What is the maximal number of knights that can be placed on the usual 8x8 chessboard so that each of then threatens at most 7 others?

The country of HMMTLand has $8$ cities. Its government decides to construct several two-way roads between pairs of distinct cities. After they finish construction, it turns out that each city can reach exactly $3$ other cities via a single road, and from any pair of distinct cities, either exactly $0$ or $2$ other cities can be reached from both cities by a single road. Compute the number of ways HMMTLand could have constructed the roads.

Among the $n$ inhabitants of an island, where $n$ is even, every two are either friends or enemies. Some day, the chief of the island orders that each inhabitant (including himself) makes and wears a necklace consisting of marbles, in such a way that two necklaces have a marble of the same type if and only if their owners are friends. (a) Show that the chief’s order can be achieved by using $n^2/4$ different types of stones. (b) Prove that this is not necessarily true with less than $n^2/4$ types.

Prove that for all $n \ge 3$ there are an infinite number of $n$-sided polygonal numbers which are also the sum of two other (not necessarily different) $n$-sided polygonal numbers! The first $n$-sided polygonal number is $1$. The kth n-sided polygonal number for $k \ge 2$ is the number of different points in a figure that consists of all of the regular $n$-sided polygons which have one common vertex, are oriented in the same direction from that vertex and their sides are $\ell$ cm long where $1 \le \ell \le k - 1$ cm and $\ell$ is an integer. [i]In this figure, what we call points are the vertices of the polygons and the points that break up the sides of the polygons into exactly $1$ cm long segments. For example, the first four pentagonal numbers are 1,5,12, and 22, like it is shown in the figure.[/i] [img]https://cdn.artofproblemsolving.com/attachments/1/4/290745d4be1888813678127e6d63b331adaa3d.png[/img]

Find the number of functions $f : \{1, 2, . . . , n\} \to \{1995, 1996\}$ such that $f(1) + f(2) + ... + f(1996)$ is odd.

One hundred and one of the squares of an $n\times n$ table are colored blue. It is known that there exists a unique way to cut the table to rectangles along boundaries of its squares with the following property: every rectangle contains exactly one blue square. Find the smallest possible $n$.

Suppose $X$ is a set with $|X| = 56$. Find the minimum value of $n$, so that for any 15 subsets of $X$, if the cardinality of the union of any 7 of them is greater or equal to $n$, then there exists 3 of them whose intersection is nonempty.

At a two-day team competition in chess, three schools with $15$ pupils each attend. Each student plays one game against each player on the other two teams, ie a total of $30$ chess games per student. a) Is it possible for each student to play exactly $15$ games after the first day? b) Show that it is possible for each student to play exactly $16$ games after the first day. c) Assume that each student has played exactly $16$ games after the first day. Show that there are three students, one from each school, who have played their three parties

Consider all products by $2, 4, 6, ..., 2000$ of the elements of the set $A =\left\{\frac12, \frac13, \frac14,...,\frac{1}{2000},\frac{1}{2001}\right\}$ . Find the sum of all these products.

Given positive integers $a,b,$ find the least positive integer $m$ such that among any $m$ distinct integers in the interval $[-a,b]$ there are three pair-wise distinct numbers that their sum is zero. [i]Proposed by Marian Tetiva, Romania[/i]

Today is Barbara's birthday, and Alberto wants to give her a gift playing the following game. The numbers 0,1,2,...,1024 are written on a blackboard. First Barbara erases $2^{9}$ numbers, then Alberto erases $2^{8}$ numbers, then Barbara $2^{7}$ and so on, until there are only two numbers a,b left. Now Barbara earns $|a-b|$ euro. Find the maximum number of euro that Barbara can always win, independently of Alberto's strategy.

All members of the senate were firstly divided into $S$ senate commissions . According to the rules, no commission has less that $5$ senators and every two commissions have different number of senators. After the first session the commissions were closed and new commissions were opened. Some of the senators now are not a part of any commission. It resulted also that every two senators that were in the same commission in the first session , are not any more in the same commission. [b](a)[/b]Prove that at least $4S+10$ senators were left outside the commissions. [b](b)[/b]Prove that this number is achievable. Albanian National Mathematical Olympiad 2010---12 GRADE Question 5.

Let $k$ be a positive integer. The organising commitee of a tennis tournament is to schedule the matches for $2k$ players so that every two players play once, each day exactly one match is played, and each player arrives to the tournament site the day of his first match, and departs the day of his last match. For every day a player is present on the tournament, the committee has to pay $1$ coin to the hotel. The organisers want to design the schedule so as to minimise the total cost of all players' stays. Determine this minimum cost.

There are 100 merchants who are selling salmon for Durer dollars around the circular shore of the island of Durerland. Since the beginning of times good and bad years have been alternating on the island. (So after a good year, the next year is bad; and after a bad year, the next year is good.) In every good year all merchants set their price as the maximum value between their own selling price from the year before and the selling price of their left-hand neighbour from the year before. In turn, in every bad year they sell it for the minimum between their own price from the year before and their left-hand neighbour’s price from the year before. Paul and Pauline are two merchants on the island. This year Paul is selling salmon for 17 Durer dollars a kilogram. Prove that there will come a year when Pauline will sell salmon for 17 Durer dollars a kilogram. [i]Note: The merchants are immortal, they have been selling salmon on the island for thousands of years and will continue to do so until the end of time.[/i]

Divide a regular $8960$-gon into non-overlapping parallelograms. Suppose that $R$ of these parallelograms are rectangles. What is the minimum possible value of $R$?

[b]p1.[/b] Nicky is studying biology and has a tank of $17$ lizards. In one day, he can either remove $5$ lizards or add $2$ lizards to his tank. What is the minimum number of days necessary for Nicky to get rid of all of the lizards from his tank? [b]p2.[/b] What is the maximum number of spheres with radius $1$ that can fit into a sphere with radius $2$? [b]p3.[/b] A positive integer $x$ is sunny if $3x$ has more digits than $x$. If all sunny numbers are written in increasing order, what is the $50$th number written? [b]p4.[/b] Quadrilateral $ABCD$ satisfies $AB = 4$, $BC = 5$, $DA = 4$, $\angle DAB = 60^o$, and $\angle ABC = 150^o$. Find the area of $ABCD$. [b]p5. [/b]Totoro wants to cut a $3$ meter long bar of mixed metals into two parts with equal monetary value. The left meter is bronze, worth $10$ zoty per meter, the middle meter is silver, worth $25$ zoty per meter, and the right meter is gold, worth $40$ zoty per meter. How far, in meters, from the left should Totoro make the cut? [b]p6.[/b] If the numbers $x_1, x_2, x_3, x_4$, and $x5$ are a permutation of the numbers $1, 2, 3, 4$, and $5$, compute the maximum possible value of $$|x_1 - x_2| + |x_2 - x_3| + |x_3 - x_4| + |x_4 - x_5|.$$ [b]p7.[/b] In a $3 \times 4$ grid of $12$ squares, find the number of paths from the top left corner to the bottom right corner that satisfy the following two properties: $\bullet$ The path passes through each square exactly once. $\bullet$ Consecutive squares share a side. Two paths are considered distinct if and only if the order in which the twelve squares are visited is different. For instance, in the diagram below, the two paths drawn are considered the same. [img]https://cdn.artofproblemsolving.com/attachments/7/a/bb3471bbde1a8f58a61d9dd69c8527cfece05a.png[/img] [b]p8.[/b] Scott, Demi, and Alex are writing a computer program that is $25$ ines long. Since they are working together on one computer, only one person may type at a time. To encourage collaboration, no person can type two lines in a row, and everyone must type something. If Scott takes $10$ seconds to type one line, Demi takes $15$ seconds, and Alex takes $20$ seconds, at least how long, in seconds, will it take them to finish the program? [b]p9.[/b] A hand of four cards of the form $(c, c, c + 1, c + 1)$ is called a tractor. Vinjai has a deck consisting of four of each of the numbers $7$, $8$, $9$ and $10$. If Vinjai shuffles and draws four cards from his deck, compute the probability that they form a tractor. [b]p10. [/b]The parabola $y = 2x^2$ is the wall of a fortress. Totoro is located at $(0, 4)$ and fires a cannonball in a straight line at the closest point on the wall. Compute the y-coordinate of the point on the wall that the cannonball hits. [b]p11. [/b]How many ways are there to color the squares of a $10$ by $10$ grid with black and white such that in each row and each column there are exactly two black squares and between the two black squares in a given row or column there are exactly [b]4[/b] white squares? Two configurations that are the same under rotations or reflections are considered different. [b]p12.[/b] In rectangle $ABCD$, points $E$ and $F$ are on sides $AB$ and $CD$, respectively, such that $AE = CF > AD$ and $\angle CED = 90^o$. Lines $AF, BF, CE$ and $DE$ enclose a rectangle whose area is $24\%$ of the area of $ABCD$. Compute $\frac{BF}{CE}$ . [b]p13.[/b] Link cuts trees in order to complete a quest. He must cut $3$ Fenwick trees, $3$ Splay trees and $3$ KD trees. If he must also cut 3 trees of the same type in a row at some point during his quest, in how many ways can he cut the trees and complete the quest? (Trees of the same type are indistinguishable.) [b]p14.[/b] Find all ordered pairs (a, b) of positive integers such that $\sqrt{64a + b^2} + 8 = 8\sqrt{a} + b$. [b]p15.[/b] Let $ABCDE$ be a convex pentagon such that $\angle ABC = \angle BCD = 108^o$, $\angle CDE = 168^o$ and $AB =BC = CD = DE$. Find the measure of $\angle AEB$ PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $n > 1$ be an integer. In a circular arrangement of $n$ lamps $L_0, \ldots, L_{n-1},$ each of of which can either ON or OFF, we start with the situation where all lamps are ON, and then carry out a sequence of steps, $Step_0, Step_1, \ldots .$ If $L_{j-1}$ ($j$ is taken mod $n$) is ON then $Step_j$ changes the state of $L_j$ (it goes from ON to OFF or from OFF to ON) but does not change the state of any of the other lamps. If $L_{j-1}$ is OFF then $Step_j$ does not change anything at all. Show that: (i) There is a positive integer $M(n)$ such that after $M(n)$ steps all lamps are ON again, (ii) If $n$ has the form $2^k$ then all the lamps are ON after $n^2-1$ steps, (iii) If $n$ has the form $2^k + 1$ then all lamps are ON after $n^2 - n + 1$ steps.

Let $k$ be a positive integer. Show that for all $n>k$ there exist convex figures $F_{1},\ldots, F_{n}$ and $F$ such that there doesn't exist a subset of $k$ elements from $F_{1},..., F_{n}$ and $F$ is covered for this elements, but $F$ is covered for every subset of $k+1$ elements from $F_{1}, F_{2},....., F_{n}$.

Consider the graph $G$ with $100$ vertices, and the minimum odd cycle goes through $13$ vertices. Prove that the vertices of the graph can be colored in $6$ colors in a way that no two adjacent vertices have the same color.

Each of the squares of an $8 \times 8$ board can be colored white or black. Find the number of colorings of the board such that every $2 \times 2$ square contains exactly 2 black squares and 2 white squares.

In a heap there are $2021$ stones. Two players $A$ and $B$ play removing stones of the pile, alternately starting with $A$. A valid move for $A$ consists of remove $1, 2$ or $7$ stones. A valid move for B is to remove $1, 3, 4$ or $6$ stones. The player who leaves the pile empty after making a valid move wins. Determine if some of the players have a winning strategy. If such a strategy exists, explain it.

The numbers $1, 2, \dots , n$ are written on a blackboard and then erased via the following process:[list] [*] Before any numbers are erased, a pair of numbers is chosen uniformly at random and circled. [*] Each minute for the next $n -1$ minutes, a pair of numbers still on the blackboard is chosen uniformly at random and the smaller one is erased. [*] In minute $n$, the last number is erased. [/list] What is the probability that the smaller circled number is erased before the larger? [i]Proposed by Ethan Tan[/i]

Each side of a triangle $ABC$ has been divided into $n+1$ equal parts. Find the number of triangles with the vertices at the division points having no side parallel to or lying at a side of $\triangle ABC$.

Given a set $X=\{x_1,\ldots,x_n\}$ of natural numbers in which for all $1< i \leq n$ we have $1\leq x_i-x_{i-1}\leq 2$, call a real number $a$ [b]good[/b] if there exists $1\leq j \leq n$ such that $2|x_j-a|\leq 1$. Also a subset of $X$ is called [b]compact[/b] if the average of its elements is a good number. Prove that at least $2^{n-3}$ subsets of $X$ are compact. [i]Proposed by Mahyar Sefidgaran[/i]

Jeff has a deck of $12$ cards: $4$ $L$s, $4$ $M$s, and $4$ $T$s. Armaan randomly draws three cards without replacement. The probability that he takes $3$ $L$s can be written as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m +n$.