0$ for $i = 1, 2, . . . , N- 1.$

Let $\mathbb{Z} _{>0}$ be the set of positive integers. Find all functions $f: \mathbb{Z} _{>0}\rightarrow \mathbb{Z} _{>0}$ such that \[ m^2 + f(n) \mid mf(m) +n \] for all positive integers $m$ and $n$.

Let $a$ and $b$ be positive integers such that $ab+1$ divides $a^{2}+b^{2}$. Show that \[\frac{a^{2}+b^{2}}{ab+1}\] is the square of an integer.

Let $a_1, a_2, \ldots, a_n$ be $n$ positive integers, and let $b_1, b_2, \ldots, b_m$ be $m$ positive integers such that $a_1 a_2 \cdots a_n = b_1 b_2 \cdots b_m$. Prove that a rectangular table with $n$ rows and $m$ columns can be filled with positive integer entries in such a way that * the product of the entries in the $i$-th row is $a_i$ (for each $i \in \left\{1,2,\ldots,n\right\}$); * the product of the entries in the $j$-th row is $b_j$ (for each $i \in \left\{1,2,\ldots,m\right\}$).

Let $n,s,t$ be three positive integers, and let $A_1,\ldots, A_s, B_1,\ldots, B_t$ be non-necessarily distinct subsets of $\{1,2,\ldots,n\}$. For any subset $S$ of $\{1,\ldots,n\}$, define $f(S)$ to be the number of $i\in\{1,\ldots,s\}$ with $S\subseteq A_i$ and $g(S)$ to be the number of $j\in\{1,\ldots,t\}$ with $S\subseteq B_j$. Assume that for any $1\leq x<y\leq n$, we have $f(\{x,y\})=g(\{x,y\})$. Show that if $t<n$, then there exists some $1\leq x\leq n$ so that $f(\{x\})\geq g(\{x\})$. [i]Proposed by usjl[/i]

[b]a)[/b] Prove that $ 0=\inf\{ |x\sqrt 2+y\sqrt 3+y\sqrt 5|\big| x,y,z\in\mathbb{Z} ,x^2+y^2+z^2>0 \} $ [b]b)[/b] Prove that there exist three positive rational numbers $ a,b,c $ such that the expression $ E(x,y,z):=xa+yb+zc $ vanishes for infinitely many integer triples $ (x,y,z), $ but it doesn´t get arbitrarily close to $ 0. $

Let $n$ be a positive integer. Find all functions $f\colon \mathbb{R}\rightarrow \mathbb{R}$ that satisfy the equation $$ (f(x))^n f(x+y) = (f(x))^{n+1} + x^n f(y) $$ for all $x ,y \in \mathbb{R}$.

Consider the polynomials \[f(x) =\sum_{k=1}^{n}a_{k}x^{k}\quad\text{and}\quad g(x) =\sum_{k=1}^{n}\frac{a_{k}}{2^{k}-1}x^{k},\] where $a_{1},a_{2},\ldots,a_{n}$ are real numbers and $n$ is a positive integer. Show that if 1 and $2^{n+1}$ are zeros of $g$ then $f$ has a positive zero less than $2^{n}$.

Compute the real solution for$ x$ to the equation $$(4^x + 8)^4 - (8^x - 4)^4 = (4 + 8^x + 4^x)^4.$$

Let $ a,b,c,d$ be rational numbers with $ a>0$. If for every integer $ n\ge 0$, the number $ an^{3} \plus{}bn^{2} \plus{}cn\plus{}d$ is also integer, then the minimal value of $ a$ will be $\textbf{(A)}\ 1 \qquad\textbf{(B)}\ \frac{1}{2} \qquad\textbf{(C)}\ \frac{1}{6} \qquad\textbf{(D)}\ \text{Cannot be found} \qquad\textbf{(E)}\ \text{None}$

Determine all functions $f : \mathbb{R}^2 \to\mathbb {R}$ for which \[f(A)+f(B)+f(C)+f(D)=0,\]whenever $A,B,C,D$ are the vertices of a square with side-length one. [i]Ilir Snopce[/i]

Let $n\geq 3$ an integer. Determine whether there exist permutations $(a_1,a_2, \ldots, a_n)$ of the numbers $(1,2,\ldots, n)$ and $(b_1, b_2, \ldots, b_n)$ of the numbers $(n+1,n+2,\ldots, 2n)$ so that $(a_1b_1, a_2b_2, \ldots a_nb_n)$ is a strictly increasing arithmetic progression.

Find the smallest real number $p(\leq 1)$ that satisfies the following condition. (Condition) For real numbers $x_1, x_2, \dots, x_{2024}, y_1, y_2, \dots, y_{2024}$, if [list] [*] $0 \leq x_1 \leq x_2 \leq \dots \leq x_{2024} \leq 1$, [*] $0 \leq y_1 \leq y_2 \leq \dots \leq y_{2024} \leq 1$, [*] $\displaystyle \sum_{i=1}^{2024}x_i = \displaystyle \sum_{i=1}^{2024}y_i = 2024p$, [/list] then the inequality $\displaystyle \sum_{i=1}^{2024}x_i(y_{2025-i}-y_{2024-i}) \geq 1 - p$ holds.

[u]Round 5 [/u] [b]p13.[/b] Suppose $\vartriangle ABC$ is an isosceles triangle with $\overline{AB} = \overline{BC}$, and $X$ is a point in the interior of $\vartriangle ABC$. If $m \angle ABC = 94^o$, $m\angle ABX = 17^o$, and $m\angle BAX = 13^o$, then what is $m\angle BXC$ (in degrees)? [b]p14.[/b] Find the remainder when $\sum^{2019}_{n=1} 1 + 2n + 4n^2 + 8n^3$ is divided by $2019$. [b]p15.[/b] How many ways can you assign the integers $1$ through $10$ to the variables $a, b, c, d, e, f, g, h, i$, and $j$ in some order such that $a < b < c < d < e, f < g < h < i$, $a < g, b < h, c < i$, $f < b, g < c$, and $h < d$? [u]Round 6 [/u] [b]p16.[/b] Call an integer $n$ equi-powerful if $n$ and $n^2$ leave the same remainder when divided by 1320. How many integers between $1$ and $1320$ (inclusive) are equi-powerful? [b]p17.[/b] There exists a unique positive integer $j \le 10$ and unique positive integers $n_j$ , $n_{j+1}$, $...$, $n_{10}$ such that $$j \le n_j < n_{j+1} < ... < n_{10}$$ and $${n_{10} \choose 10}+ {n_9 \choose 9}+ ... + {n_j \choose j}= 2019.$$ Find $n_j + n_{j+1} + ... + n_{10}$. [b]p18.[/b] If $n$ is a randomly chosen integer between $1$ and $390$ (inclusive), what is the probability that $26n$ has more positive factors than $6n$? [u]Round 7[/u] [b]p19.[/b] Suppose $S$ is an $n$-element subset of $\{1, 2, 3, ..., 2019\}$. What is the largest possible value of $n$ such that for every $2 \le k \le n$, $k$ divides exactly $n - 1$ of the elements of $S$? [b]p20.[/b] For each positive integer $n$, let $f(n)$ be the fewest number of terms needed to write $n$ as a sum of factorials. For example, $f(28) = 3$ because $4! + 2! + 2! = 28$ and 28 cannot be written as the sum of fewer than $3$ factorials. Evaluate $f(1) + f(2) + ... + f(720)$. [b]p21.[/b] Evaluate $\sum_{n=1}^{\infty}\frac{\phi (n)}{101^n-1}$ , where $\phi (n)$ is the number of positive integers less than or equal to n that are relatively prime to $n$. PS. You should use hide for answers. Rounds 1-4 have been posted [url=https://artofproblemsolving.com/community/c4h2788993p24519281]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Factor the polynomial $x^3+x^2z+xyz+y^2z-y^3$.

a) Let $x$ and $y$ be positive numbers such that $x + y = 2$. Show that $\frac{1}{x}+\frac{1}{y} \le \frac{1}{x^2}+\frac{1}{y^2}$ b) Let $x,y$ and $z$ be positive numbers such that $x + y +z= 2$. Show that $\frac{1}{x}+\frac{1}{y} +\frac{1}{z} +\frac{9}{4} \le \frac{1}{x^2}+\frac{1}{y^2}+\frac{1}{z^2}$ .

Emma is seated on a train traveling at a speed of $120$ miles per hour. She notices distance markers are placed evenly alongside the track, with a constant distance $x$ between any two consecutive ones, and during a span of 6 minutes, she sees precisely $11$ markers pass by her. Determine the difference (in miles) between the largest and smallest possible values of $x$.

Given distinct prime numbers $p$ and $q$ and a natural number $n \geq 3$, find all $a \in \mathbb{Z}$ such that the polynomial $f(x) = x^n + ax^{n-1} + pq$ can be factored into 2 integral polynomials of degree at least 1.

Determine all functions $f: \mathbb{R} \to \mathbb{R}$ satisfying \[f(x + yf(x)) + f(xy) = f(x) + f(2019y),\] for all real numbers $x$ and $y$.

a) Let $\{a_n\}_{n=1}^\infty$ is sequence of integers bigger than 1. Proove that if $x>0$ is irrational, then $\ds x_n>\frac{1}{a_{n+1}}$ for infinitely many $n$, where $x_n$ is fractional part of $a_na_{n-1}\dots a_1x$. b)Find all sequences $\{a_n\}_{n=1}^\infty$ of positive integers, for which exist infinitely many $x\in(0,1)$ such that $\ds x_n>\frac{1}{a_{n+1}}$ for all $n$. [i]Nikolai Nikolov, Emil Kolev[/i]

Prove that there exists a function $ f $ defined and differentiable in the set of all real numbers, satisfying the conditions $|f'(x) - f'(y)| \leq 4|x-y|$.

Let $ z \in \mathbb{C}$ such that for all $ k \in \overline{1, 3}$, $ |z^k \plus{} 1| \le 1$. Prove that $ z \equal{} 0$.

A linear form in $k$ variables is an expression of the form $P(x_1,...,x_k)=a_1x_1+...+a_kx_k$ with real constants $a_1,...,a_k$. Prove that there exist a positive integer $n$ and linear forms $P_1,...,P_n$ in $2017$ variables such that the equation $$x_1\cdot x_2\cdot ... \cdot x_{2017}=P_1(x_1,...,x_{2017})^{2017}+...+P_n(x_1,...,x_{2017})^{2017}$$ holds for all real numbers $x_1,...,x_{2017}$.