Two persons play a game on a board divided into $3\times 100$ squares. They move in turn: the first places tiles of size $1\times2$ lengthwise (along the long axis of the board), the second, in the perpendicular direction. The loser is the one who cannot make a move. Which of the players can always win (no matter how his opponent plays), and what is the winning strategy?

Call an $n$-tuple $(a_1, . . . , a_n)$ [i]occasionally periodic [/i] if there exist a nonnegative integer $i$ and a positive integer $p$ satisfying $i + 2p \le n$ and $a_{i+j} = a_{i+p+j}$ for every $j = 1, 2, . . . , p$. Let $k$ be a positive integer. Find the least positive integer $n$ for which there exists an $n$-tuple $(a_1, . . . , a_n)$ with elements from set $\{1, 2, . . . , k\}$, which is not occasionally periodic but whose arbitrary extension $(a_1, . . . , a_n, a_{n+1})$ is occasionally periodic for any $a_{n+1} \in \{1, 2, . . . , k\}$.

Points $ A_1,A_2,...,A_n$ and $ B_1,B_2,...,B_n$ are given on a plane. Show that the points $ B_i$ can be renumbered in such a way that the angle between vectors $ A_iA_j^{\longrightarrow}$ and $ B_iB_j^{\longrightarrow}$ is acute or right whenever $ i\neq j$.

A finite set of integers is called [i]bad[/i] if its elements add up to $2010$. A finite set of integers is a [i]Benelux-set[/i] if none of its subsets is bad. Determine the smallest positive integer $n$ such that the set $\{502, 503, 504, . . . , 2009\}$ can be partitioned into $n$ Benelux-sets. (A partition of a set $S$ into $n$ subsets is a collection of $n$ pairwise disjoint subsets of $S$, the union of which equals $S$.) [i](2nd Benelux Mathematical Olympiad 2010, Problem 1)[/i]

Suppose that every positve integer has been given one of the colors red, blue,arbitrarily. Prove that there exists an infinite sequence of positive integers $ a_{1} < a_{2} < a_{3} < \cdots < a_{n} < \cdots,$ such that inifinite sequence of positive integers $ a_{1},\frac {a_{1} \plus{} a_{2}}{2},a_{2},\frac {a_{2} \plus{} a_{3}}{2},a_{3},\frac {a_{3} \plus{} a_{4}}{2},\cdots$ has the same color.

In the plane we consider rectangles whose sides are parallel to the coordinate axes and have positive length. Such a rectangle will be called a [i]box[/i]. Two boxes [i]intersect[/i] if they have a common point in their interior or on their boundary. Find the largest $ n$ for which there exist $ n$ boxes $ B_1$, $ \ldots$, $ B_n$ such that $ B_i$ and $ B_j$ intersect if and only if $ i\not\equiv j\pm 1\pmod n$. [i]Proposed by Gerhard Woeginger, Netherlands[/i]

In how many ways can we fill the cells of a $m\times n$ board with $+1$ and $-1$ such that the product of numbers on each line and on each column are all equal to $-1$?

Five real numbers of absolute values not greater than $1$ and having the sum equal to $1$ are written on the circumference of a circle. Prove that one can choose three consecutively disposed numbers $a, b, c$, such that all the sums $a + b,b + c$ and $a + b + c$ are nonnegative.

A string of letters is called $good$ if it contains a continuous substring $IMONST$ in it. For example, the string $NSIMONSTIM$ is $good$, but the string $IMONNNST$ is not. Find the number of good strings consisting of $12$ letters from $I$, $M$, $O$, $N$, $S$, $T$ only.

Consider an $n\times n$ matrix whose entry at the intersection of the $i$-th row and the $j-$th column equals $i+ j -1$. What is the largest possible value of the product of $n$ entries of the matrix, no two of which are in the same row or column?

There are $2000$ components in a circuit, every two of which were initially joined by a wire. The hooligans Vasya and Petya cut the wires one after another. Vasya, who starts, cuts one wire on his turn, while Petya cuts one or three. The hooligan who cuts the last wire from some component loses. Who has the winning strategy?

A square is divided into rectangles. A "chain" is a subset $K$ of the set of these rectangles such that there exists a side of the square which is covered by projections of rectangles of $K$ and such that no point of this side is a projection of two inner points of two inner points of two different rectangles of $K$. (a) Prove that every two rectangles in such a division are members of a certain "chain". (b) Solve the similar problem for a cube, divided into rectangular parallelopipeds (in the definition of chain , replace "side" by"edge") . (A.I . Golberg, V.A. Gurevich)

Given a simple, connected graph with $n$ vertices and $m$ edges. Prove that one can find at least $m$ ways separating the set of vertices into two parts, such that the induced subgraphs on both parts are connected.

[i]20 problems for 25 minutes.[/i] [b]p1.[/b] Compute $\frac{2024}{2 + 0 \times 2 - 4}.$ [b]p2.[/b] Find the smallest integer that can be written as the product of three distinct positive odd integers. [b]p3.[/b] Bryan’s physics test score is a two-digit number. When Bryan reverses its digits and adds the tens digit of his test score, he once again obtains his test score. Determine Bryan’s physics test score. [b]p4.[/b] Grant took four classes today. He spent $70$ minutes in math class. Had his math class been $40$ minutes instead, he would have spent $15\%$ less total time in class today. Find how many minutes he spent in his other classes combined. [b]p5.[/b] Albert’s favorite number is a nonnegative integer. The square of Albert’s favorite number has $9$ digits. Find the number of digits in Albert’s favorite number. [b]p6.[/b] Two semicircular arcs are drawn in a rectangle, splitting it into four regions as shown below. Given the areas of two of the regions, find the area of the entire rectangle. [img]https://cdn.artofproblemsolving.com/attachments/1/a/22109b346c7bdadeaf901d62155de4c506b33c.png[/img] [b]p7.[/b] Daria is buying a tomato and a banana. She has a $20\%$-off coupon which she may use on one of the two items. If she uses it on the tomato, she will spend $\$1.21$ total, and if she uses it on the banana, she will spend $\$1.31$ total. In cents, find the absolute difference between the price of a tomato and the price of a banana. [b]p8.[/b] Celine takes an $8\times 8$ checkerboard of alternating black and white unit squares and cuts it along a line, creating two rectangles with integer side lengths, each of which contains at least $9$ black squares. Find the number of ways Celine can do this. (Rotations and reflections of the cut are considered distinct.) [b]p9.[/b] Each of the nine panes of glass in the circular window shown below has an area of $\pi$, eight of which are congruent. Find the perimeter of one of the non-circular panes. [img]https://cdn.artofproblemsolving.com/attachments/b/c/0d3644dde33b68f186ba1ff0602e08ce7996f5.png[/img] [b]p10.[/b] In Alan’s favorite book, pages are numbered with consecutive integers starting with $1$. The average of the page numbers in Chapter Five is $95$ and the average of the page numbers in Chapter Six is $114$. Find the number of pages in Chapters Five and Six combined. [b]p11.[/b] Find the number of ordered pairs $(a, b)$ of positive integers such that $a + b = 2024$ and $$\frac{a}{b}>\frac{1000}{1025}.$$ [b]p12.[/b] A square is split into three smaller rectangles $A$, $B$, and $C$. The area of $A$ is $80$, $B$ is a square, and the area of $C$ is $30$. Compute the area of $B$. [img]https://cdn.artofproblemsolving.com/attachments/d/5/43109b964eacaddefd410ddb8bf4e4354a068b.png[/img] [b]p13.[/b] A knight on a chessboard moves two spaces horizontally and one space vertically, or two spaces vertically and one space horizontally. Two knights attack each other if each knight can move onto the other knight’s square. Find the number of ways to place a white knight and a black knight on an $8 \times 8$ chessboard so that the two knights attack each other. One such possible configuration is shown below. [img]https://cdn.artofproblemsolving.com/attachments/2/2/b4a83fbbab7e54dda81ac5805728d268b6db9f.png[/img] [b]p14.[/b] Find the sum of all positive integers $N$ for which the median of the positive divisors of $N$ is $9$. [b]p15.[/b] Let $x$, $y$, and $z$ be nonzero real numbers such that $$\begin{cases} 20x + 24y = yz \\ 20y + 24x = xz \end{cases}$$ Find the sum of all possible values of $z$. [b]p16.[/b] Ava glues together $9$ standard six-sided dice in a $3 \times 3$ grid so that any two touching faces have the same number of dots. Find the number of dots visible on the surface of the resulting shape. (On a standard six-sided die, opposite faces sum to $7$.) [img]https://cdn.artofproblemsolving.com/attachments/5/5/bc71dac9b8ae52a4456154000afde2c89fd83a.png[/img] [b]p17.[/b] Harini has a regular octahedron of volume $1$. She cuts off its $6$ vertices, turning the triangular faces into regular hexagons. Find the volume of the resulting solid. [b]p18.[/b] Each second, Oron types either $O$ or $P$ with equal probability, forming a growing sequence of letters. Find the probability he types out $POP$ before $OOP$. [b]p19.[/b] For an integer $n \ge 10$, define $f(n)$ to be the number formed after removing the first digit from $n$ (and removing any leading zeros) and define $g(n)$ to be the number formed after removing the last digit from $n$. Find the sum of the solutions to the equation $f(n) + g(n) = 2024$. [b]p20.[/b] In convex trapezoid $ABCD$ with $\overline{AB} \parallel \overline{CD}$ and $AD = BC$, let $M$ be the midpoint of $\overline{BC}$. If $\angle AMB = 24^o$ and $\angle CMD = 66^o$, find $\angle ABC$, in degrees. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $a_1,a_2,...,a_9$ be nine real numbers, not necessarily distinct, with average $m$. Let $A$ denote the number of triples $1 \le i < j < k \le 9$ for which $a_i + a_j + a_k \ge 3m$. What is the minimum possible value of $A$? [i]Proposed by Ray Li[/i]

p1. The pattern $ABCCCDDDDABBCCCDDDDABBCCCDDDD...$ repeats to infinity. Which letter ranks in place $2533$ ? p2. Prove that if $a > 2$ and $b > 3$ then $ab + 6 > 3a + 2b$. p3. Given a rectangle $ABCD$ with size $16$ cm $\times 25$ cm, $EBFG$ is kite, and the length of $AE = 5$ cm. Determine the length of $EF$. [img]https://cdn.artofproblemsolving.com/attachments/2/e/885af838bcf1392eb02e2764f31ae83cb84b78.png[/img] p4. Consider the following series of statements. It is known that $x = 1$. Since $x = 1$ then $x^2 = 1$. So $x^2 = x$. As a result, $x^2 - 1 = x- 1$ $(x -1) (x + 1) = (x - 1) \cdot 1$ Using the rule out, we get $x + 1 = 1$ $1 + 1 = 1$ $2 = 1$ The question. a. If $2 = 1$, then every natural number must be equal to $ 1$. Prove it. b. The result of $2 = 1$ is something that is impossible. Of course there's something wrong in the argument above? Where is the fault? Why is that you think wrong? p5. To calculate $\sqrt{(1998)(1996)(1994)(1992)+16}$ . someone does it in a simple way as follows: $2000^2-2 \times 5\times 2000 + 5^2 - 5$? Is the way that person can justified? Why? p6. To attract customers, a fast food restaurant give gift coupons to everyone who buys food at the restaurant with a value of more than $25,000$ Rp.. Behind every coupon is written one of the following numbers: $9$, $12$, $42$, $57$, $69$, $21$, 15, $75$, $24$ and $81$. Successful shoppers collect coupons with the sum of the numbers behind the coupon is equal to 100 will be rewarded in the form of TV $21''$. If the restaurant owner provides as much as $10$ $21''$ TV pieces, how many should be handed over to the the customer? p7. Given is the shape of the image below. [img]https://cdn.artofproblemsolving.com/attachments/4/6/5511d3fb67c039ca83f7987a0c90c652b94107.png[/img] The centers of circles $B$, $C$, $D$, and $E$ are placed on the diameter of circle $A$ and the diameter of circle $B$ is the same as the radius of circle $A$. Circles $C$, $D$, and $E$ are equal and the pairs are tangent externally such that the sum of the lengths of the diameters of the three circles is the same with the radius of the circle $A$. What is the ratio of the circumference of the circle $A$ with the sum of the circumferences of circles $B$, $C$, $D$, and $E$? p8. It is known that $a + b + c = 0$. Prove that $a^3 + b^3 + c^3 = 3abc$.

In each unit square of an infinite square grid a natural number is written. The polygons of area $n$ with sides going along the gridlines are called [i]admissible[/i], where $n > 2$ is a given natural number. The [i]value [/i] of an admissible polygon is defined as the sum of the numbers inside it. Prove that if the values of any two congruent admissible polygons are equal, then all the numbers written in the unit squares of the grid are equal. (We recall that a symmetric image of polygon $P$ is congruent to $P$.)

What is the smallest number of sides that an polygon can have (not necessarily convex), which can be cut into parallelograms?

Let $N=\overline{abc}$ be a three-digit number. It is known that we can construct an isoceles triangle with $a,b$ and $c$ as the length of sides. Determine how many possible three-digit number $N$ there are. ($N=\overline{abc}$ means that $a,b$ and $c$ are digits of $N$, and not $N=a\times b\times c$.)

Submit a positive integer $n$ less than $10^5$. Let the sum of the valid submissions from all teams to this question be $S$. If you submit an invalid answer, you will receive $0$ points. Otherwise, your score will be $ \max \left(0,\lfloor 25 - \frac{|S'-n|}{10} \rfloor \right)$ , where $S'$ is the sum of the squares of the digits of $S$.

In the coordinate plane, a set of $2000$ points $\{(x_1, y_1), (x_2, y_2), . . . , (x_{2000}, y_{2000})\}$ is called [i]good[/i] if $0\leq x_i \leq 83$, $0\leq y_i \leq 83$ for $i = 1, 2, \dots, 2000$ and $x_i \not= x_j$ when $i\not=j$. Find the largest positive integer $n$ such that, for any good set, the interior and boundary of some unit square contains exactly $n$ of the points in the set on its interior or its boundary.

In a simple graph $G$, an operation is defined as taking two neighbor vertices $u,v$ which have a common neighbor, deleting the edge between $u,v$ and adding a new vertex $w$ whose neighbors are exactly the common neighbors of $u$ and $v$. Starting with the complete graph $G=K_n$ where $n\ge 3$ is a positive integer, find the maximum number of operations that can be applied. Proposed by[i] Deniz Can Karaçelebi[/i]

[b]p1.[/b] A Slavic dragon has three heads. A knight fights the dragon. If the knight cuts off one dragon’s head three new heads immediately grow. Is it possible that the dragon has $2018$ heads at some moment of the fight? [b]p2.[/b] Peter has two squares $3\times 3$ and $4\times 4$. He must cut one of them or both of them in no more than four parts in total. Is Peter able to assemble a square using all these parts? [b]p3.[/b] Usually, dad picks up Constantine after his music lessons and they drive home. However, today the lessons have ended earlier and Constantine started walking home. He met his dad $14$ minutes later and they drove home together. They arrived home $6$ minutes earlier than usually. Home many minutes earlier than usual have the lessons ended? Please, explain your answer. [b]p4.[/b] All positive integers from $1$ to $2018$ are written on a blackboard. First, Peter erased all numbers divisible by $7$. Then, Natalie erased all remaining numbers divisible by $11$. How many numbers did Natalie remove? Please, explain your answer. [b]p5.[/b] $30$ students took part in a mathematical competition consisting of four problems. $25$ students solved the first problem, $24$ students solved the second problem, $22$ students solved the third, and, finally, $21$ students solved the fourth. Show that there are at least two students who solved all four problems. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

For a positive integer $n$, there are two countries $A$ and $B$ with $n$ airports each and $n^2-2n+ 2$ airlines operating between the two countries. Each airline operates at least one flight. Exactly one flight by one of the airlines operates between each airport in $A$ and each airport in $B$, and that flight operates in both directions. Also, there are no flights between two airports in the same country. For two different airports $P$ and $Q$, denote by "[i]$(P, Q)$-travel route[/i]" the list of airports $T_0, T_1, \ldots, T_s$ satisfying the following conditions. [list] [*] $T_0=P,\ T_s=Q$ [*] $T_0, T_1, \ldots, T_s$ are all distinct. [*] There exists an airline that operates between the airports $T_i$ and $T_{i+1}$ for all $i = 0, 1, \ldots, s-1$. [/list] Prove that there exist two airports $P, Q$ such that there is no or exactly one [i]$(P, Q)$-travel route[/i]. [hide=Graph Wording]Consider a complete bipartite graph $G(A, B)$ with $\vert A \vert = \vert B \vert = n$. Suppose there are $n^2-2n+2$ colors and each edge is colored by one of these colors. Define $(P, Q)-path$ a path from $P$ to $Q$ such that all of the edges in the path are colored the same. Prove that there exist two vertices $P$ and $Q$ such that there is no or only one $(P, Q)-path$. [/hide]

[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].