Consider the sequence of integers $ \left( a_n\right)_{n\ge 0} $ defined as $$ a_n=\left\{\begin{matrix}n^6-2017, & 7|n\\ \frac{1}{7}\left( n^6-2017\right) , & 7\not | n\end{matrix}\right. . $$ Determine the largest length a string of consecutive terms from this sequence sharing a common divisor greater than $ 1 $ may have.

Suppose $N$ is any positive integer. Add the digits of $N$ to obtain a smaller integer. Repeat this process of digit-addition till you get a single digit numbem. Find the number of positive integers $N \le 1000$, such that the final single-digit number $n$ is equal to $5$. Example: $N = 563\to (5 + 6 + 3) = 14 \to(1 + 4) = 5$ will be counted as one such integer.

1. Given $n \geq 1$, let $A_{n}$ denote the set of the first $n$ positive integers. We say that a bijection $f: A_{n} \rightarrow A_{n}$ has a hump at $m \in A_{n} \backslash\{1, n\}$ if $f(m)>f(m+1)$ and $f(m)>f(m-1)$. We say that $f$ has a hump at 1 if $f(1)>f(2)$, and $f$ has a hump at $n$ if $f(n)>f(n-1)$. Let $P_{n}$ be the probability that a bijection $f: A_{n} \rightarrow A_{n}$, when selected uniformly at random, has exactly one hump. For how many positive integers $n \leq 2020$ is $P_{n}$ expressible as a unit fraction?

Find all irrational real numbers $\alpha$ such that \[ \alpha^3 - 15 \alpha \text{ and } \alpha^4 - 56 \alpha \] are both rational numbers.

For how many pairs $(a,b)$ from $(1,2)$, $(3,5)$, $(5,7)$, $(7,11)$, the polynomial $P(x)=x^5+ax^4+bx^3+bx^2+ax+1$ has exactly one real root? $ \textbf{(A)}\ 4 \qquad\textbf{(B)}\ 3 \qquad\textbf{(C)}\ 2 \qquad\textbf{(D)}\ 1 \qquad\textbf{(E)}\ 0 $

A mushroom is called [i]bad [/i] if it contains at least $10$ worms. A basket contains $90$ bad and $10$ good mushrooms. Can all mushrooms become good after some worms crawl from bad mushrooms to good ones? [hide=original wording]Гриб называется плохим, если в нем не менее 10 червей. В лукошке 90 плохих и 10 хороших грибов. Могут ли все грибы стать хорошими после того, как некоторые черви переползут из плохих грибов в хорошие?[/hide]

Let $R_+=(0,+\infty)$. Find all functions $f: R_+ \to R_+$ such that $f(xf(y))+f(yf(z))+f(zf(x))=xy+yz+zx$, for all $x,y,z \in R_+$. by Athanasios Kontogeorgis (aka socrates)

Find all functions $f : R \to R$ such that $$f(xf(y))= (1 - y)f(xy) + x^2y^2f(y)$$ for all reals $x, y$.

Let $a$ and $b$ be real numbers for which the equation $2x^4 + 2ax^3 + bx^2 + 2ax + 2 = 0$ has at least one real solution. For all such pairs $(a, b)$, find the minimum value of $8a^2 + b^2$.

Determine all polynomials $P$ such that for every real number $x$, $P(x)^2 +P(-x) = P(x^2)+P(x)$

a) Find : $A=\{(a,b,c) \in \mathbb{R}^{3} | a+b+c=3 , (6a+b^2+c^2)(6b+c^2+a^2)(6c+a^2+b^2) \neq 0\}$ b) Prove that for any $(a,b,c) \in A$ next inequality hold : \begin{align*} \frac{a}{6a+b^2+c^2}+\frac{b}{6b+c^2+a^2}+\frac{c}{6c+a^2+b^2} \le \frac{3}{8} \end{align*}

Bobo the clown was juggling his spherical cows again when he realized that when he drops a cow is related to how many cows he started off juggling. If he juggles $1$, he drops it after $64$ seconds. When juggling $2$, he drops one after $55$ seconds, and the other $55$ seconds later. In fact, he was able to create the following table: [img]https://cdn.artofproblemsolving.com/attachments/1/0/554a9bace83b4b3595c6012dfdb42409465829.png[/img] He can only juggle up to $22$ cows. To juggle the cows the longest, what number of cows should he start off juggling? How long (in minutes) can he juggle for?

Let be the polynomial $ f=X^4+X^2\in\mathbb{Z}_2[X] $ Find: a) its degree.. b) the splitting field of $ f $ c) the Galois group of $ f $ (Galois group of its splitting field)

$p(x)$ is a polynomial of degree $n$ with leading coefficient $c$, and $q(x)$ is a polynomial of degree $m$ with leading coefficient $c$, such that \[ p(x)^2 = \left(x^2 - 1\right)q(x)^2 + 1 \] Show that $p'(x) = nq(x)$.

Suppose $\{x_n\}$ is a decreasing sequence that $\displaystyle\lim_{n \rightarrow\infty}x_n=0$. Prove that $\sum(-1)^nx_n$ is convergent

How many integers $ x $ satisfy $ x^2 - 9x + 18 < 0 $?

Let $n\ge3$ be a positive integer. Find the real numbers $x_1\ge0,\ldots,x_n\ge 0$, with $x_1+x_2+\ldots +x_n=n$, for which the expression \[(n-1)(x_1^2+x_2^2+\ldots+x_n^2)+nx_1x_2\ldots x_n\] takes a minimal value.

Let $n \geq 2023$ be an integer. For each real $x$, we say that $\lfloor x \rceil$ is the closest integer to $x$, and if there are two closest integers then it is the greater of the two. Suppose there is a positive real $a$ such that $$\lfloor an \rceil = n + \bigg\lfloor\frac{n}{a} \bigg\rceil.$$ Show that $|a^2 - a - 1| < \frac{n\varphi+1}{n^2}$.

Find the remainder when $2^{1990}$ is divided by $1990.$

p1. A quadratic equation has the natural roots $a$ and $ b$. Another quadratic equation has roots $ b$ and $c$ with $a\ne c$. If $a$, $ b$, and $c$ are prime numbers less than $15$, how many triplets $(a,b,c)$ that might meet these conditions are there (provided that the coefficient of the quadratic term is equal to $ 1$)? p2. In Indonesia, was formerly known the "Archipelago Fraction''. The [i]Archipelago Fraction[/i] is a fraction $\frac{a}{b}$ such that $a$ and $ b$ are natural numbers with $a < b$. Find the sum of all Archipelago Fractions starting from a fraction with $b = 2$ to $b = 1000$. p3. Look at the following picture. The letters $a, b, c, d$, and $e$ in the box will replaced with numbers from $1, 2, 3, 4, 5, 6, 7, 8$, or $9$, provided that $a,b, c, d$, and $e$ must be different. If it is known that $ae = bd$, how many arrangements are there? [img]https://cdn.artofproblemsolving.com/attachments/f/2/d676a57553c1097a15a0774c3413b0b7abc45f.png[/img] p4. Given a triangle $ABC$ with $A$ as the vertex and $BC$ as the base. Point $P$ lies on the side $CA$. From point $A$ a line parallel to $PB$ is drawn and intersects extension of the base at point $D$. Point $E$ lies on the base so that $CE : ED = 2 :3$. If $F$ is the midpoint between $E$ and $C$, and the area of ​​triangle ABC is equal with $35$ cm$^2$, what is the area of ​​triangle $PEF$? p5. Each side of a cube is written as a natural number. At the vertex of each angle is given a value that is the product of three numbers on three sides that intersect at the vertex. If the sum of all the numbers at the points of the angle is equal to $1001$, find the sum of all the numbers written on the sides of the cube.

Solve the following system of equations in real numbers: \[ \begin{cases} \sqrt{x^2-2x+6}\cdot \log_{3}(6-y) =x \\ \sqrt{y^2-2y+6}\cdot \log_{3}(6-z)=y \\ \sqrt{z^2-2z+6}\cdot\log_{3}(6-x)=z \end{cases}. \]

Do there exist a infinite sequence of positive integers $(a_{n})$ ,such that for any $n\ge 1$ the relation $ a_{n+2}=\sqrt{a_{n+1}}+a_{n} $?

When $(a-b)^n$, $n\geq 2$, $ab\neq 0$, is expanded by the binomial theorem, it is found that , when $a=kb$, where $k$ is a positive integer, the sum of the second and third terms is zero. Then $n$ equals: $\textbf{(A) }\tfrac12k(k-1)\qquad \textbf{(B) }\tfrac12k(k+1)\qquad \textbf{(C) }2k-1\qquad \textbf{(D) }2k\qquad \textbf{(E) }2k+1$

Conside the continuous $ f: \mathbb{R}\to\mathbb{R}$ . It is also know that equation $f(f(f(x)))=x$ has solution in $\mathbb{R}$. Prove that equation $f(x)=x$ has solution in $\mathbb{R}$.

$\boxed{\text{A5}}$ Let $x,y,z$ be positive reals. Prove that $(x^2+y+1)(x^2+z+1)(y^2+x+1)(y^2+z+1)(z^2+x+1)(z^2+y+1)\geq (x+y+z)^6$