Fix a prime number $ p > 5$. Let $ a,b,c$ be integers no two of which have their difference divisible by $ p$. Let $ i,j,k$ be nonnegative integers such that $ i \plus{} j \plus{} k$ is divisible by $ p \minus{} 1$. Suppose that for all integers $ x$, the quantity \[ (x \minus{} a)(x \minus{} b)(x \minus{} c)[(x \minus{} a)^i(x \minus{} b)^j(x \minus{} c)^k \minus{} 1]\] is divisible by $ p$. Prove that each of $ i,j,k$ must be divisible by $ p \minus{} 1$. [i]Kiran Kedlaya and Peter Shor.[/i]

Let $$f(X)=a_{n}X^{n}+a_{n-1}X^{n-1}+\cdots +a_{1}X+a_{0}$$be a polynomial with real coefficients which satisfies $$a_{n}\geq a_{n-1}\geq \cdots \geq a_{1}\geq a_{0}>0.$$Prove that for every complex root $z$ of this polynomial, we have $|z|\leq 1$.

We say that a set $S$ of positive integers is special when there exists a function $f : \mathbb{N}\to \mathbb{N}$ satisfying that: $\bullet$ $f(k)\in S, \forall k\in\mathbb{N}$ $\bullet$ No integer $k$ with $2\le k \le 2021$ can be written as $\frac{af(a)}{bf(b)}$ with $a,b\in \mathbb{N}$ Find the smallest positive integer $n$ such that the set $S = \{ 1, 2021, 2021^2, \ldots , 2021^n \}$ is special or prove that no such integer exists. Note: $\mathbb{N}$ represents the set $\{ 1, 2, 3, \ldots \}$

Define the sequence $a_1 = 2$ and $a_n = 2^{a_{n-1}} + 2$ for all integers $n \ge 2$. Prove that $a_{n-1}$ divides $a_n$ for all integers $n \ge 2$. [i]Proposed by Sam Korsky[/i]

Let $a, b, c$ be positive real numbers such that $a + b + c = 1$. Prove that $$2 \left( \frac{ab}{a + b} +\frac{bc}{b + c} +\frac{ca}{c+ a}\right)+ 1 \ge 6(ab + bc + ca)$$ Trần Nam Dũng

Define the sequence of real numbers $\{x_n\}_{n \geq 1}$, where $x_1$ is any real number and \[x_n = 1 - x_1x_2\ldots x_{n-1} \text{ for all } n > 1.\] Show that $x_{2011} > \frac{2011}{2012}$.

For every sequence $ (x_1, x_2, \ldots, x_n)$ of non-zero natural prime numbers, $ \{1, 2, \ldots, n\}$ arranged in any order, denote by $ f(s)$ the sum of absolute values of the differences between two consecutive members of $ s.$ Find the maximum value of $ f(s)$ where $ s$ runs through the set of all such sequences, i.e. for all sequences $ s$ with the given properties.

[b]p1.[/b] Twenty five horses participate in a competition. The competition consists of seven runs, five horse compete in each run. Each horse shows the same result in any run it takes part. No two horses will give the same result. After each run you can decide what horses participate in the next run. Could you determine the three fastest horses? (You don’t have stopwatch. You can only remember the order of the horses.) [b]p2.[/b] Prove that the equation $x^6-143x^5-917x^4+51x^3+77x^2+291x+1575=0$ does not have solutions in integer numbers. [b]p3.[/b] Show how we can cut the figure shown in the picture into two parts for us to be able to assemble a square out of these two parts. Show how we can assemble a square. [img]https://cdn.artofproblemsolving.com/attachments/7/b/b0b1bb2a5a99195688638425cf10fe4f7b065b.png[/img] [b]p4.[/b] The city of Vyatka in Russia produces local drink, called “Vyatka Cola”. «Vyatka Cola» is sold in $1$, $3/4$, and $1/2$-gallon bottles. Ivan and John bought $4$ gallons of “Vyatka Cola”. Can we say for sure, that they can split the Cola evenly between them without opening the bottles? [b]p5.[/b] Positive numbers a, b and c satisfy the condition $a + bc = (a + b)(a + c)$. Prove that $b + ac = (b + a)(b + c)$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Does there exist a function $f:\mathbb{N}\rightarrow\mathbb{N}$ such that \[f(f(n-1)=f(n+1)-f(n)\quad\text{for all}\ n\ge 2\text{?} \]

If $a,b,x,y$ are real numbers then find all solutions of $a,b$ such that $\left \lfloor ax+by \right \rfloor + \left \lfloor bx+ay \right \rfloor = (a+b) \left \lfloor x+y \right \rfloor$

If $$s_n = 1 + q + q^2 +... + q^n$$ and $$ S_n = 1 +\frac{1 + q}{2}+ \left( \frac{1 + q}{2}\right)^2 +... + \left( \frac{1 + q}{2}\right)^n,$$ prove that $${n + 1 \choose 1}+{n + 1 \choose 2} s_1 + {n + 1 \choose 3} s_2 + ... + {n + 1 \choose n + 1} s_n = 2^nS_n$$

The sequence $(x_n)_{n\in N}$ satisfies $x_1 = 2015$ and $x_{n+1} =\sqrt[3]{13x_n - 18}$ for all $n \ge 1$. Determine $\lim_{n\to \infty} x_n$.

Let $a \neq b a,b \in \mathbb{R}$ such that $(x^2+20ax+10b)(x^2+20bx+10a)=0$ has no roots for $x$. Prove that $20(b-a)$ is not an integer.

Let $G$ be a non-empty set of non-constant functions $f$ such that $f(x)=ax + b$ (where $a$ and $b$ are two reals) and satisfying the following conditions: 1) if $f \in G$ and $g \in G$ then $gof \in G$, 2) if $f \in G$ then $f^ {-1} \in G$, 3) for all $f \in G$ there exists $x_f \in \mathbb{R}$ such that $f(x_f)=x_f$. Prove that there is a real $k$ such that for all $f \in G$ we have $f(k)=k$

Determine all monic polynomials $P(x)$ of degree $3$ with coefficients integers, which are divisible by $x-1$, when divided by $ x-5$ leave the same remainder as when divided by$ x+5$ and have a root between $2$ and $3$.

Prove that, for any natural number $n$, the expression $$A = 2903^n-803^n-464^n+261^n$$ is divisible by $1897$.

A function $f: \mathbb{R}\to \mathbb{R}$ is [i]essentially increasing[/i] if $f(s)\leq f(t)$ holds whenever $s\leq t$ are real numbers such that $f(s)\neq 0$ and $f(t)\neq 0$. Find the smallest integer $k$ such that for any 2022 real numbers $x_1,x_2,\ldots , x_{2022},$ there exist $k$ essentially increasing functions $f_1,\ldots, f_k$ such that \[f_1(n) + f_2(n) + \cdots + f_k(n) = x_n\qquad \text{for every } n= 1,2,\ldots 2022.\]

Let real numbers $x_1, x_2, \cdots , x_n$ satisfy $0 < x_1 < x_2 < \cdots< x_n < 1$ and set $x_0 = 0, x_{n+1} = 1$. Suppose that these numbers satisfy the following system of equations: \[\sum_{j=0, j \neq i}^{n+1} \frac{1}{x_i-x_j}=0 \quad \text{where } i = 1, 2, . . ., n.\] Prove that $x_{n+1-i} = 1- x_i$ for $i = 1, 2, . . . , n.$

Let $x, y, z$ be nonnegative real numbers such that $x + y + z = 3$. Prove that $$\frac{x}{4-y}+\frac{y}{4-z}+\frac{z}{4-x}+\frac{1}{16}(1-x)^2(1-y)^2(1-z)^2\leq 1,$$ and determine all such triples $(x, y, z)$ where the equality holds.

Let $p$ be an[color=#FF0000] odd [/color]prime number. Determine all pairs of polynomials $f$ and $g$ from $\mathbb{Z}[X]$ such that \[f(g(X))=\sum_{k=0}^{p-1} X^k = \Phi_p(X).\]

[b]p10.[/b] Square $ABCD$ has side length $n > 1$. Points $E$ and $F$ lie on $\overline{AB}$ and $\overline{BC}$ such that $AE = BF = 1$. Suppose $\overline{DE}$ and $\overline{AF}$ intersect at $X$ and $\frac{AX}{XF} = \frac{11}{111}$ . What is $n$? [b]p11.[/b] Let $x$ be the positive root of $x^2 - 10x - 10 = 0$. Compute $\frac{1}{20}x^4 - 6x^2 - 45$. [b]p12.[/b] Francesca has $7$ identical marbles and $5$ distinctly labeled pots. How many ways are there for her to distribute at least one (but not necessarily all) of the marbles into the pots such that at most two pots are nonempty? PS. You should use hide for answers.

Let $a_1$, $a_2$, ..., $a_{2004}$ be non-negative real numbers such that $a_1+...+ a_{2004} \le 25$. Prove that among them there exist at least two numbers $a_i$ and $a_j$ ($i \neq j$) such that $|\sqrt{a_i}-\sqrt{a_j}| \le \frac{5}{2003}$.

Does there exist a surjective function $f:R \to R$ such that the expression $f(x + y) - f(x) - f(y)$ takes exactly two values $0$ and $1$ for various real $x$ and $y$ ? (E. Barabanov)

For what a from the interval $[0,\pi]$ do there exist $a$ and $b$ that are not simultaneously equal to zero, for which the inequality $$a \cos x + b \cos 2x \le 0$$ is satisfied for all $x$ belonging to the segment $[a, \pi]$?

Find all positive integers $n$ with the following property: there exists a polynomial $P_n(x)$ of degree $n$, with integer coefficients, such that $P_n(0)=0$ and $P_n(x)=n$ for $n$ distinct integer solutions.