Rational numbers are written in the following sequence: $\frac{1}{1},\frac{2}{1},\frac{1}{2},\frac{3}{1},\frac{2}{2},\frac{1}{3},\frac{4}{1},\frac{3}{2},\frac{2}{3},\frac{1}{4}, . . .$ In which position of this sequence is $\frac{2005}{2004}$ ?

Given a natural number $n,$ find all polynomials $P(x)$ of degree less than $n$ satisfying the following condition \[ \sum^n_{i=0} P(i) \cdot (-1)^i \cdot \binom{n}{i} = 0. \]

Let an integer $k > 1$ be given. For each integer $n > 1$, we put \[f(n) = k \cdot n \cdot \left(1-\frac{1}{p_1}\right) \cdot \left(1-\frac{1}{p_2}\right) \cdots \left(1-\frac{1}{p_r}\right)\] where $p_1, p_2, \ldots, p_r$ are all distinct prime divisors of $n$. Find all values $k$ for which the sequence $\{x_m\}$ defined by $x_0 = a$ and $x_{m+1} = f(x_m), m = 0, 1, 2, 3, \ldots$ is bounded for all integers $a > 1$.

Define the function $f: \mathbb{N} \rightarrow \mathbb{N}$ which satisfies, for any $s, n \in \mathbb{N}$, the following conditions: $f(1) = f(2^s)$ and if $n < 2^s$, then $f(2^s + n) = f(n) + 1.$ Calculate the maximum value of $f(n)$ when $n \leq 2001$ and find the smallest natural number $n$ such that $f(n) = 2001.$

Suppose that $a,b,c,d$ are positive real numbers satisfying $(a+c)(b+d)=ac+bd$. Find the smallest possible value of $$\frac{a}{b}+\frac{b}{c}+\frac{c}{d}+\frac{d}{a}.$$ [i]Israel[/i]

Let $a, b, c, d$ be positive real numbers. Prove the following inequality and determine all cases in which the equality holds : $$\frac{a - b}{b + c}+\frac{b - c}{c + d}+\frac{c - d}{d + a}+\frac{d - a}{a + b} \ge 0.$$

Complex numbers ${x_i},{y_i}$ satisfy $\left| {{x_i}} \right| = \left| {{y_i}} \right| = 1$ for $i=1,2,\ldots ,n$. Let $x=\frac{1}{n}\sum\limits_{i=1}^n{{x_i}}$, $y=\frac{1}{n}\sum\limits_{i=1}^n{{y_i}}$ and $z_i=x{y_i}+y{x_i}-{x_i}{y_i}$. Prove that $\sum\limits_{i=1}^n{\left| {{z_i}}\right|}\leqslant n$.

Let $f, g$ be functions from the positive integers to the integers. Vlad the impala is jumping around the integer grid. His initial position is $x_0 = (0, 0)$, and for every $n \ge 1$, his jump is $x_n - x_{n - 1} = (\pm f(n), \pm g(n))$ or $(\pm g(n), \pm f(n)),$ with eight possibilities in total. Is it always possible that Vlad can choose his jumps to return to his initial location $(0, 0)$ infinitely many times when (a) $f, g$ are polynomials with integer coefficients? (b) $f, g$ are any pair of functions from the positive integers to the integers?

For $a,b,c$ be real numbers greater than $1$, prove that \[\frac{a+b+c}{4} \geq \frac{\sqrt{ab-1}}{b+c}+\frac{\sqrt{bc-1}}{c+a}+\frac{\sqrt{ca-1}}{a+b}.\]

Given a real number $C\leqslant 2$. Prove that for every positive real number $x,y$ with $xy=1$, the following inequality holds: \[ \sqrt{\frac{x^2+y^2}{2}} + \frac{C}{x+y} \geqslant 1 + \frac{C}{2}.\] [i]Proposed by Fajar Yuliawan, Indonesia[/i]

The function f has the following properties : $f(x + y) = f(x) + f(y) + xy$ for all real $x$ and $y$ $f(4) = 10$ Calculate $f(2001)$.

a) Let $a$ and $b$ be two non-negative real numbers. Show that $a+b \ge \sqrt{\frac{a^2+b^2}{2}}+ \sqrt{ab}$ b) Let $a$ and $b$ be two real numbers in $[0, 3]$. Show that $\sqrt{\frac{a^2+b^2}{2}}+ \sqrt{ab} \ge \frac{(a+b)^2}{2}$

Let $x$ and $y$ be positive integers such that $xy$ divides $x^{2}+y^{2}+1$. Show that \[\frac{x^{2}+y^{2}+1}{xy}=3.\]

[b]a)[/b] Prove that, for any integer $ k, $ the equation $ x^3-24x+k=0 $ has at most an integer solution. [b]b)[/b] Show that the equation $ x^3+24x-2016=0 $ has exactly one integer solution.

Find all real numbers $x$ such that $$x^3 = \{(x + 1)^3\}$$ where $\{y\}$ denotes the fractional part of $y$, i.e. the difference between $y$ and the largest integer less than or equal to $y$.

Show that if the last digit of the number $x^2+xy+y^2$ is $0$ (where $x,y\in\mathbb{N}$ ) then last two digits are zero.

For a positive integer $n,$ let $a_n$ be the integer closest to $\sqrt{n}.$ Compute $$ \frac{1}{a_1 } + \frac{1}{a_2 }+ \cdots + \frac{1}{a_{2004}}.$$

Let $ a,b,c\in\mathbb{R} , E=(a-b)^2(b-c)^2(c-a)^2, $ and $ S_k=a^k+b^k+c^k,\forall k\in\{ 1,2,3,4\} . $ Write $ E $ in terms of $ S_k. $

Given the system of equations \[ ax \plus{} (a \minus{} 1)y \equal{} 1 \\ (a \plus{} 1)x \minus{} ay \equal{} 1. \] For which one of the following values of $ a$ is there no solution $ x$ and $ y$? $ \textbf{(A)}\ 1\qquad \textbf{(B)}\ 0\qquad \textbf{(C)}\ \minus{} 1\qquad \textbf{(D)}\ \frac {\pm \sqrt {2}}{2}\qquad \textbf{(E)}\ \pm\sqrt {2}$

Let $ a,b,c,d $ be four real numbers such that $ |ax^3+bx^2+cx+d|\le 1,\forall x\in [0,1] . $ Prove that $ |dx^2+cx^2+bx+a|\le 9/2,\forall x\in [0,1] . $ [i]Lavinia Savu[/i]

Find all functions $f: \mathbb{Q}\to\mathbb{R}$ which fulfill the following conditions: a) $f(1)+1>0$; b) $f(x+y) -xf(y) -yf(x) = f(x)f(y) -x-y +xy$, for all $x,y\in\mathbb{Q}$; c) $f(x) = 2f(x+1) +x+2$, for every $x\in\mathbb{Q}$.

Given is a sequence $a_1, a_2, \ldots$, such that $a_1=1$ and $a_{n+1}=\frac{9a_n+4}{a_n+6}$ for any $n \in \mathbb{N}$. Which terms of this sequence are positive integers?

Show that there are real numbers $A,B$ such that the identity \[\sum_{k=1}^n\tan(k)\tan(k-1)=A\tan(n)+Bn\] holds for every positive integer $n.$

Find all solutions for positive integers $(x,y,k,m)$ such that \[ 20x^k+24y^m = 2024\] with $k, m > 1$

[b]p1.[/b] In triangle $ABC$, the median $BM$ is drawn. The length $|BM| = |AB|/2$. The angle $\angle ABM = 50^o$. Find the angle $\angle ABC$. [b]p2.[/b] Is there a positive integer $n$ which is divisible by each of $1, 2,3,..., 2018$ except for two numbers whose difference is$ 7$? [b]p3.[/b] Twenty numbers are placed around the circle in such a way that any number is the average of its two neighbors. Prove that all of the numbers are equal. [b]p4.[/b] A finite number of frogs occupy distinct integer points on the real line. At each turn, a single frog jumps by $1$ to the right so that all frogs again occupy distinct points. For some initial configuration, the frogs can make $n$ moves in $m$ ways. Prove that if they jump by $1$ to the left (instead of right) then the number of ways to make $n$ moves is also $m$. [b]p5.[/b] A square box of chocolates is divided into $49$ equal square cells, each containing either dark or white chocolate. At each move Alex eats two chocolates of the same kind if they are in adjacent cells (sharing a side or a vertex). What is the maximal number of chocolates Alex can eat regardless of distribution of chocolates in the box? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].