For any set $A=\{a_1,a_2,\cdots,a_m\}$, let $P(A)=a_1a_2\cdots a_m$. Let $n={2010\choose99}$, and let $A_1, A_2,\cdots,A_n$ be all $99$-element subsets of $\{1,2,\cdots,2010\}$. Prove that $2010|\sum^{n}_{i=1}P(A_i)$.

Determine, with proof, the maximum and minimum among the numbers $$\sqrt5 - \lfloor \sqrt5 \rfloor, 2\sqrt5 - \lfloor 2\sqrt5 \rfloor, 3\sqrt5 - \lfloor 3 \sqrt5\rfloor, ..., 2013\sqrt5 - \lfloor 2013\sqrt5\rfloor, 2014\sqrt5 - \lfloor 2014\sqrt5\rfloor $$

Find the minimum and the maximum value of the expression $\sqrt{4 -a^2} +\sqrt{4 -b^2} +\sqrt{4 -c^2}$ where $a,b, c$ are positive real numbers satisfying the condition $a^2 + b^2 + c^2=6$

If $x_1,x_2,...,x_{2001}$ are real numbers with $0 \le x_n \le 1$ for $n = 1,2,...,2001$, find the maximum value of $$\left(\frac{1}{2001}\sum_{n=1}^{2001}x_n^2\right)-\left(\frac{1}{2001}\sum_{n=1}^{2001}x_n\right)^2$$ Where is this maximum attained?

Solve the system of equations for real $x$ and $y$: \begin{eqnarray*} 5x \left( 1 + \frac{1}{x^2 + y^2}\right) &=& 12 \\ 5y \left( 1 - \frac{1}{x^2+y^2} \right) &=& 4 . \end{eqnarray*}

Prove that $\frac{1}{25} \sum_{k=0}^{2001} \left[ \frac{2^k}{25}\right]$ is a positive integer.

Let $n$ be a natural number and $f_1$, $f_2$, ..., $f_n$ be polynomials with integers coeffcients. Show that there exists a polynomial $g(x)$ which can be factored (with at least two terms of degree at least $1$) over the integers such that $f_i(x)+g(x)$ cannot be factored (with at least two terms of degree at least $1$) over the integers for every $i$.

[b]p1.[/b] The oil pipeline passes by three villages $A$, $B$, $C$. In the first village, $30\%$ of the initial amount of oil is drained, in the second - $40\%$ of the amount that will reach village $B$, and in the third - $50\%$ of the amount that will reach village $C$ What percentage of the initial amount of oil reaches the end of the pipeline? [b]p2.[/b] There are several ordinary irreducible fractions (not necessarily proper) with natural numerators and denominators (and the denominators are greater than $1$). The product of all fractions is equal to $10$. All numerators and denominators are increased by $1$. Can the product of the resulting fractions be greater than $10$? [b]p3.[/b] The garland consists of $10$ light bulbs connected in series. Exactly one of the light bulbs has burned out, but it is not known which one. There is a suitable light bulb available to replace a burnt out one. To unscrew a light bulb, you need $10$ seconds, to screw it in - also $10$ seconds (the time for other actions can be neglected). Is it possible to be guaranteed to find a burnt out light bulb: a) in $10$ minutes, b) in $5$ minutes? [b]p4.[/b] When fast and slow athletes run across the stadium in one direction, the fast one overtakes the slow one every $15$ minutes, and when they run towards each other, they meet once every $5$ minutes. How many times is the speed of a fast runner greater than the speed of a slow runner? [b]p5.[/b] Petya was $35$ minutes late for school. Then he decided to run to the kiosk for ice cream. But when he returned, the second lesson had already begun. He immediately ran for ice cream a second time and was gone for the same amount of time. When he returned, it turned out that he was late again, and he had to wait $50$ minutes before the start of the fourth lesson. How long does it take to run from school to the ice cream stand and back if each lesson, including recess after it, lasts $55$ minutes? [b]p6.[/b] In a convex heptagon, draw as many diagonals as possible so that no three of them are sides of the same triangle, the vertices of which are at the vertices of the original heptagon. [b]p7.[/b] In the writing of the antipodes, numbers are also written with the digits $0, ..., 9$, but each of the numbers has different meanings for them and for us. It turned out that the equalities are also true for the antipodes $5 * 8 + 7 + 1 = 48$ $2 * 2 * 6 = 24$ $5* 6 = 30$ a) How will the equality $2^3 = ...$ in the writing of the antipodes be continued? b) What does the number 9 mean among the Antipodes? Clarifications: a) It asks to convert $2^3$ in antipodes language, and write with what number it is equal and find a valid equality in both numerical systems. b) What does the digit $9$ mean among the antipodes, i.e. with which digit is it equal in our number system? [b]p8.[/b] They wrote the numbers $1, 2, 3, 4, ..., 1996, 1997$ in a row. Which digits were used more when writing these numbers - ones or twos? How long? [b]p9.[/b] On the number axis there lives a grasshopper who can jump $1$ and $4$ to the right and left. Can he get from point $1$ to point $2$ of the numerical axis $in 1996$ jumps if he must not get to points with coordinates divisible by $ 4$ (points $0$, $\pm 4$, $\pm 8$, etc.)? [b]p10.[/b] Is there a convex quadrilateral that can be cut along a straight line into two parts of the same size and shape, but neither the diagonal nor the straight line passing through the midpoints of opposite sides divides it into two equal parts? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c2416727_soros_olympiad_in_mathematics]here.[/url]

There are n cars waiting at distinct points of a circular race track. At the starting signal each car starts. Each car may choose arbitrarily which of the two possible directions to go. Each car has the same constant speed. Whenever two cars meet they both change direction (but not speed). Show that at some time each car is back at its starting point.

Let \(P(x)=x^{2}+ax+b\) be a quadratic polynomial where \(a\) is real and \(b \neq 2\), is rational. Suppose \(P(0)^{2},P(1)^{2},P(2)^{2}\) are integers, prove that \(a\) and \(b\) are integers.

Let $a$ and $b$ be two fixed positive real numbers. Find all real numbers $x$, such that inequality holds $$\frac{1}{\sqrt{x}}+\frac{1}{\sqrt{a+b-x}} < \frac{1}{\sqrt{a}} + \frac{1}{\sqrt{b}}$$

Father moves $3$ steps forward just as son moves $5$ steps, but this while the father takes $6$ steps, the son does $7$ steps. At first, father and son are together, then the son begins to walk away from his father in a straight line. When the son has done $30$ steps, the father starts to follow him. In a few steps, Dad arrives after the son?

It is known that $a^{2005} + b^{2005}$ can be expressed as the polynomial of $a + b$ and $ab$. Find the coefficients' sum of this polynomial.

Find all pairs $ (m, n)$ of positive integers that have the following property: For every polynomial $P (x)$ of real coefficients and degree $m$, there exists a polynomial $Q (x)$ of real coefficients and degree $n$ such that $Q (P (x))$ is divisible by $Q (x)$.

Suppose that a sequence $a_1,a_2,\ldots$ of positive real numbers satisfies \[a_{k+1}\geq\frac{ka_k}{a_k^2+(k-1)}\] for every positive integer $k$. Prove that $a_1+a_2+\ldots+a_n\geq n$ for every $n\geq2$.

Let $\mathbb{R}^{+}$ be the set of positive real numbers. Find all real numbers $a$ for which there exists a function $f :\mathbb{R}^{+} \to \mathbb{R}^{+}$ such that $3(f(x))^{2}=2f(f(x))+ax^{4}$, for all $x \in \mathbb{R}^{+}$.

Let the real numbers $a$, $b$, and $c$ satisfy the equations $(a+b)(b+c)(c+a)=abc$ and $(a^9+b^9)(b^9+c^9)(c^9+a^9)=(abc)^9$. Prove that at least one of $a$, $b$, and $c$ equals $0$.

Prove that if positive numbers $ x, y, z $ satisfy the inequality $$ \frac{x^2+y^2-z^2}{2xy} + \frac{y^2+z^2-x^2}{2yz} + \frac{z^2+x^2-y ^2}{2xz} > 1,$$ then they are the lengths of the sides of a certain triangle.

Find $x$ such that trigonometric \[\frac{\sin 3x \cos (60^\circ -4x)+1}{\sin(60^\circ - 7x) - \cos(30^\circ + x) + m}=0\] where $m$ is a fixed real number.

By $rad(x)$ we denote the product of all distinct prime factors of a positive integer $n$. Given $a \in N$, a sequence $(a_n)$ is defined by $a_0 = a$ and $a_{n+1} = a_n+rad(a_n)$ for all $n \ge 0$. Prove that there exists an index $n$ for which $\frac{a_n}{rad(a_n)} = 2022$

Let $n\in\mathbb{Z}$, $n\geq 2$. Find all functions $f:\mathbb{R}_{>0}\rightarrow\mathbb{R}_{>0}$ such that $$f(x_1+\dots +x_n)^2=\sum_{i=1}^nf(x_i) ^2+ 2\sum_{i<j}f(x_ix_j),$$ for all $x_1,\dots ,x_n\in\mathbb{R}_{>0}$. [i]Proposed by Andrei Vila[/i]

Let $x$, $y$, $z$ be positive reals satisfying $\left(x+y+z\right)^{3}=32xyz$ Find the minimum and the maximum of $P=\frac{x^{4}+y^{4}+z^{4}}{\left(x+y+z\right)^{4}}$

Let $a,b, c, d,e$ be the roots of $p(x) = 2x^5 - 3x^3 + 2x -7$. Find the value of $$(a^3 - 1)(b^3 - 1)(c^3 - 1)(d^3 - 1)(e^3 - 1).$$

Let $k$ be a constant number larger than $1$. Find all polynomials $P(x)$ such that $P({x^k}) = {\left( {P(x)} \right)^k}$ for all real $x$.

The symbol $N_m$ denotes the set of all integers not less than the given integer $m$. Find all functions $f: N_m \to N_m$ such that $f(x^2+f(y))=y^2+f(x)$ for all $x,y \in N_m$.