Let $f$ be a function that associates a positive integer $(x, y)$ with each pair of positive integers $f(x, y)$. Suppose that $f(x, y) \le xy$ for all positive integers $x$, $y$. Show that there are $2023$ different pairs $(x_1, y_1)$,$...$, $ (x_{2023}, y_{2023}$) such that $$f(x_1, y_1) = f(x_2, y_2) = ....= f(x_{2023}, y_{2023}).$$

Prove that $\frac{a^3}{bc} + \frac{b^3}{ca} + \frac{c^3}{ba} \ge a + b + c$, for all positive real numbers $a, b$, and $c$.

Let $ABC$ be a triangle, and let $O$ be its circumcenter. Let $\overline{CO}\cap AB\equiv D$. Let $\angle BAC=\alpha$, and $\angle CBA=\beta$. Prove that $$\dfrac{OD}{OC}=\Bigg|\dfrac{\cos(\alpha+\beta)}{\cos(\alpha-\beta)}\Bigg|$$\\ For clarification, $\overline{CO}$ represents the line $CO$, and $AC$ represents the segment $AC$. Cases in which $D$ doesn't exist should be ignored.

Prove that if the product $1\cdot 2\cdot ...\cdot n$ ($n> 3$) is not divisible by $n + 1$, then $n + 1$ is prime.

Let $C: y=\ln x$. For each positive integer $n$, denote by $A_n$ the area of the part enclosed by the line passing through two points $(n,\ \ln n),\ (n+1,\ \ln (n+1))$ and denote by $B_n$ that of the part enclosed by the tangent line at the point $(n,\ \ln n)$, $C$ and the line $x=n+1$. Let $g(x)=\ln (x+1)-\ln x$. (1) Express $A_n,\ B_n$ in terms of $n,\ g(n)$ respectively. (2) Find $\lim_{n\to\infty} n\{1-ng(n)\}$.

There are several contestants at a math olympiad. We say that two contestants $ A$ and $ B$ are [i]indirect friends[/i] if there are contestants $ C_1, C_2, ..., C_n$ such that $ A$ and $ C_1$ are friends, $ C_1$ and $ C_2$ are friends, $ C_2$ and $ C_3$ are friends, ..., $ C_n$ and $ B$ are friends. In particular, if $ A$ and $ B$ are friends themselves, they are [i]indirect friends[/i] as well. Some of the contestants were friends before the olympiad. During the olympiad, some contestants make new friends, so that after the olympiad every contestant has at least one friend among the other contestants. We say that a contestant is [i]special[/i] if, after the olympiad, he has exactly twice as indirect friends as he had before the olympiad. Prove that the number of special contestants is less or equal than $ \frac{2}{3}$ of the total number of contestants.

Find all polynomials $P(x)$ which have the properties: 1) $P(x)$ is not a constant polynomial and is a mononic polynomial. 2) $P(x)$ has all real roots and no duplicate roots. 3) If $P(a)=0$ then $P(a|a|)=0$ [i](nooonui)[/i]

In rectangular coodinate system $Oxy$, consider the line $y=3x$ and point $A(4,3)$. Find on the line $y=3x$, point $B\ne O$ such that the area of triangle $OBC$ is the minimum possible, where $C= AB\cap Ox$.

Farmer Boso has a busy farm with lots of animals. He tends to $5b$ cows, $5a +7$ chickens, and $b^{a-5}$ insects. Note that each insect has $6$ legs. The number of cows is equal to the number of insects. The total number of legs present amongst his animals can be expressed as $\overline{LLL }+1$, where $L$ stands for a digit. Find $L$.

If $ a$, $ b$, $ c$, and $ d$ are positive real numbers such that $ a$, $ b$, $ c$, $ d$ form an increasing arithmetic sequence and $ a$, $ b$, $ d$ form a geometric sequence, then $ \frac{a}{d}$ is $ \textbf{(A)}\ \frac{1}{12} \qquad \textbf{(B)}\ \frac{1}{6} \qquad \textbf{(C)}\ \frac{1}{4} \qquad \textbf{(D)}\ \frac{1}{3} \qquad \textbf{(E)}\ \frac{1}{2}$

Let $ABC$ be an acute triangle, and let $F_A$ and $F_B$ be the midpoints of sides $BC$ and $CA$, respectively. Let $E$ and $F$ be the intersection points of the circle centered at $F_A$ and passing through $A$ and the circle centered at $F_B$ and passing through $B$. Prove that if segments $CE$ and $CF$ have midpoints $N$ and $M$, respectively, then the intersection points of the circle centered at $M$ and passing through $E$ and the circle centered at $N$ and passing through $F$ lie on the line $AB$.

Determine all the functions $f : \mathbb{R}\rightarrow\mathbb{R}$ that satisfies the following. $f(xf(x)+f(x)f(y)+y-1)=f(xf(x)+xy)+y-1$

Let $ABC$ be an acute triangle with $|AB| > |AC|.$ Let $D$ be a point on the side $AB$, such that the angles $\angle ACD$ and $\angle CBD$ are equal. Let $E$ denote the midpoint of $BD,$ and let $S$ be the circumcenter of the triangle $BCD.$ Prove that the points $A, E, S$ and $C$ lie on the same circle.

Let $ p=\sum\limits_{k=0}^n a_kX^k\in R[X] $ a polynomial such that all his roots lie in the half plane $ \{z\in C| Re(z)<0 \}. $ Prove that $ a_ka_{k+3}<a_{k+1}a_{k+2}, $ for every k=0,1,2...,n-3.

Let $ABCD$ be a quadrilateral inscribed in circle $k$. Lines $AB$ and $CD$ intersect at point $E$ such that $AB=BE$. Let $F$ be the intersection point of tangents on circle $k$ in points $B$ and $D$, respectively. If the lines $AB$ and $DF$ are parallel, prove that $A$, $C$ and $F$ are collinear.

Let $A,B,C,D$ be points on a circle with radius $r$ in this order such that $|AB|=|BC|=|CD|=s$ and $|AD|=s+r$. Find all possible values of the interior angles of the quadrilateral $ABCD$.

Given $5$ points in the plane, no three of them being collinear. Show that among these $5$ points, we can always find $4$ points forming a convex quadrilateral.

In an obtuse triangle $ABC$, for which $AC < AB$, the radius of the inscribed circle is $R$, the midpoint of its arc $BC$ (which does not contain $A$) is $S$. A point $T$ is placed on the extension of altitude $AD$ such that $D$ is between $ A$ and $T$ and $AT = 2R$. Prove that $\angle AST = 90^o$.

There are $100$ cards on a table, flipped face down. Madina knows that on each card a single number is written and that the numbers are different integers from $1$ to $100$. In a move, Madina is allowed to choose any $3$ cards, and she is told a number that is written on one of the chosen cards, but not which specific card it is on. After several moves, Madina must determine the written numbers on as many cards as possible. What is the maximum number of cards Madina can ensure to determine? [i]Shubin Yakov, Russia[/i]

\item Two circles $ \Gamma_1$ and $ \Gamma_2$ intersect at points $ A$ and $ B$. Consider a circle $ \Gamma$ contained in $ \Gamma_1$ and $ \Gamma_2$, which is tangent to both of them at $ D$ and $ E$ respectively. Let $ C$ be one of the intersection points of line $ AB$ with $ \Gamma$, $ F$ be the intersection of line $ EC$ with $ \Gamma_2$ and $ G$ be the intersection of line $ DC$ with $ \Gamma_1$. Let $ H$ and $ I$ be the intersection points of line $ ED$ with $ \Gamma_1$ and $ \Gamma_2$ respectively. Prove that $ F$, $ G$, $ H$ and $ I$ are on the same circle.

Let $p$ be a prime number and $n, k$ and $q$ natural numbers, where $q\le \frac{n -1}{p-1}$ should be. Let $M$ be the set of all integers $m$ from $0$ to $n$, for which $m-k$ is divisible by $p$. Show that $$\sum_{m \in M} (-1) ^m {n \choose m}$$ is divisible by $p^q$.

Compute the number of monic polynomials $q(x)$ with integer coefficients of degree $12$ such that there exists an integer polynomial $p(x)$ satisfying $q(x)p(x) = q(x^2).$ [i]Proposed by Yang Liu[/i]

Let $[AB]$ be a chord of the circle $\Gamma$ not passing through its center and let $M$ be the midpoint of $[AB].$ Let $C$ be a variable point on $\Gamma$ different from $A$ and $B$ and $P$ be the point of intersection of the tangent lines at $A$ of circumcircle of $CAM$ and at $B$ of circumcircle of $CBM.$ Show that all $CP$ lines pass through a fixed point.

Let $n$ be a positive integer. David has six $n\times n$ chessboards which he arranges in an $n\times n\times n$ cube. Two cells are "aligned" if they can be connected by a path of cells $a=c_1,\ c_2,\ \dots,\ c_m=b$ such that all consecutive cells in the path share a side, and the sides that the cell $c_i$ shares with its neighbors are on opposite sides of the square for $i=2,\ 3,\ \dots\ m-1$. Two towers attack each other if the cells they occupy are aligned. What is the maximum amount of towers he can place on the board such that no two towers attack each other?

Determine the minimum value of $$x^{2014} + 2x^{2013} + 3x^{2012} + 4x^{2011} +\ldots + 2014x + 2015$$ where $x$ is a real number.