Ayala and Barvaz play a game: Ayala initially gives Barvaz two $100\times100$ tables of positive integers, such that the product of numbers in each table is the same. In one move, Barvaz may choose a row or column in one of the tables, and change the numbers in it (to some positive integers), as long as the total product remains the same. Barvaz wins if after $N$ such moves, he manages to make the two tables equal to each other, and otherwise Ayala wins. a. For which values of $N$ does Barvaz have a winning strategy? b. For which values of $N$ does Barvaz have a winning strategy, if all numbers in Ayalah’s tables must be powers of $2$?

A sequence of integers $a_{1}, a_{2}, a_{3}, \cdots$ is defined as follows: $a_{1}=1$, and for $n \ge 1$, $a_{n+1}$ is the smallest integer greater than $a_{n}$ such that $a_{i}+a_{j} \neq 3a_{k}$ for any $i, j, $ and $k$ in $\{1, 2, 3, \cdots, n+1 \}$, not necessarily distinct. Determine $a_{1998}$.

Albert writes $2025$ numbers $a_1, \ldots, a_{2025}$ in a circle on a blackboard. Initially, each of the numbers is uniformly and independently sampled at random from the interval $[0,1].$ Then, each second, he [i]simultaneously[/i] replaces $a_i$ with $\max(a_{i-1},a_i,a_{i+1})$ for all $i = 1, 2, \ldots, 2025$ (where $a_0 = a_{2025}$ and $a_{2026} = a_1$). Compute the expected value of the number of distinct values remaining after $100$ seconds.

There are $100$ cities in a country, some of them being joined by roads. Any four cities are connected to each other by at least two roads. Assume that there is no path passing through every city exactly once. Prove that there are two cities such that every other city is connected to at least one of them.

[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]

Find the sum of all positive integers $n$ such that $$\frac{n^3+8n^2+8n+80}{n+7}$$ is an integer. $\text{(A) }31\qquad\text{(B) }57\qquad\text{(C) }66\qquad\text{(D) }87\qquad\text{(E) }112$

Let $ABC$ be an acute triangle such that $\angle BAC = 45^o$. Let $D$ a point on $AB$ such that $CD \perp AB$. Let $P$ be an internal point of the segment $CD$. Prove that $AP\perp BC$ if and only if $|AP| = |BC|$.

If integers $m,n,k$ satisfy $m^2+n^2+1=kmn$, what values can $k$ have?

Find the infinite integrals as follows. (1) 2013 Hiroshima City University entrance exam/Informatic Science $\int \frac{x^2}{2-x^2}dx$ (2) 2013 Kanseigakuin University entrance exam/Science and Technology $\int x^4\ln x\ dx$ (3) 2013 Shinsyu University entrance exam/Textile Science and Technology, Second-exam $\int \frac{\cos ^ 3 x}{\sin ^ 2 x}\ dx$

Prove that there exist an infinite number of odd natural numbers $m_1<m_2<...$ and an infinity of natural numbers $n_1<n_2<...$ ,such that $(m_k,n_k)=1$ and $m_k^4-2n_k^4$ is a perfect square,for all $k\in\mathbb{N}$.

Define $ x \heartsuit y$ to be $ |x\minus{}y|$ for all real numbers $ x$ and $ y$. Which of the following statements is [b]not[/b] true? $\textbf{(A)}\ x \heartsuit y \equal{} y \heartsuit x \text{ for all } x \text{ and } y$ $\textbf{(B)}\ 2(x \heartsuit y) \equal{} (2x) \heartsuit (2y) \text{ for all } x \text{ and } y$ $\textbf{(C)}\ x \heartsuit 0 \equal{} x \text{ for all } x$ $\textbf{(D)}\ x \heartsuit x \equal{} 0 \text{ for all } x$ $\textbf{(E)}\ x \heartsuit y > 0 \text{ if } x \ne y$

Let $ABC$ be an isosceles triangle. Suppose that points $K$ and $L$ are chosen on lateral sides $AB$ and $AC$ respectively so that $AK = CL$ and $\angle ALK + \angle LKB = 60^o$. Prove that $KL = BC$.

Let $(v_1, ..., v_{2^n})$ be the vertices of an $n$-dimensional hypercube. Label each vertex $v_i$ with a real number $x_i$. Label each edge of the hypercube with the product of labels of the two vertices it connects. Let $S$ be the sum of the labels of all the edges. Over all possible labelings, find the minimum possible value of $\frac{S}{x^2_1+ x^2_2+ ...+ x^2_n}$ in terms of $ n$. Note: an $n$ dimensional hypercube is a graph on $2^n$ vertices labeled labeled with the binary strings of length $n$, where two vertices have an edge between them if and only if their labels differ in exactly one place. For instance, the vertices $100$ and $101$ on the $3$ dimensional hypercube are connected, but the vertices $100$ and $111$ are not.

Let's call it a [i] staircase of height [/i]$n$, a figure consisting from all square cells $n\times n$ lying no higher diagonals (the figure shows a [i]staircase of height [/i] $4$ ). In how many different ways can a [i]staircase of height[/i] $n$ can be divided into several rectangles whose sides go along the grid lines, but the areas are different in pairs? [img]https://cdn.artofproblemsolving.com/attachments/f/0/f66d7e9ada0978e8403fbbd8989dc1b201f2cd.png[/img]

Let $AH_1, BH_2$ and $CH_3$ be the altitudes of a triangle $ABC$. Point $M$ is the midpoint of $H_2H_3$. Line $AM$ meets $H_2H_1$ at point $K$. Prove that $K$ lies on the medial line of $ABC$ parallel to $AC$.

Let a circle through $B$ and $C$ of a triangle $ABC$ intersect $AB$ and $AC$ in $Y$ and $Z$ , respectively. Let $P$ be the intersection of $BZ$ and $CY$ , and let $X$ be the intersection of $AP$ and $BC$ . Let $M$ be the point that is distinct from $X$ and on the intersection of the circumcircle of the triangle $XYZ$ with $BC$. Prove that $M$ is the midpoint of $BC$

Let $a,b$ be positive integers and $p=\frac{b}{4}\sqrt{\frac{2a-b}{2a+b}}$ be a prime number. Find the maximum value of $p$ and justify your answer.

Let $M$ be an arbitrary positive real number greater than $1$, and let $a_1,a_2,...$ be an infinite sequence of real numbers with $a_n\in [1,M]$ for any $n\in \mathbb{N}$. Show that for any $\epsilon\ge 0$, there exists a positive integer $n$ such that $$\frac{a_n}{a_{n+1}}+\frac{a_{n+1}}{a_{n+2}}+\cdots+\frac{a_{n+t-1}}{a_{n+t}}\ge t-\epsilon$$ holds for any positive integer $t$.

Let $ABC$ be an acute triangle with altitude $\overline{AH}$, and let $P$ be a variable point such that the angle bisectors $k$ and $\ell$ of $\angle PBC$ and $\angle PCB$, respectively, meet on $\overline{AH}$. Let $k$ meet $\overline{AC}$ at $E$, $\ell$ meet $\overline{AB}$ at $F$, and $\overline{EF}$ meet $\overline{AH}$ at $Q$. Prove that as $P$ varies, line $PQ$ passes through a fixed point.

Find all ordered pairs $(a,b)$ of positive integers for which the numbers $\dfrac{a^3b-1}{a+1}$ and $\dfrac{b^3a+1}{b-1}$ are both positive integers.

Let $R$ be the rectangle in the Cartesian plane with vertices at $(0,0)$, $(2,0)$, $(2,1)$, and $(0,1)$. $R$ can be divided into two unit squares, as shown. [asy]size(120); defaultpen(linewidth(0.7)); draw(origin--(2,0)--(2,1)--(0,1)--cycle^^(1,0)--(1,1));[/asy] Pro selects a point $P$ at random in the interior of $R$. Find the probability that the line through $P$ with slope $\frac{1}{2}$ will pass through both unit squares.

Let one of the intersection points of two circles with centres $O_1,O_2$ be $P$. A common tangent touches the circles at $A,B$ respectively. Let the perpendicular from $A$ to the line $BP$ meet $O_1O_2$ at $C$. Prove that $AP\perp PC$.

Consider a triangle $ABC$ with circumcenter $O$ and let $A_1,B_1,C_1$ be the midpoints of the sides $BC,AC,AB,$ respectively. Points $A_2,B_2,C_2$ are defined as $\overrightarrow{OA_2}=\lambda \cdot \overrightarrow{OA_1}, \ \overrightarrow{OB_2}=\lambda \cdot \overrightarrow{OB_1}, \ \overrightarrow{OC_2}=\lambda \cdot \overrightarrow{OC_1},$ where $\lambda >0.$ Prove that lines $AA_2,BB_2,CC_2$ are concurrent.

Prove that the functional equations \[f(x + y) = f(x) + f(y),\] \[ \text{and} \qquad f(x + y + xy) = f(x) + f(y) + f(xy) \quad (x, y \in \mathbb R)\] are equivalent.

A quadrilateral is rotated clockwise, and the sides are extended its length in the direction of the movement. It turns out the endpoints of the segments constructed form a square. Prove the initial quadrilateral must also be a square. [b]Generalization:[/b] Prove that if the same process is applied to any $n$-gon and the result is a regular $n$-gon, then the intial $n$-gon must also be regular.