For each five distinct variables from the set $x_1,..., x_{10}$ there is a single card on which their product is written. Peter and Basil play the following game. At each move, each player chooses a card, starting with Peter. When all cards have been taken, Basil assigns values to the variables as he wants, so that $0 \le x_1 \le ... \le x_10$. Can Basil ensure that the sum of the products on his cards is greater than the sum of the products on Peter's cards? (Ilya Bogdanov)

Consider a complete graph of $2020$ vertices. What is the least number of edges that need to be marked such that each triangle ($3$-vertex subgraph) has an odd number of marked edges?

Let \(f:\{0, 1\}^n \to \{0, 1\} \subseteq \mathbb{R}\) be a function. Call such a function a Boolean function. Let \(\wedge\) denote the component-wise multiplication in \(\{0,1\}^n\). For example, for \(n = 4\), \[(0,0,1,1) \wedge (0,1,0,1) = (0,0,0,1).\] Let \(S = \left\{i_1,i_2,\ldots ,i_k\right\} \subseteq \left\{1,2,\ldots ,n\right\}\). \(f\) is called the oligarchy function over \(S\) if \[f (x) = x_{i_1},x_{i_2},\ldots,x_{i_k}\ \text{ (with the usual multiplication),}\] where \(x_i\) denotes the \(i\)-th component of \(x\). By convention, \(f\) is called the oligarchy function over \(\emptyset\) if \(f\) is constantly 1. (i) Suppose \(f\) is not constantly zero. Show that \(f\) is an oligarchy function [u]if and only if[/u] \(f\) satisfies \[f(x\wedge y)=f(x)f(y),\ \forall x,y\in\left\{0,1\right\}^n.\] Let \(Y\) be a uniformly distributed random variable over \(\left\{0, 1\right\}^n\). Let \(T\) be an operator that maps Boolean functions to functions \(\left\{0, 1\right\}^n\to\mathbb{R}\), such that \[(Tf)(x)=E_Y(f(x\wedge Y)),\ \forall x\in\left\{0,1\right\}^n\] where \(E_Y()\) denotes the expectation over \(Y\). \(f\) is called an eigenfunction of \(T\) if \(\exists\lambda\in\mathbb{R}\backslash\left\{0\right\}\) such that \[(Tf)(x)=\lambda f(x),\ \forall x\in\left\{0,1\right\}^n\] (ii) Prove that \(f\) is an eigenfunction of \(T\) [u]if and only if[/u] \(f\) is an oligarchy function.

Let $g(t)$ be the minimum value of $f(x)=x2^{-x}$ in $t\leq x\leq t+1$. Evaluate $\int_0^2 g(t)dt$. [i]2010 Kumamoto University entrance exam/Science[/i]

Find the locus of points $P$ in the plane of a square $ABCD$ such that $$\max\{ PA,\ PC\}=\frac12(PB+PD).$$ [i](Anonymous314)[/i]

In a certain country there are two tennis clubs consisting of $1000 $ and $1001$ members respectively. All the members have different playing strength, and the descending order of palying strengths in each club is known. Find a procedure which determines, within $ 11$ games, who is in the $1001$st place among the $ 2001$ players in these clubs. It is assumed that a stronger player always beats a weaker one.

The freshman class of a school consists of $229$ boys and $271$ girls, and is divided into $10$ rooms of $50$ students each, the students in each room are numbered from $1$ to $50$. The physical education teacher wants to select a relay running team consisting of $1$ boy and $3$ girls or $1$ girl and $3$ boys, so that the four students must be two pairs of students with the same number from two rooms. Show that the number of possible teams is odd.

A collection of cardboard circles, each with a diameter of at most $1$, lie on a $5\times 8$ table without overlapping or overhanging the edge of the table. A cardboard circle of diameter $2$ is added to the collection. Prove that this new collection of cardboard circles can be placed on a $7\times 7$ table without overlapping or overhanging the edge.

For function $ f: \mathbb{R} \to \mathbb{R}$ given that $ f(x^2 +x +3) +2 \cdot f(x^2 - 3x + 5) = 6x^2 - 10x +17$, calculate $ f(2009)$.

Let $ABC$ be a triangle with $\angle BAC=120^o$, which the median $AD$ is perpendicular to side $AB$ and intersects the circumscribed circle of triangle $ABC$ at point $E$. Lines $BA$ and $EC$ intersect at $Z$. Prove that a) $ZD \perp BE$ b) $ZD=BC$

Parallelogram ${ABCD}$ is given with ${AC>BD}$, and ${O}$ intersection point of ${AC}$ and ${BD}$. Circle with center at ${O}$and radius ${OA}$ intersects extensions of ${AD}$and ${AB}$at points ${G}$ and ${L}$, respectively. Let ${Z}$ be intersection point of lines ${BD}$and ${GL}$. Prove that $\angle ZCA={{90}^{{}^\circ }}$.

A committee of $3366$ film critics are voting for the Oscars. Every critic voted just an actor and just one actress. After the voting, it was found that for every positive integer $n \in \left \{1, 2, \ldots, 100 \right \}$, there is some actor or some actress who was voted exactly $n$ times. Prove that there are two critics who voted the same actor and the same actress. [i](Cyprus)[/i]

Say that a sequence $a_1$, $a_2$, $a_3$, $a_4$, $a_5$, $a_6$, $a_7$, $a_8$ is [i]cool[/i] if * the sequence contains each of the integers 1 through 8 exactly once, and * every pair of consecutive terms in the sequence are relatively prime. In other words, $a_1$ and $a_2$ are relatively prime, $a_2$ and $a_3$ are relatively prime, $\ldots$, and $a_7$ and $a_8$ are relatively prime. How many cool sequences are there?

Suppose $\alpha, \beta$ are two positive rational numbers. Assume for some positive integers $m,n$, it is known that $\alpha^{\frac 1n}+\beta^{\frac 1m}$ is a rational number. Prove that each of $\alpha^{\frac 1n}$ and $\beta^{\frac 1m}$ is a rational number.

There is a point inside an equilateral triangle side $d$ whose distance from the vertices is $3, 4, 5$. Find $d$.

Find all real numbers $a, b, c$ that satisfy $$ 2a - b =a^2b, \qquad 2b-c = b^2 c, \qquad 2c-a= c^2 a.$$

Let $D$ be the midpoint of the base $AB$ of the isosceles acute triangle $ABC$. Choose point $E$ on segment $AB$, and let $O$ be the circumcenter of triangle $ACE$. Prove that the line through $D$ perpendicular to $DO$, the line through $E$ perpendicular to $BC$, and the line through $B$ parallel to $AC$ are concurrent.

Find all prime numbers $ p $ such that $ 4p^2 + 1 $ and $ 6p^2 + 1 $ are also prime numbers.

Prove that the numbers from $1$ to $ 15$ cannot be divided into two groups: $A$ of $2$ numbers and $B$ of $13$ numbers such that the sum of the numbers in group $B$ is equal to product of numbers in group $A$.

Let $n \geq 2$ be a positive integer and $a_{1},\ldots , a_{n}$ be positive real numbers such that $a_{1}+...+a_{n}= 1$. Prove that: \[\frac{a_{1}}{1+a_{2}+\cdots +a_{n}}+\cdots +\frac{a_{n}}{1+a_{1}+a_{2}+\cdots +a_{n-1}}\geq \frac{n}{2n-1}\]

Prove that: (a) if $y<\frac12$ and $n\ge3$ is a natural number then $(y+1)^n\ge y^n+(1+2y)^\frac n2$; (b) if $x,y,z$ and $n\ge3$ are natural numbers for which $x^2-1\le2y$ then $x^n+y^n\ne z^n$.

Let $ f:\mathbb{Z}_{>0}\rightarrow\mathbb{R} $ be a function such that for all $n > 1$ there is a prime divisor $p$ of $n$ such that \[ f(n)=f\left(\frac{n}{p}\right)-f(p). \] Furthermore, it is given that $ f(2^{2014})+f(3^{2015})+f(5^{2016})=2013 $. Determine $ f(2014^2)+f(2015^3)+f(2016^5) $.

Real numbers $x$ and $y$ satisfy $$xy+\frac{x}{y} = 3$$ $$\frac{1}{x^2y^2}+\frac{y^2}{x^2} = 4$$ If $x^2$ can be expressed in the form $\frac{a+\sqrt{b}}{c}$ for integers $a$, $b$, and $c$. Find $a+b+c$. [i]Proposed by Andy Xu[/i]

Al has three bags, each with three marbles each. Bag $1$ has two blue marbles and one red marble, Bag $2$ has one blue marble and two red marbles, and Bag $3$ has three red marbles. He chooses two distinct bags at random, then one marble at random from each of the chosen bags. What is the probability that he chooses two blue marbles?

Each point of a circle is painted in one of the $ N$ colors ($N \geq 2$). Prove that there exists an inscribed trapezoid such that all its vertices are painted the same color.