Let the sequence $a_n$, $n\geq2$, $a_n=\frac{\sqrt[3]{n^3+n^2-n-1}}{n} $. Find the greatest natural number $k$ ,such that $a_2 \cdot a_3 \cdot . . .\cdot a_k <8$

$f$ is a function whose domain is the set of nonnegative integers and whose range is contained in the set of nonnegative integers. $f$ satisfies the condition that $f(f(n))+f(n)=2n+3$ for all nonnegative integers $n$. Find $f(2014)$.

A quadruplet of distinct positive integers $(a, b, c, d)$ is called $k$-good if the following conditions hold: 1. Among $a, b, c, d$, no three form an arithmetic progression. 2. Among $a+b, a+c, a+d, b+c, b+d, c+d$, there are $k$ of them, forming an arithmetic progression. $a)$ Find a $4$-good quadruplet. $b)$ What is the maximal $k$, such that there is a $k$-good quadruplet?

Find all functions $ f: \mathbb {R} \rightarrow \mathbb {R} $ such that $ f (x + y) \le f (xy) $ for every pair of real $ x $, $ y$.

Let $f(x)$ be a periodic function with periods $T$ and $1$($0<T<1$).Prove that: (1)If $T$ is rational,then there exists a prime $p$ such that $\frac{1}{p}$ is also a period of $f$; (2)If $T$ is irrational,then there exists a strictly decreasing infinite sequence {$a_n$},with $1>a_n>0$ for all positive integer $n$,such that all $a_n$ are periods of $f$.

Given $n$ positive real numbers $a_1, a_2, \ldots , a_n$ such that $a_1a_2 \cdots a_n = 1$, prove that \[(1 + a_1)(1 + a_2) \cdots (1 + a_n) \geq 2^n.\]

Suppose $1, 2, 3$ are the roots of the equation $x^4 + ax^2 + bx = c$. Find the value of $c$.

Let $(A,+,\cdot)$ be a finite unit ring, with $n\geq 3$ elements in which there exist [b]exactly[/b] $\dfrac {n+1}2$ perfect squares (e.g. a number $b\in A$ is called a perfect square if and only if there exists an $a\in A$ such that $b=a^2$). Prove that a) $1+1$ is invertible; b) $(A,+,\cdot)$ is a field. [i]Proposed by Marian Andronache[/i]

Prove such $a$, that all the numbers $\cos a, \cos 2a, \cos 4a, ... , \cos (2^na)$ are negative.

Find the sum of the roots of $\tan^2x-9\tan x+1=0$ that are between $x=0$ and $x=2\pi$ radians. $ \textbf{(A)}\ \frac{\pi}{2} \qquad\textbf{(B)}\ \pi \qquad\textbf{(C)}\ \frac{3\pi}{2} \qquad\textbf{(D)}\ 3\pi \qquad\textbf{(E)}\ 4\pi $

Determine the least possible value of $f(1998),$ where $f:\Bbb{N}\to \Bbb{N}$ is a function such that for all $m,n\in {\Bbb N}$, \[f\left( n^{2}f(m)\right) =m\left( f(n)\right) ^{2}. \]

Consider the equation $x^2+(a+b+c)x+\lambda (ab+bc+ca)=0$ with $a,b,c>0$ and $\lambda\in \mathbb{R}$. Prove that: a)If $\lambda\le \frac{3}{4}$, the equation has real roots. b)If $a,b,c$ are the side lengths of a triangle and $\lambda\ge 1$, then the equation doesn't have real roots. [i]***[/i]

Prove that for each $x,y,z \in R$ it holds that $$\frac{x^2-y^2}{2x^2+1} +\frac{y^2-z^2}{2y^2+1}+\frac{z^2-x^2}{2z^2+1}\le 0$$

Dunno wrote down $11$ natural numbers in a circle. For every two adjacent numbers, he calculated their difference. As a result among the differences found there were four units, four twos and three threes. Prove that Dunno made a mistake somewhere an error.

We define the sequences $a_n =\frac{n (n + 1)}{2}$ and $b_n = a_1 + a_2 +… + a_n$. Prove that there is no integer $n$ such that $b_n = 2017$.

Let $n\geq 2$ and $x_{1},x_{2},\ldots,x_{n}$ positive real numbers. Prove that \[\frac{1+x_{1}^2}{1+x_{1}x_{2}}+\frac{1+x_{2}^2}{1+x_{2}x_{3}}+\cdots+\frac{1+x_{n}^2}{1+x_{n}x_{1}}\geq n.\]

The graphs of four functions of the form $y = x^2 + ax + b$, where a and b are real coefficients, are plotted on the coordinate plane. These graphs have exactly four points of intersection, and at each one of them, exactly two graphs intersect. Prove that the sum of the largest and the smallest $x$-coordinates of the points of intersection is equal to the sum of the other two. [i](3 points)[/i]

For each value of parameter $a$, solve the the equation $$ x - \sqrt{x^2-a^2} = \frac{(x-a)^2}{2(x+a)}$$

Let $p_1,p_2,\ldots,p_n$ be distinct primes greater than $3$. Show that $2^{p_1p_2\cdots p_n}+1$ has at least $4^n$ divisors.

Let $a,b$ and $c$ be positive integers that $$\frac{a\sqrt{3}+b}{b\sqrt3+c}$$ is a rational number, show that $$\frac{a^2+b^2+c^2}{a+b+ c}$$ is an integer. [i](Anonymous314)[/i]

Let $z$ be a complex number such that $z,z^2,z^3$ are all collinear in the complex plane . Show that $z$ is a real number .

[b]p1.[/b] If $a \diamond b = ab - a + b$, find $(3 \diamond 4) \diamond 5$ [b]p2.[/b] If $5$ chickens lay $5$ eggs in $5$ days, how many chickens are needed to lay $10$ eggs in $10$ days? [b]p3.[/b] As Alissa left her house to go to work one hour away, she noticed that her odometer read $16261$ miles. This number is a "special" number for Alissa because it is a palindrome and it contains exactly $1$ prime digit. When she got home that evening, it had changed to the next greatest "special" number. What was Alissa's average speed, in miles per hour, during her two hour trip? [b]p4.[/b] How many $1$ in by $3$ in by $8$ in blocks can be placed in a $4$ in by $4$ in by $9$ in box? [b]p5.[/b] Apple loves eating bananas, but she prefers unripe ones. There are $12$ bananas in each bunch sold. Given any bunch, if there is a $\frac13$ probability that there are $4$ ripe bananas, a $\frac16$ probability that there are $6$ ripe bananas, and a $\frac12$ probability that there are $10$ ripe bananas, what is the expected number of unripe bananas in $12$ bunches of bananas? [b]p6.[/b] The sum of the digits of a $3$-digit number $n$ is equal to the same number without the hundreds digit. What is the tens digit of $n$? [b]p7.[/b] How many ordered pairs of positive integers $(a, b)$ satisfy $a \le 20$, $b \le 20$, $ab > 15$? [b]p8.[/b] Let $z(n)$ represent the number of trailing zeroes of $n!$. What is $z(z(6!))?$ (Note: $n! = n\cdot (n-1) \cdot\cdot\cdot 2 \cdot 1$) [b]p9.[/b] On the Cartesian plane, points $A = (-1, 3)$, $B = (1, 8)$, and $C = (0, 10)$ are marked. $\vartriangle ABC$ is reflected over the line $y = 2x + 3$ to obtain $\vartriangle A'B'C'$. The sum of the $x$-coordinates of the vertices of $\vartriangle A'B'C'$ can be expressed as $\frac{a}{b}$ for relatively prime positive integers $a$, $b$. Compute $a + b$. [b]p10.[/b] How many ways can Bill pick three distinct points from the figure so that the points form a non-degenerate triangle? [img]https://cdn.artofproblemsolving.com/attachments/6/a/8b06f70d474a071b75556823f70a2535317944.png[/img] [b]p11.[/b] Say piece $A$ is attacking piece $B$ if the piece $B$ is on a square that piece $A$ can move to. How many ways are there to place a king and a rook on an $8\times 8$ chessboard such that the rook isn't attacking the king, and the king isn't attacking the rook? Consider rotations of the board to be indistinguishable. (Note: rooks move horizontally or vertically by any number of squares, while kings move $1$ square adjacent horizontally, vertically, or diagonally). [b]p12.[/b] Let the remainder when $P(x) = x^{2020} - x^{2017} - 1$ is divided by $S(x) = x^3 - 7$ be the polynomial $R(x) = ax^2 + bx + c$ for integers $a$, $b$, $c$. Find the remainder when $R(1)$ is divided by $1000$. [b]p13.[/b] Let $S(x) = \left \lfloor \frac{2020}{x} \right\rfloor + \left \lfloor \frac{2020}{x + 1} \right\rfloor$. Find the number of distinct values $S(x)$ achieves for integers $x$ in the interval $[1, 2020]$. [b]p14.[/b] Triangle $\vartriangle ABC$ is inscribed in a circle with center $O$ and has sides $AB = 24$, $BC = 25$, $CA = 26$. Let $M$ be the midpoint of $\overline{AB}$. Points $K$ and $L$ are chosen on sides $\overline{BC}$ and $\overline{CA}$, respectively such that $BK < KC$ and $CL < LA$. Given that $OM = OL = OK$, the area of triangle $\vartriangle MLK$ can be expressed as $\frac{a\sqrt{b}}{c}$ where $a, b, c$ are positive integers, $gcd(a, c) = 1$ and $b$ is not divisible by the square of any prime. Find $a + b + c$. [b]p15.[/b] Euler's totient function, $\phi (n)$, is defined as the number of positive integers less than $n$ that are relatively prime to $n$. Let $S(n)$ be the set of composite divisors of $n$. Evaluate $$\sum^{50}_{k=1}\left( k - \sum_{d\in S(k)} \phi (d) \right)$$ PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $m$ be a positive integer. Find polynomials $P(x)$ with real coefficients such that $$(x-m)P(x+2023) = xP(x)$$ is satisfied for all real numbers $x.$

Let $x_i \in \{0, 1\} (i = 1, 2, \cdots, n)$. If the function $f = f(x_1, x_2, \cdots, x_n)$ only equals $0$ or $1$, then define $f$ as an "$n$-variable Boolean function" and denote $$D_n (f) = \{ (x_1, x_2, \cdots, x_n) | f(x_1, x_2, \cdots, x_n) = 0 \}$$. $(1)$ Determine the number of $n$-variable Boolean functions; $(2)$ Let $g$ be a $10$-variable Boolean function satisfying $$g(x_1, x_2, \cdots, x_{10}) \equiv 1 + x_1 + x_1 x_2 + x_1 x_2 x_3 + \cdots + x_1 x_2\cdots x_{10} \pmod{2}$$ Evaluate the size of the set $D_{10} (g)$ and $\sum\limits_{(x_1, x_2, \cdots, x_{10}) \in D_{10} (g)} (x_1 + x_2 + x_3 + \cdots + x_{10})$.

What is more $\frac{2}{201}$ or $\ln\frac{101}{100}$? (No differential calculus allowed).