Given a chessboard that is infinite in all directions. Is it possible to place an infinite number of queens on it so that on each horizontally, on each vertical and on each diagonal of both directions (i.e. on a set of cells located at an angle of $45^o$ or $135^o$ to the horizontal) was exactly one queen?

You are playing a game in which you have $3$ envelopes, each containing a uniformly random amount of money between $0$ and $1000$ dollars. (That is, for any real $0 \leq a < b \leq 1000$, the probability that the amount of money in a given envelope is between $a$ and $b$ is $\frac{b-a}{1000}$.) At any step, you take an envelope and look at its contents. You may choose either to keep the envelope, at which point you finish, or discard it and repeat the process with one less envelope. If you play to optimize your expected winnings, your expected winnings will be $E$. What is $\lfloor E\rfloor,$ the greatest integer less than or equal to $E$? [i]Author: Alex Zhu[/i]

Define the [i]distance [/i] between two $5$-digit numbers $\overline{a_1a_2a_3a_4a_5}$ and $\overline{b_1b_2b_3b_4b_5}$ to be the largest integer $j$ such that $a_j \ne b_j$ . (Example: the distance between $16523$ and $16452$ is $5$.) Suppose all $5$-digit numbers are written in a line in some order. What is the minimal possible sum of the distances of adjacent numbers in that written order?

One dimension of a cube is increased by $ 1$, another is decreased by $ 1$, and the third is left unchanged. The volume of the new rectangular solid is $ 5$ less than that of the cube. What was the volume of the cube? $ \textbf{(A)}\ 8 \qquad \textbf{(B)}\ 27 \qquad \textbf{(C)}\ 64 \qquad \textbf{(D)}\ 125 \qquad \textbf{(E)}\ 216$

For $a, b, c, d, e$ in the interval $[0,1]$, prove that $(1+a+b+c+d+e)^2=>4(a^2+b^2+c^2+d^2+e^2)$

Let $f(x)$ and $g(x)$ be polynomials with non-negative integer coefficients, and let m be the largest coefficient of $f.$ Suppose that there exist natural numbers $a < b$ such that $f(a) = g(a)$ and $f(b) = g(b)$. Show that if $b > m,$ then $f = g.$

Find all real numbers $a$ for which the equation $x^2a- 2x + 1 = 3 |x|$ has exactly three distinct real solutions in $x$.

Find the maximal integer $ M$ with nonzero last digit (in its decimal representation) such that after crossing out one of its digits (not the first one) we can get an integer that divides $M$. (A Galochkin)

10.4, 11.3 Given $N\geq 3$ points enumerated with 1, 2, ..., $N$. Each two numbers are connected by mean of arrow from a lesser number to a greater one. A coloring of all arrows into red and blue is called [i]monochromatic[/i] iff for any numbers $A$ and $B$ there are [color=red]no[/color] two monochromatic paths from $A$ to $B$ of different colors. Find the number of monochromatic colorings. ([i]I. Bogdanov, G. Chelnokov[/i])

Find all values of $A$ such that $0^o < A < 360^o$ and also $\frac{\sin A}{\cos A - 1} \ge 1$ and $\frac{3\cos A - 1}{\sin A} \ge 1.$

Let us consider a convex hexagon ABCDEF. Let $A_1, B_1,C_1, D_1, E_1, F_1$ be midpoints of the sides $AB, BC, CD, DE, EF,FA$ respectively. Denote by $p$ and $p_1$, respectively, the perimeter of the hexagon $ A B C D E F $ and hexagon $ A_1B_1C_1D_1E_1F_1 $. Suppose that all inner angles of hexagon $ A_1B_1C_1D_1E_1F_1 $ are equal. Prove that \[ p \geq \frac{2 \cdot \sqrt{3}}{3} \cdot p_1 .\] When does equality hold ?

For a positive integer $n$, $\sigma(n)$ denotes the sum of the positive divisors of $n$. Determine $$\limsup\limits_{n\rightarrow \infty} \frac{\sigma(n^{2023})}{(\sigma(n))^{2023}}$$ [b]Note:[/b] Given a sequence ($a_n$) of real numbers, we say that $\limsup\limits_{n\rightarrow \infty} a_n = +\infty$ if ($a_n$) is not upper bounded, and, otherwise, $\limsup\limits_{n\rightarrow \infty} a_n$ is the smallest constant $C$ such that, for every real $K > C$, there is a positive integer $N$ with $a_n < K$ for every $n > N$.

For an integer $n \geq 5,$ two players play the following game on a regular $n$-gon. Initially, three consecutive vertices are chosen, and one counter is placed on each. A move consists of one player sliding one counter along any number of edges to another vertex of the $n$-gon without jumping over another counter. A move is legal if the area of the triangle formed by the counters is strictly greater after the move than before. The players take turns to make legal moves, and if a player cannot make a legal move, that player loses. For which values of $n$ does the player making the first move have a winning strategy?

Find the angles of the cyclic quadrilateral if you know that each of its diagonals is a bisector of one angle and a trisector of the opposite one (the trisector of the angle is one of the two rays that lie in the interior of the angle and divide it into three equal parts). (Vyacheslav Yasinsky)

[b]p1.[/b] The numerator of the fraction was increased by 20%. By what percentage should its denominator be reduced so that the resulting fraction doubles? [b]p2.[/b] From point $O$ on the plane there are four rays $OA$, $OB$, $OC$ and $OD$ (not necessarily in that order). It is known that $\angle AOB =40^o$, $\angle BOC = 70^o$, $\angle COD = 80^o$. What values can the angle between rays $OA$ and $OD$ take? (The angle between the rays is from $0^o$ to $180^o$.) [b]p3.[/b] Three equal circles have a common interior point. Prove that there is a circle of the same radius containing the centers of these three circles. [b]p4.[/b] Two non-leap years are consecutive. The first one has more Mondays than Wednesdays. Which of the seven days of the week will occur most often in the second year? [b]p5.[/b] The difference between two four-digit numbers is $7$. How much can the sums of their digits differ? [b]p6.[/b] The numbers $1, 2, 3, 4, 5, 6, 7, 8, 9$ are written on the board. In one move you can increase any of the numbers by $3$ or $5$. What is the minimum number of moves you need to make for all the numbers to become equal? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c2416727_soros_olympiad_in_mathematics]here.[/url]

Show that in a right-angled triangle the bisector of the right angle divides into equal parts the angle between the altitude and the median, drawn from the same vertex.

Let $D$ be a point on the side $[BC]$ of $\triangle ABC$ such that $|BD|=2$ and $|DC|=6$. If $|AB|=4$ and $m(\widehat{ACB})=20^\circ$, then what is $m(\widehat {BAD})$? $ \textbf{(A)}\ 10^\circ \qquad\textbf{(B)}\ 18^\circ \qquad\textbf{(C)}\ 20^\circ \qquad\textbf{(D)}\ 22^\circ \qquad\textbf{(E)}\ 25^\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]

suppose that $\mathcal F\subseteq \bigcup_{j=k+1}^{n}X^{(j)}$ and $|X|=n$. we know that $\mathcal F$ is a sperner family and it's also $H_k$. prove that: $\sum_{B\in \mathcal F}\frac{1}{\dbinom{n-1}{|B|-1}}\le 1$ (15 points)

Evaluate $\int^2_0\frac{x(16-x^2)}{16-x^2+\sqrt{(4-x)(4+x)(12+x^2)}}dx$.

Find the remainder when \[\prod_{i=0}^{100}(1-i^2+i^4)\] is divided by $101$. [i]Victor Wang[/i]

Given $f:\mathbb{R}\to \mathbb{R}$ an arbitrary function and $g:\mathbb{R}\to \mathbb{R}$ a function of the second degree, with the property: for any real numbers m and n equation $f\left( x \right)=mx+n$ has solutions if and only if the equation $g\left( x \right)=mx+n$ has solutions Show that the functions $f$ and $g$ are equal.

Let $ABCD$ be a non-cyclic convex quadrilateral. The feet of perpendiculars from $A$ to $BC,BD,CD$ are $P,Q,R$ respectively, where $P,Q$ lie on segments $BC,BD$ and $R$ lies on $CD$ extended. The feet of perpendiculars from $D$ to $AC,BC,AB$ are $X,Y,Z$ respectively, where $X,Y$ lie on segments $AC,BC$ and $Z$ lies on $BA$ extended. Let the orthocenter of $\triangle ABD$ be $H$. Prove that the common chord of circumcircles of $\triangle PQR$ and $\triangle XYZ$ bisects $BH$.

Let $r$ be a positive constant. If 2 curves $C_1: y=\frac{2x^2}{x^2+1},\ C_2: y=\sqrt{r^2-x^2}$ have each tangent line at their point of intersection and at which their tangent lines are perpendicular each other, then find the area of the figure bounded by $C_1,\ C_2$.

The Fibonacci numbers $F_0, F_1, F_2, . . .$ are defined inductively by $F_0=0, F_1=1$, and $F_{n+1}=F_n+F_{n-1}$ for $n \ge 1$. Given an integer $n \ge 2$, determine the smallest size of a set $S$ of integers such that for every $k=2, 3, . . . , n$ there exist some $x, y \in S$ such that $x-y=F_k$. [i]Proposed by Croatia[/i]