Ali and Naqi are playing a game. At first, they have Polynomial $P(x) = 1+x^{1398}$. Naqi starts. In each turn one can choice natural number $k \in [0,1398]$ in his trun, and add $x^k$ to the polynomial. For example after 2 moves $P$ can be : $P(x) = x^{1398} + x^{300} + x^{100} +1$. If after Ali's turn, there exist $t \in R$ such that $P(t)<0$ then Ali loses the game. Prove that Ali can play forever somehow he never loses the game!

Prove that, for all sufficiently large integers $n$, there exists $n$ numbers $a_1, a_2, \dots, a_n$ satisfying the following three conditions: [list] [*] Each number $a_i$ is equal to either $-1, 0$ or $1$. [*] At least $\frac{2n}{5}$ of the numbers $a_1, a_2, \dots, a_n$ are non-zero. [*] The sum $\frac{a_1}{1} + \frac{a_2}{2} + \dots + \frac{a_n}{n}$ is $0$. [/list] $\textit{Note: Results with 2/5 replaced by a constant } c \textit{ will be awarded points depending on the value of } c$ [i]Proposed by Navneel Singhal, India; Kyle Hess, USA; and Vincent Jugé, France[/i]

Consider a matrix of size $n\times n$ whose entries are real numbers of absolute value not exceeding $1$. The sum of all entries of the matrix is $0$. Let $n$ be an even positive integer. Determine the least number $C$ such that every such matrix necessarily has a row or a column with the sum of its entries not exceeding $C$ in absolute value. [i]Proposed by Marcin Kuczma, Poland[/i]

Solve the equation $x^2+ 2 = 4\sqrt{x^3+1}$

Let $N$ be a positive integer and $A = a_1, a_2, ... , a_N$ be a sequence of real numbers. Define the sequence $f(A)$ to be $$f(A) = \left( \frac{a_1 + a_2}{2},\frac{a_2 + a_3}{2}, ...,\frac{a_{N-1} + a_N}{2},\frac{a_N + a_1}{2}\right)$$ and for $k$ a positive integer define $f^k (A)$ to be$ f$ applied to $A$ consecutively $k$ times (i.e. $f(f(... f(A)))$) Find all sequences $A = (a_1, a_2,..., a_N)$ of integers such that $f^k (A)$ contains only integers for all $k$.

Let $a_1,a_2,\ldots a_n,k$, and $M$ be positive integers such that $$\frac{1}{a_1}+\frac{1}{a_2}+\cdots+\frac{1}{a_n}=k\quad\text{and}\quad a_1a_2\cdots a_n=M.$$ If $M>1$, prove that the polynomial $$P(x)=M(x+1)^k-(x+a_1)(x+a_2)\cdots (x+a_n)$$ has no positive roots.

a) Given a chain of $60$ links each weighing $1$ g. Find the smallest number of links that need to be broken if we want to be able to get from the obtained parts all weights $1$ g, $2$ g, . . . , $59$ g, $60$ g? A broken link also weighs $1$ g. b) Given a chain of $150$ links each weighing $1$ g. Find the smallest number of links that need to be broken if we want to be able to get from the obtained parts all weights $1$ g, $2$ g, . . . , $149$ g, $150$ g? A broken link also weighs $1$ g.

Let $n, m, k$ be natural numbers, with $m > n$. Which of the numbers is greater: $$\sqrt{n+\sqrt{m+\sqrt{n+...}}}\,\,\, or \,\,\,\, \sqrt{m+\sqrt{n+\sqrt{m+...}}}\,\, ?$$ Note: Each of the expressions contains $k$ square root signs; $n, m$ alternate within each expression. (N. Kurlandchik)

Let $A,B,C\in \mathcal{M}_3(\mathbb{R})$ such that $\det A=\det B=\det C$ and $\det(A+iB)=\det(C+iA)$. Prove that $\det (A+B)=\det (C+A)$.

Prove that there exists a polynomial $P(x)$ with real coefficients and degree greater than 1 such that both of the following conditions are true $\cdot$ $P(a)+P(b)+P(c)\ge 2021$ holds for all nonnegative real numbers $a,b,c$ such that $a+b+c=3$ $\cdot$ There are infinitely many triples $(a,b,c)$ of nonnegative real numbers such that $a+b+c=3$ and $P(a)+P(b)+P(c)= 2021$

Consider a function $f: \mathbb Z \to \mathbb Z$ such that for every integer $n \ge 0$, there are at most $0.001n^2$ pairs of integers $(x,y)$ for which $f(x+y) \neq f(x)+f(y)$ and $\max\{ \lvert x \rvert, \lvert y \rvert \} \le n$. Is it possible that for some integer $n \ge 0$, there are more than $n$ integers $a$ such that $f(a) \neq a \cdot f(1)$ and $\lvert a \rvert \le n$? [i]Proposed by David Yang[/i]

$d$ is a positive integer and $f : [0,d] \rightarrow \mathbb{R}$ is a continuous function with $f(0) = f(d)$. Show that there exists $x \in [0,d-1]$ such that $f(x) = f(x+1)$.

Given $\cos (\alpha + \beta) + sin (\alpha - \beta) = 0$, $\tan \beta =\frac{1}{2000}$, find $\tan \alpha$.

Find all the continuous functions $f : \mathbb{R} \mapsto\mathbb{R}$ such that $\forall x,y \in \mathbb{R}$, $(1+f(x)f(y))f(x+y)=f(x)+f(y)$.

[u]Round 5[/u] [b]p13.[/b] In coordinate space, a lattice point is a point all of whose coordinates are integers. The lattice points $(x, y, z)$ in three-dimensional space satisfying $0 \le x, y, z \le 5$ are colored in n colors such that any two points that are $\sqrt3$ units apart have different colors. Determine the minimum possible value of $n$. [b]p14.[/b] Determine the number of ways to express $121$ as a sum of strictly increasing positive Fibonacci numbers. [b]p15.[/b] Let $ABCD$ be a rectangle with $AB = 7$ and $BC = 15$. Equilateral triangles $ABP$, $BCQ$, $CDR$, and $DAS$ are constructed outside the rectangle. Compute the area of quadrilateral $P QRS$. [u] Round 6[/u] Each of the three problems in this round depends on the answer to one of the other problems. There is only one set of correct answers to these problems; however, each problem will be scored independently, regardless of whether the answers to the other problems are correct. [b]p16.[/b] Let $C$ be the answer to problem $18$. Suppose that $x$ and $y$ are real numbers with $y > 0$ and $$x + y = C$$ $$x +\frac{1}{y} = -2.$$ Compute $y +\frac{1}{y}$. [b]p17.[/b] Let $A$ be the answer to problem $16$. Let $P QR$ be a triangle with $\angle P QR = 90^o$, and let $X$ be the foot of the perpendicular from point $Q$ to segment $P R$. Given that $QX = A$, determine the minimum possible area of triangle $PQR$. [b]p18.[/b] Let $B$ be the answer to problem $17$ and let $K = 36B$. Alice, Betty, and Charlize are identical triplets, only distinguishable by their hats. Every day, two of them decide to exchange hats. Given that they each have their own hat today, compute the probability that Alice will have her own hat in $K$ days. [u]Round 7[/u] [b]p19.[/b] Find the number of positive integers a such that all roots of $x^2 + ax + 100$ are real and the sum of their squares is at most $2013$. [b]p20.[/b] Determine all values of $k$ such that the system of equations $$y = x^2 - kx + 1$$ $$x = y^2 - ky + 1$$ has a real solution. [b]p21.[/b] Determine the minimum number of cuts needed to divide an $11 \times 5 \times 3$ block of chocolate into $1\times 1\times 1$ pieces. (When a block is broken into pieces, it is permitted to rotate some of the pieces, stack some of the pieces, and break any set of pieces along a vertical plane simultaneously.) [u]Round 8[/u] [b]p22.[/b] A sequence that contains the numbers $1, 2, 3, ... , n$ exactly once each is said to be a permutation of length $n$. A permutation $w_1w_2w_3... w_n$ is said to be sad if there are indices $i < j < k$ such that $w_j > w_k$ and $w_j > w_i$. For example, the permutation $3142756$ is sad because $7 > 6$ and $7 > 1$. Compute the number of permutations of length $11$ that are not sad. [b]p23.[/b] Let $ABC$ be a triangle with $AB = 39$, $BC = 56$, and $CA = 35$. Compute $\angle CAB - \angle ABC$ in degrees. [b]p24.[/b] On a strange planet, there are $n$ cities. Between any pair of cities, there can either be a one-way road, two one-way roads in different directions, or no road at all. Every city has a name, and at the source of every one-way road, there is a signpost with the name of the destination city. In addition, the one-way roads only intersect at cities, but there can be bridges to prevent intersections at non-cities. Fresh Mann has been abducted by one of the aliens, but Sophy Moore knows that he is in Rome, a city that has no roads leading out of it. Also, there is a direct one-way road leading from each other city to Rome. However, Rome is the secret police’s name for the so-described city; its official name, the name appearing on the labels of the one-way roads, is unknown to Sophy Moore. Sophy Moore is currently in Athens and she wants to head to Rome in order to rescue Fresh Mann, but she does not know the value of $n$. Assuming that she tries to minimize the number of roads on which she needs to travel, determine the maximum possible number of roads that she could be forced to travel in order to find Rome. Express your answer as a function of $n$. PS. You should use hide for answers. Rounds 1-4 have been posted [url=https://artofproblemsolving.com/community/c4h2809419p24782489]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $n$ be a positive integer, and consider a sequence $a_1 , a_2 , \dotsc , a_n $ of positive integers. Extend it periodically to an infinite sequence $a_1 , a_2 , \dotsc $ by defining $a_{n+i} = a_i $ for all $i \ge 1$. If \[a_1 \le a_2 \le \dots \le a_n \le a_1 +n \] and \[a_{a_i } \le n+i-1 \quad\text{for}\quad i=1,2,\dotsc, n, \] prove that \[a_1 + \dots +a_n \le n^2. \]

Calculate $f(\sqrt[3]{2}-1) $, where $$f(x) = x^{1999} + 3x^{1998} + 4x^{1997} + 2x^{1996} + 4x^{1995} + 2x^{1994} + ...$$ $$... + 4x^3 + 2x^2 + 3x+ 1.$$

Find all functions $ f: \mathbb{R} \to \mathbb{R}$, such that \[ f(xf(y)) \plus{} f(f(x) \plus{} f(y)) \equal{} yf(x) \plus{} f(x \plus{} f(y))\] holds for all $ x$, $ y \in \mathbb{R}$, where $ \mathbb{R}$ denotes the set of real numbers.

Polynomial $P(x)$ has integer coefficients. Prove, that if polynomials $P(x)$ and $P(P(P(x)))$ have common real root, they also have a common integer root.

Let $P(x,y)$ be a polynomial with real coefficients in the variables $x,y$ that is not identically zero. Suppose that $P(\lfloor 2a \rfloor, \lfloor 3a\rfloor) = 0$ for all real numbers $a.$ If $P$ has the minimum possible degree and the coefficient of the monomial $y$ is $4,$ find the coefficient of $x^2y^2$ in $P.$ (The [i]degree[/i] of a monomial $x^my^n$ is $m + n.$ The [i]degree[/i] of a polynomial $P(x,y)$ is then the maximum degree of any of its monomials.)

Let $a, b, c$ be reals satisfying $a^2+b^2+c^2=6$. Find the maximal values of the expressions a) $(a-b)^2+(b-c)^2+(c-a)^2$; b) $(a-b)^2 \cdot (b-c)^2 \cdot (c-a)^2$. In both cases, describe all triples for which equality holds.

Let $\mathbb{Q}$ denote the set of rational numbers. Determine all functions $f:\mathbb{Q}\longrightarrow\mathbb{Q}$ such that, for all $x, y \in \mathbb{Q}$, $$f(x)f(y+1)=f(xf(y))+f(x)$$ [i]Nicolás López Funes and José Luis Narbona Valiente, Spain[/i]

The coefficient of $x^7$ in the polynomial expansion of \[ (1+2x-x^2)^4 \] is $ \textbf{(A)}\ -8 \qquad\textbf{(B)}\ 12 \qquad\textbf{(C)}\ 6 \qquad\textbf{(D)}\ -12 \qquad\textbf{(E)}\ \text{none of these} $

Let $a,b,c$ be positive real numbers such that $a+b+c=3$. Determine, with certainty, the largest possible value of the expression $$ \frac{a}{a^3+b^2+c}+\frac{b}{b^3+c^2+a}+\frac{c}{c^3+a^2+b}$$

Let $a, b, c$ be distinct real numbers such that $a+b+c=0$ and $$ a^{2}-b=b^{2}-c=c^{2}-a. $$ Evaluate all the possible values of $a b+a c+b c$. [i]Proposed by Nguyen Anh Vu, Vietnam[/i]