Let $ n \ge 2$ be a fixed integer and let $ a_{i,j} (1 \le i < j \le n)$ be some positive integers. For a sequence $ x_1, ... , x_n$ of reals, let $ K(x_1, .... , x_n)$ be the product of all expressions $ (x_i \minus{} x_j)^{a_{i,j}}$ where $ 1 \le i < j \le n$. Prove that if the inequality $ K(x_1, .... , x_n) \ge 0$ holds independently of the choice of the sequence $ x_1, ... , x_n$ then all integers $ a_{i,j}$ are even.

Show that it is impossible to cover a unit square with five equal squares with side $s<\frac{1}{2}$.

The sum of several consecutive natural numbers is $20$ times greater than the largest of them and $30$ times greater than the smallest. Find these numbers.

A function $f : I \to \mathbb R$, defined on an interval $I$, is called concave if $f(\theta x + (1 - \theta)y) \geq \theta f(x) + (1 - \theta)f(y)$ for all $x, y \in I$ and $0 \leq \theta \leq 1$. Assume that the functions $f_1, \ldots , f_n$, having all nonnegative values, are concave. Prove that the function $(f_1f_2 \cdots f_n)^{1/n}$ is concave.

function $f:\mathbb{N}^{\star} \times \mathbb{N}^{\star} \rightarrow \mathbb{N}^{\star}$ ($\mathbb{N}^{\star}=\mathbb{N}\cup \{0\}$)with these conditon: 1- $f(0,x)=x+1$ 2- $f(x+1,0)=f(x,1)$ 3- $f(x+1,y+1)=f(x,f(x+1,y))$(romania 1997) find $f(3,1997)$

Let be a natural number $ n $ and $ n $ real numbers $ a_1,a_2,\ldots ,a_n $ such that $$ a_m+a_{m+1} +\cdots +a_n\ge \frac{(m+n)(n-m+1)}{2} ,\quad\forall m\in\{ 1,2,\ldots ,n \} . $$ Prove that $ a_1^2+a_2^2+\cdots +a_n^2\ge\frac{n(n+1)(2n+1)}{6} . $

Let $A$ and $B$ infinite sets of positive real numbers such that: 1. For any pair of elements $u \ge v$ in $A$, it follows that $u+v$ is an element of $B$. 2. For any pair of elements $s>t$ in $B$, it follows that $s-t$ is an element of $A$. Prove that $A=B$ or there exists a real number $r$ such that $B=\{2r, 3r, 4r, 5r, \dots\}$.

Get rid of irrationality in the denominator of a fraction $$\frac{1}{\sqrt[3]{4}+\sqrt[3]{2}+2}$$.

Given a (fixed) positive integer $N$, solve the functional equation \[f \colon \mathbb{Z} \to \mathbb{R}, \ f(2k) = 2f(k) \textrm{ and } f(N-k) = f(k), \ \textrm{for all } k \in \mathbb{Z}.\] [i](Dan Schwarz)[/i]

Let $ a, b \in \mathbb{N}$ with $ 1 \leq a \leq b,$ and $ M \equal{} \left[\frac {a \plus{} b}{2} \right].$ Define a function $ f: \mathbb{Z} \mapsto \mathbb{Z}$ by \[ f(n) \equal{} \begin{cases} n \plus{} a, & \text{if } n \leq M, \\ n \minus{} b, & \text{if } n >M. \end{cases} \] Let $ f^1(n) \equal{} f(n),$ $ f_{i \plus{} 1}(n) \equal{} f(f^i(n)),$ $ i \equal{} 1, 2, \ldots$ Find the smallest natural number $ k$ such that $ f^k(0) \equal{} 0.$

Find all functions $f : \mathbb{R} \mapsto \mathbb{R}$ such that $f(xy+f(x)) = xf(y) +f(x)$ for all $x,y \in \mathbb{R}$.

An arbitrary positive number $a$ is given. A sequence ${a_n}$ is defined by equalities $a_1=\frac{a}{a+1}$ and $a_{n+1}=\frac{aa_n}{a^2+a_n-aa_n}$ for all $n \geq 1$ Find the minimal constant $C$ such that inequality $$a_1+a_1a_2+\ldots+a_1\ldots a_m<C$$ holds for all positive integers $m$ regardless of $a$

Let $\mathbb{R}$ denote the set of real numbers. Find all functions $f:\mathbb{R}\rightarrow\mathbb{R}$ such that \[f(x^2)+f(xy)=f(x)f(y)+yf(x)+xf(x+y)\] for all $x,y\in\mathbb{R}$.

Find all function $ f: \mathbb R\to \mathbb R$ satisfying the condition: \[ f(f(x \minus{} y)) \equal{} f(x)\cdot f(y) \minus{} f(x) \plus{} f(y) \minus{} xy \]

A natural number $n$ is given. Find the longest interval of a real line such that for numbers taken arbitrarily from it $a_0$, $a_1$, $a_2$, $...$, $a_{2n-1}$ the polynomial $x^{2n}+a_{2n-1}x^{2n-1}+...+a_1x + a_0$ has no roots on the entire real axis. (The left and right ends of the interval do not belong to the interval.)

Suppose $a$, $b$, $c$, and $d$ are non-negative integers such that \[(a+b+c+d)(a^2+b^2+c^2+d^2)^2=2023.\] Find $a^3+b^3+c^3+d^3$. [i]Proposed by Connor Gordon[/i]

Assume that $a$ is a given irrational number. (a) Prove that for each positive real number $\epsilon$ there exists at least one integer $q\ge0$ such that $aq-\lfloor aq\rfloor<\epsilon$. (b) Prove that for given $\epsilon>0$ there exist infinitely many rational numbers $\frac pq$ such that $q>0$ and $\left|a-\frac pq\right|<\frac\epsilon q$.

Given two positive numbers $a,b$ such that $a^3 +b^3 = a^5 +b^5$, then the greatest value of $M = a^2 + b^2 - ab$ is (A): $\frac14$ (B): $\frac12$ (C): $2$ (D): $1$ (E): None of the above.

Solve the system of equations for positive real numbers: $$\frac{1}{xy}=\frac{x}{z}+ 1,\frac{1}{yz} = \frac{y}{x} + 1, \frac{1}{zx} =\frac{z}{y}+ 1$$

A monkey in Zoo becomes lucky if he eats three different fruits. What is the largest number of monkeys one can make lucky, by having $20$ oranges, $30$ bananas, $40$ peaches and $50$ tangerines? Justify your answer. (A): $30$ (B): $35$ (C): $40$ (D): $45$ (E): None of the above.

A function $ f$ defined on the positive integers (and taking positive integers values) is given by: $ \begin{matrix} f(1) \equal{} 1, f(3) \equal{} 3 \\ f(2 \cdot n) \equal{} f(n) \\ f(4 \cdot n \plus{} 1) \equal{} 2 \cdot f(2 \cdot n \plus{} 1) \minus{} f(n) \\ f(4 \cdot n \plus{} 3) \equal{} 3 \cdot f(2 \cdot n \plus{} 1) \minus{} 2 \cdot f(n), \end{matrix}$ for all positive integers $ n.$ Determine with proof the number of positive integers $ \leq 1988$ for which $ f(n) \equal{} n.$

Let $P,Q$ be the polynomials: $$x^3-4x^2+39x-46, x^3+3x^2+4x-3,$$ respectively. 1. Prove that each of $P, Q$ has an unique real root. Let them be $\alpha,\beta$, respectively. 2. Prove that $\{ \alpha\}>\{ \beta\} ^2$, where $\{ x\}=x-\lfloor x\rfloor$ is the fractional part of $x$.

Find a costant $C$, such that $$ \frac{S}{ab+bc+ca}\le C$$ where $a,b,c$ are the side lengths of an arbitrary triangle, and $S$ is the area of the triangle. (The maximal number of points is given for the best possible constant, with proof.)

[b]p1.[/b] Ivan Ivanovich came to the store with $20$ rubles. The store sold brooms for $1$ ruble. $17$ kopecks and basins for $1$ rub. $66$ kopecks (there are no other products left in the store). How many brooms and how many basins does he need to buy in order to spend as much money as possible? (Note: $1$ ruble = $100$ kopecks) [b]p2.[/b] On the road from city A to city B there are kilometer posts. On each pillar, on one side, the distance to city A is written, and on the other, to B. In the morning, a tourist passed by a pillar on which one number was twice the size of the other. After walking another $10$ km, the tourist saw a post on which the numbers differed exactly three times. What is the distance from A to B? List all possibilities. [b]p3.[/b] On New Year's Eve, geraniums, crocuses and cacti stood in a row (from left to right) on the windowsill. Every morning, Masha, wiping off the dust, swaps the places of the flower on the right and the flower in the center. During the day, Tanya, while watering flowers, swaps places between the one in the center and the one on the left. In what order will the flowers be in 365 days on the next New Year's Eve? [b]p4.[/b] What is the smallest number of digits that must be written in a row so that by crossing out some digits you can get any three-digit natural number from $100$ to $999$? [b]p5.[/b] An ordinary irreducible fraction was written on the board, the numerator and denominator of which were positive integers. The numerator was added to its denominator and a new fraction was obtained. The denominator was added to the numerator of the new fraction to form a third fraction. When the numerator was added to the denominator of the third fraction, the result was $13/23$. What fraction was written on the board? [b]p6.[/b] The number $x$ is such that $15\%$ of it and $33\%$ of it are positive integers. What is the smallest number $x$ (not necessarily an integer!) with this property? [b]p7.[/b] A radio-controlled toy leaves a certain point. It moves in a straight line, and on command can turn left exactly $17^o$ (relative to the previous direction of movement). What is the smallest number of commands required for the toy to pass through the starting point again? [b]p8.[/b] The square is divided by straight lines into $25$ rectangles (fig. 1). The areas of some of them are indicated in the figure (not to scale). Find the area of the rectangle marked with a question mark. [img]https://cdn.artofproblemsolving.com/attachments/0/9/591c93421067123d50382744f9d28357acf83a.png[/img] [b]p9.[/b] Petya multiplied all natural numbers from $1$ to his age inclusive. The result is a number $$8 \,\, 841 \,\,761993 \,\,739 \,\,701954 \,\,543 \,\,616 \,\,000 \,\,000.$$ How old is Petya? [b]p10.[/b] There are $100$ integers written in a line, and the sum of any three in a row is equal to $10$ or $11$. The first number is equal to one. What could the last number be? List all possibilities. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c2416727_soros_olympiad_in_mathematics]here.[/url]

Let $\mathbb{Q}_{>0}$ denote the set of all positive rational numbers. Determine all functions $f:\mathbb{Q}_{>0}\to \mathbb{Q}_{>0}$ satisfying $$f(x^2f(y)^2)=f(x)^2f(y)$$ for all $x,y\in\mathbb{Q}_{>0}$