Let $m$ and $n$ be positive integers. Player $A$ has a field of $m \times n$, and player $B$ has a $1 \times n$ field (the first is the number of rows). On the first move, each player places on each square of his field white or black chip as he pleases. At each next on the move, each player can change the color of randomly chosen pieces on your field to the opposite, provided that in no row for this move will not change more than one chip (it is allowed not to change not a single chip). The moves are made in turn, player $A$ starts. Player $A$ wins if there is such a position that in the only row player $B$'s squares, from left to right, are the same as in some row of player's field $A$. Prove that player $A$ has the ability to win for any game of player $B$ if and only if $n <2m$.

[b]p1.[/b] Cities A, B, C, D and E are located next to each other along the highway at a distance of $5$ km from each other. The bus runs along the highway from city A to city E and back. The bus consumes $20$ liters of gasoline for every $100$ kilometers. In which city will a bus run out of gas if it initially had $150$ liters of gasoline in its tank? [b]p2.[/b] Find the minimum four-digit number whose product of all digits is $729$. Explain your answer. [b]p3.[/b] At the parade, soldiers are lined up in two lines of equal length, and in the first line the distance between adjacent soldiers is $ 20\%$ greater than in the second (there is the same distance between adjacent soldiers in the same line). How many soldiers are in the first rank if there are $85$ soldiers in the second rank? [b]p4.[/b] It is known about three numbers that the sum of any two of them is not less than twice the third number, and the sum of all three is equal to $300$. Find all triplets of such (not necessarily integer) numbers. [b]p5.[/b] The tourist fills two tanks of water using two hoses. $2.9$ liters of water flow out per minute from the first hose, $8.7$ liters from the second. At that moment, when the smaller tank was half full, the tourist swapped the hoses, after which both tanks filled at the same time. What is the capacity of the larger tank if the capacity of the smaller one is $12.5$ liters? [b]p6.[/b] Is it possible to mark 6 points on a plane and connect them with non-intersecting segments (with ends at these points) so that exactly four segments come out of each point? [b]p7.[/b] Petya wrote all the natural numbers from $1$ to $1000$ and circled those that are represented as the difference of the squares of two integers. Among the circled numbers, which numbers are more even or odd? [b]p8.[/b] On a sheet of checkered paper, draw a circle of maximum radius that intersects the grid lines only at the nodes. Explain your answer. [b]p9.[/b] Along the railway there are kilometer posts at a distance of $1$ km from each other. One of them was painted yellow and six were painted red. The sum of the distances from the yellow pillar to all the red ones is $14$ km. What is the maximum distance between the red pillars? [b]p10.[/b] The island nation is located on $100$ islands connected by bridges, with some islands also connected to the mainland by a bridge. It is known that from each island you can travel to each (possibly through other islands). In order to improve traffic safety, one-way traffic was introduced on all bridges. It turned out that from each island you can leave only one bridge and that from at least one of the islands you can go to the mainland. Prove that from each island you can get to the mainland, and along a single route. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c2416727_soros_olympiad_in_mathematics]here.[/url]

Let $a$ and $S$ be the length of the side and the area of regular triangle inscribed in a circle of radius $1$. A closed broken line $A_1A_2...A_{51}A_1$ consisting of $51$ segments of the same length $a$ is placed inside the circle. Prove that the sum of areas of the $ 51$ triangles between the neighboring segments $$A_1A_2A_3, A_2A_3A_4,..., A_{49}A_{50}A_{51}, A_{50}A_{51}A_1, A_{51}A_1A_2$$ is not less than $3S$. (A. Berzinsh, Riga)

In a football tournament there are $8$ teams, each of which plays exacly one match against every other team. If a team $A$ defeats team $B$, then $A$ is awarded $3$ points and $B$ gets $0$ points. If they end up in a tie, they receive $1$ point each. It turned out that in this tournament, whenever a match ended up in a tie, the two teams involved did not finish with the same final score. Find the maximum number of ties that could have happened in such a tournament.

Show that we cannot tile a $10 x 10$ board with $25$ pieces of type $A$, or with $25$ pieces of type $B$, or with $25$ pieces of type $C$.

The princess wishes to have a bracelet with $r$ rubies and $s$ emeralds arranged in such order that there exist two jewels on the bracelet such that starting with these and enumerating the jewels in the same direction she would obtain identical sequences of jewels. Prove that it is possible to fulfill the princess’s wish if and only if $r$ and $s$ have a common divisor.

For each positive integer $\,n,\;S(n)\,$ is defined to be the greatest integer such that, for every positive integer $\,k\leq S(n),\;n^{2}\,$ can be written as the sum of $\,k\,$ positive squares. [b]a.)[/b] Prove that $\,S(n)\leq n^{2}-14\,$ for each $\,n\geq 4$. [b]b.)[/b] Find an integer $\,n\,$ such that $\,S(n)=n^{2}-14$. [b]c.)[/b] Prove that there are infintely many integers $\,n\,$ such that $S(n)=n^{2}-14.$

Assume that $M \subset N$ has the property that every two numbers $m,n$ of $M$ satisfy $|m-n| \ge mn/25$. Prove that the set $M$ contains no more than $9$ elements. Decide whether there exists such set M.

There are $3n$ $A$s and $2n$ $B$s in a string, where $n$ is a positive integer, prove that you can find a substring in this string that contains $3$ $A$s and $2$ $B$s.

let $ V$ be a finitive set and $ g$ and $ f$ be two injective surjective functions from $ V$to$ V$.let $ T$ and $ S$ be two sets such that they are defined as following" $ S \equal{} \{w \in V: f(f(w)) \equal{} g(g(w))\}$ $ T \equal{} \{w \in V: f(g(w)) \equal{} g(f(w))\}$ we know that $ S \cup T \equal{} V$, prove: for each $ w \in V : f(w) \in S$ if and only if $ g(w) \in S$

A convex $n$-gon is divided into triangles by diagonals which do not intersect except at vertices of the n-gon. Each vertex belongs to an odd number of triangles. Show that $n$ must be a multiple of $3$.

a) Let $f,g:\mathbb{Z}\rightarrow\mathbb{Z}$ be one to one maps. Show that the function $h:\mathbb{Z}\rightarrow\mathbb{Z}$ defined by $h(x)=f(x)g(x)$, for all $x\in\mathbb{Z}$, cannot be a surjective function. b) Let $f:\mathbb{Z}\rightarrow\mathbb{Z}$ be a surjective function. Show that there exist surjective functions $g,h:\mathbb{Z}\rightarrow\mathbb{Z}$ such that $f(x)=g(x)h(x)$, for all $x\in\mathbb{Z}$.

[u]Round 5[/u] [b]5.1.[/b] Julia baked a pie for herself to celebrate pi day this year. If Julia bakes anyone pie on pi day, the following year on pi day she bakes a pie for herself with $1/3$ probability, she bakes her friend a pie with $1/6$ probability, and she doesn't bake anyone a pie with $1/2$ probability. However, if Julia doesn't make pie on pi day, the following year on pi day she bakes a pie for herself with $1/2$ probability, she bakes her friend a pie with $1/3$ probability, and she doesn't bake anyone a pie with $1/6$ probability. The probability that Julia bakes at least $2$ pies on pi day in the next $5$ years can be expressed as $p/q$, for relatively prime positive integers $p$ and $q$. Compute $p + q$. [b]5.2.[/b] Steven is flipping a coin but doesn't want to appear too lucky. If he ips the coin $8$ times, the probability he only gets sequences of consecutive heads or consecutive tails that are of length $4$ or less can be expressed as $p/q$, for relatively prime positive integers $p$ and $q$. Compute $p + q$. [b]5.3.[/b] Let $ABCD$ be a square with side length $3$. Further, let $E$ be a point on side$ AD$, such that $AE = 2$ and $DE = 1$, and let $F$ be the point on side $AB$ such that triangle $CEF$ is right with hypotenuse $CF$. The value $CF^2$ can be expressed as $m/n$ , where $m$ and $n$ are relatively prime positive integers. Compute $m + n$. [u]Round 6[/u] [b]6.1.[/b] Let $P$ be a point outside circle $\omega$ with center $O$. Let $A,B$ be points on circle $\omega$ such that $PB$ is a tangent to $\omega$ and $PA = AB$. Let $M$ be the midpoint of $AB$. Given $OM = 1$, $PB = 3$, the value of $AB^2$ can be expressed as $m/n$ for relatively prime positive integers $m, n$. Find $m + n$. [b]6.2.[/b] Let $a_0, a_1, a_2,...$with each term defined as $a_n = 3a_{n-1} + 5a_{n-2}$ and $a_0 = 0$, $a_1 = 1$. Find the remainder when $a_{2020}$ is divided by $360$. [b]6.3.[/b] James and Charles each randomly pick two points on distinct sides of a square, and they each connect their chosen pair of points with a line segment. The probability that the two line segments intersect can be expressed as $m/n$ for relatively prime positive integers $m, n$. Find $m + n$. [u]Round 7[/u] [b]7.1.[/b] For some positive integers $x, y$ let $g = gcd (x, y)$ and $\ell = lcm (2x, y)$: Given that the equation $xy+3g+7\ell = 168$ holds, find the largest possible value of $2x + y$. [b]7.2.[/b] Marco writes the polynomials $$f(x) = nx^4 +2x^3 +3x^2 +4x+5$$ and $$g(x) = a(x-1)^4 +b(x-1)^3 +6(x-1)^2 + d(x - 1) + e,$$ where $n, a, b, d, e$ are real numbers. He notices that $g(i) = f(i) - |i|$ for each integer $i$ satisfying $-5 \le i \le -1$. Then $n^2$ can be expressed as $p/q$ for relatively prime positive integers $p, q$. Find $p + q$. [b]7.3. [/b]Equilateral $\vartriangle ABC$ is inscribed in a circle with center $O$. Points $D$ and $E$ are chosen on minor arcs $AB$ and $BC$, respectively. Segment $\overline{CD}$ intersects $\overline{AB}$ and $\overline{AE}$ at $Y$ and $X$, respectively. Given that $\vartriangle DXE$ and $\vartriangle AXC$ have equal area, $\vartriangle AXY$ has area $ 1$, and $\vartriangle ABC$ has area $52$, find the area of $\vartriangle BXC$. [u]Round 8[/u] [b]8.[/b] Let $A$ be the number of total webpage visits our website received last month. Let $B$ be the number photos in our photo collection from ABMC onsite 2017. Let $M$ be the mean speed round score. Further, let $C$ be the number of times the letter c appears in our problem bank. Estimate $$A \cdot B + M \cdot C.$$Your answer will be scored according to the following formula, where $X$ is the correct answer and $I$ is your input. $$max \left\{ 0, \left\lceil min \left\{13 - \frac{|I-X|}{0.05 |I|}, 13 - \frac{|I-X|}{0.05 |I-2X|} \right\} \right\rceil \right\}$$ PS. You should use hide for answers. Rounds 1-4 have been posted [url=https://artofproblemsolving.com/community/c3h2766251p24226451]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

A deck of eight cards has cards numbered $1, 2, 3, 4, 5, 6, 7, 8$, in that order, and a deck of five cards has cards numbered $1, 2, 3, 4, 5$, in that order. The two decks are riffle-shuffled together to form a deck with $13$ cards with the cards from each deck in the same order as they were originally. Thus, numbers on the cards might end up in the order $1122334455678$ or $1234512345678$ but not $1223144553678$. Find the number of possible sequences of the $13$ numbers.

Let $\pi$ be a permutation of the numbers from $ 2$ through $2012$. Find the largest possible value of $$\log_2 \pi(2) \cdot \log_3 \pi(3) ...\log_{2012} \pi(2012).$$

No two of $20$ students in a class have the same scores on both written and oral examinations in mathematics. We say that student $A$ is better than $B$ if his two scores are greater than or equal to the corresponding scores of $B$. The scores are integers between $1$ and $10$. (a) Show that there exist three students $A,B,C$ such that $A$ is better than $B$ and $B$ is better than $C$. (b) Would the same be true for a class of $19$ students?

In a group of kids there are $2022$ boys and $2023$ girls. Every girl is a friend with exactly $2021$ boys. Friendship is a symmetric relation: if A is a friend of B, then B is also a friend of A. Prove that it is not possible that all boys have the same number of girl friends. [i]Proposed by the JMMO Problem Selection Committee[/i]

Let $m$ and $n$ be positive integers. A child walks the Cartesian plane taking a few steps. The child begins its journey at the point $(0, n)$ and ends at the point $(m, 0)$ in such a way that: $\bullet$ Each step has length $1$ and is parallel to either the $X$ or $Y$ axis. $\bullet$ For each point $(x, y)$ of its path it is true that $x\ge 0$ and $y\ge 0$. For each step of the child, the distance between the child and the axis to which said step is parallel is calculated. If the step causes the child to be further from the point $(0, 0)$ than before, we consider that distance as positive, otherwise, we consider that distance as negative. Prove that at the end of the boy's journey, the sum of all the distances is $0$.

At the $69$ Thailand-Yaranaikian meeting attended by $96$ Thai delegates and a number (unknown) from the Yaranakian country. Some time after the meeting took place, the meeting also discovered something amazing that happened in this meeting!! That is, regardless of whether we select at least $69$ of Thai participants and select all the Yaranikian country participants who are known to Thais in the initial selection group, there is at least $1$ person fo form a minority. They found in that minority, there was always $1$ more Yaranikhians than Thais. Prove that there must be at least $28$ of the Yaranaikian attendees who know the Thai delegates. (Note: In this meeting, none of the attendees were half-breeds. Thai-Yara Nikian) [i](tatari/nightmare)[/i]

Given an integer $n\ge3$, prove that the set $X=\{1,2,3,\ldots,n^2-n\}$ can be divided into two non-intersecting subsets such that neither of them contains $n$ elements $a_1,a_2,\ldots,a_n$ with $a_1<a_2<\ldots<a_n$ and $a_k\le\frac{a_{k-1}+a_{k+1}}2$ for all $k=2,\ldots,n-1$.

The venusian prophet Zabruberson sent to his pupils a $ 10000$-letter word, each letter being $ A$ or $ E$: the [i]Zabrubic word[/i]. Their pupils consider then that for $ 1 \leq k \leq 10000$, each word comprised of $ k$ consecutive letters of the Zabrubic word is a [i]prophetic word[/i] of length $ k$. It is known that there are at most $ 7$ prophetic words of lenght $ 3$. Find the maximum number of prophetic words of length $ 10$.

[b]a)[/b] Let's consider a finite number of big circles of a sphere that do not pass all from a point. Show that there exists such a point that is found only in two of the circles. (With big circle we understand the circles with radius equal to the radius of the sphere.) [b]b)[/b] Using the result of part $a)$ show that, for a set of $n$ points in a plane, that are not all in a line, there exists a line that passes through only two points of the given set.

Anna and Bob, with Anna starting first, alternately color the integers of the set $S = \{1, 2, ..., 2022 \}$ red or blue. At their turn each one can color any uncolored number of $S$ they wish with any color they wish. The game ends when all numbers of $S$ get colored. Let $N$ be the number of pairs $(a, b)$, where $a$ and $b$ are elements of $S$, such that $a$, $b$ have the same color, and $b - a = 3$. Anna wishes to maximize $N$. What is the maximum value of $N$ that she can achieve regardless of how Bob plays?

In the illustration, a regular hexagon and a regular octagon have been tiled with rhombuses. In each case, the sides of the rhombuses are the same length as the sides of the regular polygon. (a) Tile a regular decagon ($10$-gon) into rhombuses in this manner. (b) Tile a regular dodecagon ($12$-gon) into rhombuses in this manner. (c) How many rhombuses are in a tiling by rhombuses of a $2002$-gon? Justify your answer. [img]https://cdn.artofproblemsolving.com/attachments/8/a/8413e4e2712609eba07786e34ba2ce4aa72888.png[/img]

Using several white edge cubes of side $ 1$, Guille builds a large cube. Then he chooses $4$ faces of the big cube and paints them red. Finally, he takes apart the large cube and observe that the cubes with at least a face painted red is $431$. Find the number of cubes that he used to assemble the large cube. Analyze all the possibilities.