Find all functions $f: \mathbb{N} \rightarrow [0, +\infty)$ such that $f(1000)=10$ and $f(n+1)= \sum_{k=1}^n \frac{1}{f^2(k) + f(k)f(k+1) + f^2(k+1)}$ for all $n \in \mathbb{N}$. (Here, $f^2(i)$ means $(f(i))^2$.)

Given an integer $n\ge 2$, a function $f:\mathbb{Z}\rightarrow \{1,2,\ldots,n\}$ is called [i]good[/i], if for any integer $k,1\le k\le n-1$ there exists an integer $j(k)$ such that for every integer $m$ we have \[f(m+j(k))\equiv f(m+k)-f(m) \pmod{n+1}. \] Find the number of [i]good[/i] functions.

$x^3+(\log_2{5}+\log_3{2}+\log_5{3})x=(\log_2{3}+\log_3{5}+\log_5{2})x^2+1$

There is no sequence $x_n$ strictly increasing with terms natural numbers such that : $$ x_n+x_{k}=x_{nk}, \ \ for \, any \,\,\, n, k \in \mathbb{N}^*$$

For every complex number $z$ different from 0 and 1 we define the following function \[ f(z) := \sum \frac 1 { \log^4 z } \] where the sum is over all branches of the complex logarithm. a) Prove that there are two polynomials $P$ and $Q$ such that $f(z) = \displaystyle \frac {P(z)}{Q(z)} $ for all $z\in\mathbb{C}-\{0,1\}$. b) Prove that for all $z\in \mathbb{C}-\{0,1\}$ we have \[ f(z) = \frac { z^3+4z^2+z}{6(z-1)^4}. \]

Let f be a function such that $f(0) = 0, f(1) = 1$, and $f(n) = 2f(n-1)- f(n- 2) + (-1)^n(2n - 4)$ for all integers $n \ge 2$. Find f(n) in terms of $n$.

Does there exist an integer $n \ge 3$ and an arithmetic sequence $a_0, a_1, ... , a_n$ such that the polynomial $a_nx^n +... + a_1x + a_0$ has $n$ roots which also form an arithmetic sequence?

Consider the minimum positive real number $\lambda$ such that for any two squares $A,B$ satisfying $\text{Area}(A) + \text{Area}(B)=1$, there always exists some rectangle $C$ of area $\lambda$, such that $A,B$ can be put inside $C$ and satisfy the following two constraints: 1. $A,B$ are non-overlapping; 2. the sides of $A$ and $B$ are parallel to some side of $C$. $\lambda$ can be written as $\frac{\sqrt{m}+n}{p}$ for positive integers $m$, $n$, and $p$ where $n$ and $p$ are relatively prime. Find $m+n+p$.

[b]p1.[/b] Given two irreducible fractions. The denominator of the first fraction is $4$, the denominator of the second fraction is $6$. What can the denominator of the product of these fractions be equal to if the product is represented as an irreducible fraction? [b]p2.[/b] Three horses compete in the race. The player can bet a certain amount of money on each horse. Bets on the first horse are accepted in the ratio $1: 4$. This means that if the first horse wins, then the player gets back the money bet on this horse, and four more times the same amount. Bets on the second horse are accepted in the ratio $1:3$, on the third -$ 1:1$. Money bet on a losing horse is not returned. Is it possible to bet in such a way as to win whatever the outcome of the race? [b]p3.[/b] A quadrilateral is inscribed in a circle, such that the center of the circle, point $O$, is lies inside it. Let $K$, $L$, $M$, $N$ be the midpoints of the sides of the quadrilateral, following in this order. Prove that the bisectors of angles $\angle KOM$ and $\angle LOC$ are perpendicular (Fig.). [img]https://cdn.artofproblemsolving.com/attachments/b/8/ea4380698eba7f4cc2639ce20e3057e0294a7c.png[/img] [b]p4.[/b] Prove that the number$$\underbrace{33...33}_{1999 \,\,\,3s}1$$ is not divisible by $7$. [b]p5.[/b] In triangle $ABC$, the median drawn from vertex $A$ to side $BC$ is four times smaller than side $AB$ and forms an angle of $60^o$ with it. Find the greatest angle of this triangle. [b]p6.[/b] Given a $7\times 8$ rectangle made up of 1x1 cells. Cut it into figures consisting of $1\times 1$ cells, so that each figure consists of no more than $5$ cells and the total length of the cuts is minimal (give an example and prove that this cannot be done with a smaller total length of the cuts). You can only cut along the boundaries of the cells. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c2416727_soros_olympiad_in_mathematics]here.[/url]

Which positive integers $n$ make the equation \[\sum_{i=1}^n \sum_{j=1}^n \left\lfloor \frac{ij}{n+1} \right\rfloor=\frac{n^2(n-1)}{4}\] true?

Determine all continuous, strictly monotone functions $\phi : \mathbb{R}^+\to\mathbb{R}$ such that $$F(x,y)=\phi^{-1} \left(\frac{x\phi(x)+y\phi(y)}{x+y}\right) + \phi^{-1} \left(\frac{y\phi(x)+x\phi(y)}{x+y}\right) $$ is homogeneous of degree 1, ie $F(tx,ty)=tF(x,y) , \forall x,y,t\in\mathbb{R}^+$ [hide=Note]F(x,y)=F(y,x) and F(x,x)=2x[/hide]

Find the positive constant $c_0$ such that the series \[ \displaystyle\sum_{n = 0}^{\infty} \dfrac {n!}{(cn)^n} \] converges for $c>c_0$ and diverges for $0<c<c_0$.

Let $a_n = \frac{\pi}{2n}$, where $n$ is a natural number. Prove that for any $k = 1$,$2$,$...$, $n$ holds the equality $$\frac{\sin ka_n}{1-\cos a_n}+\frac{\sin 5ka_n}{1-\cos 5a_n}+\frac{\sin 9ka_n}{1-\cos 9a_n}+...+\frac{\sin (4n-3)a_n}{1-\cos (4n-3)a_n}=kn$$

Let $c$ be a fixed real number. Show that a root of the equation \[x(x+1)(x+2)\cdots(x+2009)=c\] can have multiplicity at most $2$. Determine the number of values of $c$ for which the equation has a root of multiplicity $2$.

Let $ n\ge 3$ be a natural number. A set of real numbers $ \{x_1,x_2,\ldots,x_n\}$ is called [i]summable[/i] if $ \sum_{i\equal{}1}^n \frac{1}{x_i}\equal{}1$. Prove that for every $ n\ge 3$ there always exists a [i]summable[/i] set which consists of $ n$ elements such that the biggest element is: a) bigger than $ 2^{2n\minus{}2}$ b) smaller than $ n^2$

Let $a, b$ and $c$ be real numbers such that $\frac{a}{b + c}+\frac{b}{c + a}+\frac{c}{a + b}= 1$. Prove that $\frac{a^2}{b + c}+\frac{b^2}{c + a}+\frac{c^2}{a + b}= 0$.

Determine all sequences of equal ratios of the form \[ \frac{a_1}{a_2} = \frac{a_3}{a_4} = \frac{a_5}{a_6} = \frac{a_7}{a_8} \] which simultaneously satisfy the following conditions: $\bullet$ The set $\{ a_1, a_2, \ldots , a_8 \}$ represents all positive divisors of $24$. $\bullet$ The common value of the ratios is a natural number.

Sequences $(a_n)_{n=0}^{\infty}$ and $(b_n)_{n=0}^{\infty}$ are defined with recurrent relations : $$a_0=0 , \;\;\; a_1=1, \;\;\;\; a_{n+1}=\frac{2018}{n} a_n+ a_{n-1}\;\;\; \text {for }\;\;\; n\geq 1$$ and $$b_0=0 , \;\;\; b_1=1, \;\;\;\; b_{n+1}=\frac{2020}{n} b_n+ b_{n-1}\;\;\; \text {for }\;\;\; n\geq 1$$ Prove that :$$\frac{a_{1010}}{1010}=\frac{b_{1009}}{1009}$$

Let $f_n = 2^{2^n}+ 1$, $n = 1,2,3,...$, be the Fermat’s numbers. Find the least real number $C$ such that $$\frac{1}{f_1}+\frac{2}{f_2}+\frac{2^2}{f_3}+...+\frac{2^{n-1}}{f_n} <C$$ for all positive integers $n$

Let $x, y,$ and $z$ be positive real numbers such that $xy + yz + zx = 27$. Prove that $x + y + z \ge \sqrt{3xyz}$. When does equality hold?

Consider a sequence $\{a_n\}_{n\geq 0}$ such that $a_{n+1}=a_n-\lfloor{\sqrt{a_n}}\rfloor\ (n\geq 0),\ a_0\geq 0$. (1) If $a_0=24$, then find the smallest $n$ such that $a_n=0$. (2) If $a_0=m^2\ (m=2,\ 3,\ \cdots)$, then for $j$ with $1\leq j\leq m$, express $a_{2j-1},\ a_{2j}$ in terms of $j,\ m$. (3) Let $m\geq 2$ be integer and for integer $p$ with $1\leq p\leq m-1$, let $a\0=m^2-p$. Find $k$ such that $a_k=(m-p)^2$, then find the smallest $n$ such that $a_n=0$.

There are $456$ people around a circle, denoted as $X_1, X_2, \dots, X_{456}$, and each one of them thought of a number. Every time Laura says an integer $k$ with $2 \leqslant k \leqslant 100$, the announcer announces all the numbers $p_1, p_2, \dots, p_{456}$, which are the averages of the numbers thought by the people in all the groups of $k$ consecutive people: $p_1$ is the average of the numbers thought by the people from $X_1$ to $X_k$, $p_2$ is the average of the numbers thought by the people from $X_2$ to $X_{k+1}$, and so on until $p_{456}$, the average of the numbers thought by the people from $X_{456}$ to $X_{k-1}$. Determine how many numbers $k$ Laura must say at a minimum so that, with certainty, the announcer can know the number thought by the person $X_{456}$.

Let $ \prod_{n=1}^{1996}{(1+nx^{3^n})}= 1+ a_{1}x^{k_{1}}+ a_{2}x^{k_{2}}+...+ a_{m}x^{k_{m}}$ where $a_{1}, a_{1}, . . . , a_{m}$ are nonzero and $k_{1} < k_{2} <...< k_{m}$. Find $a_{1996}$.

Determine all functions $f: \mathbb{Q^+} \rightarrow \mathbb{Q^+}$ that satisfy: \[f\left(x+\frac{y}{x}\right) = f(x)+f\left(\frac{y}{x}\right)+2y \:\text{for all}\: x, y \in \mathbb{Q^+}\]

let $m,n\in \mathbb{N}$ and $p(x),q(x),h(x)$ are polynomials with real Coefficients such that $p(x)$ is Descending. and for all $x\in \mathbb{R}$ $p(q(nx+m)+h(x))=n(q(p(x))+h(x))+m$ . prove that dont exist function $f:\mathbb{R}\rightarrow \mathbb{R}$ such that for all $x\in \mathbb{R}$ $f(q(p(x))+h(x))=f(x)^{2}+1$