Some of the people attending a meeting greet each other. Let $n$ be the number of people who greet an odd number of people. Prove that $n$ is even.

The sets $A_1,A_2,...,A_n$ are finite. With $d$ we denote the number of elements in $\bigcup_{i=1}^n A_i$ which are in odd number of the sets $A_i$. Prove that the number: $D(k)=d-\sum_{i=1}^n|A_i|+2\sum_{i<j}|A_i\cap A_j |+...+(-1)^k2^{k-1}\sum_{i_1<i_2<...<i_k}|A_{i_1}\cap A_{i_2}\cap ...\cap A_{i_k}|$ is divisible by $2^k$.

In a group of $90$ children each has at least $30$ friends (friendship is mutual). Prove that they can be divided into three $30$-member groups so that each child has its own a group of at least one friend.

Given a set $ \mathcal{H}$ of points in the plane, $ P$ is called an "intersection point of $ \mathcal{H}$" if distinct points $ A,B,C,D$ exist in $ \mathcal{H}$ such that lines $ AB$ and $ CD$ are distinct and intersect in $ P$. Given a finite set $ \mathcal{A}_{0}$ of points in the plane, a sequence of sets is defined as follows: for any $ j\geq0$, $ \mathcal{A}_{j+1}$ is the union of $ \mathcal{A}_{j}$ and the intersection points of $ \mathcal{A}_{j}$. Prove that, if the union of all the sets in the sequence is finite, then $ \mathcal{A}_{i}=\mathcal{A}_{1}$ for any $ i\geq1$.

Given $m,n\geq 2$.Paint each cell of a $m\times n$ board $S$ red or blue so that:for any two red cells in a row,one of the two columns they belong to is all red,and the other column has at least one blue cell in it.Find the number of ways to paint $S$ like this.

Each integer is colored with one of two colors, red or blue. It is known that, for every finite set $A$ of consecutive integers, the absolute value of the difference between the number of red and blue integers in the set $A$ is at most $1000$. Prove that there exists a set of $2000$ consecutive integers in which there are exactly $1000$ red numbers and $1000$ numbers blue.

Prove the following theorem: If there are $n$ pairs of different points $P_i$, $i = 1, 2, ..., n$, $n > 2$ in three dimensions space, such that each of them is at a smaller distance from one and the same point $Q$ than any other $P_i$, then $n < 15$.

[b]p1.[/b] Compute $17^2 + 17 \cdot 7 + 7^2$. [b]p2.[/b] You have $\$1.17$ in the minimum number of quarters, dimes, nickels, and pennies required to make exact change for all amounts up to $\$1.17$. How many coins do you have? [b]p3.[/b] Suppose that there is a $40\%$ chance it will rain today, and a $20\%$ chance it will rain today and tomorrow. If the chance it will rain tomorrow is independent of whether or not it rained today, what is the probability that it will rain tomorrow? (Express your answer as a percentage.) [b]p4.[/b] A number is called boxy if the number of its factors is a perfect square. Find the largest boxy number less than $200$. [b]p5.[/b] Alice, Bob, Carl, and Dave are either lying or telling the truth. If the four of them make the following statements, who has the coin? [i]Alice: I have the coin. Bob: Carl has the coin. Carl: Exactly one of us is telling the truth. Dave: The person who has the coin is male.[/i] [b]p6.[/b] Vicky has a bag holding some blue and some red marbles. Originally $\frac23$ of the marbles are red. After Vicky adds $25$ blue marbles, $\frac34$ of the marbles are blue. How many marbles were originally in the bag? [b]p7.[/b] Given pentagon $ABCDE$ with $BC = CD = DE = 4$, $\angle BCD = 90^o$ and $\angle CDE = 135^o$, what is the length of $BE$? [b]p8.[/b] A Berkeley student decides to take a train to San Jose, stopping at Stanford along the way. The distance from Berkeley to Stanford is double the distance from Stanford to San Jose. From Berkeley to Stanford, the train's average speed is $15$ meters per second. From Stanford to San Jose, the train's average speed is $20$ meters per second. What is the train's average speed for the entire trip? [b]p9.[/b] Find the area of the convex quadrilateral with vertices at the points $(-1, 5)$, $(3, 8)$, $(3,-1)$, and $(-1,-2)$. [b]p10.[/b] In an arithmetic sequence $a_1$, $a_2$, $a_3$, $...$ , twice the sum of the first term and the third term is equal to the fourth term. Find $a_4/a_1$. [b]p11.[/b] Alice, Bob, Clara, David, Eve, Fred, Greg, Harriet, and Isaac are on a committee. They need to split into three subcommittees of three people each. If no subcommittee can be all male or all female, how many ways are there to do this? [b]p12.[/b] Usually, spaceships have $6$ wheels. However, there are more advanced spaceships that have $9$ wheels. Aliens invade Earth with normal spaceships, advanced spaceships, and, surprisingly, bicycles (which have $2$ wheels). There are $10$ vehicles and $49$ wheels in total. How many bicycles are there? [b]p13.[/b] If you roll three regular six-sided dice, what is the probability that the three numbers showing will form an arithmetic sequence? (The order of the dice does matter, but we count both $(1,3, 2)$ and $(1, 2, 3)$ as arithmetic sequences.) [b]p14.[/b] Given regular hexagon $ABCDEF$ with center $O$ and side length $6$, what is the area of pentagon $ABODE$? [b]p15.[/b] Sophia, Emma, and Olivia are eating dinner together. The only dishes they know how to make are apple pie, hamburgers, hotdogs, cheese pizza, and ice cream. If Sophia doesn't eat dessert, Emma is vegetarian, and Olivia is allergic to apples, how many dierent options are there for dinner if each person must have at least one dish that they can eat? [b]p16.[/b] Consider the graph of $f(x) = x^3 + x + 2014$. A line intersects this cubic at three points, two of which have $x$-coordinates $20$ and $14$. Find the $x$-coordinate of the third intersection point. [b]p17.[/b] A frustum can be formed from a right circular cone by cutting of the tip of the cone with a cut perpendicular to the height. What is the surface area of such a frustum with lower radius $8$, upper radius $4$, and height $3$? [b]p18.[/b] A quadrilateral $ABCD$ is dened by the points $A = (2,-1)$, $B = (3, 6)$, $C = (6, 10)$ and $D = (5,-2)$. Let $\ell$ be the line that intersects and is perpendicular to the shorter diagonal at its midpoint. What is the slope of $\ell$? [b]p19.[/b] Consider the sequence $1$, $1$, $2$, $2$, $3$, $3$, $3$, $5$, $5$, $5$, $5$, $5$, $...$ where the elements are Fibonacci numbers and the Fibonacci number $F_n$ appears $F_n$ times. Find the $2014$th element of this sequence. (The Fibonacci numbers are defined as $F_1 = F_2 = 1$ and for $n > 2$, $F_n = F_{n-1}+F_{n-2}$.) [b]p20.[/b] Call a positive integer top-heavy if at least half of its digits are in the set $\{7, 8, 9\}$. How many three digit top-heavy numbers exist? (No number can have a leading zero.) PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $n$ be a positive integer. Two players $A$ and $B$ play a game in which they take turns choosing positive integers $k \le n$. The rules of the game are: (i) A player cannot choose a number that has been chosen by either player on any previous turn. (ii) A player cannot choose a number consecutive to any of those the player has already chosen on any previous turn. (iii) The game is a draw if all numbers have been chosen; otherwise the player who cannot choose a number anymore loses the game. The player $A$ takes the first turn. Determine the outcome of the game, assuming that both players use optimal strategies. [i]Proposed by Finland[/i]

Unconventional dice are to be designed such that the six faces are marked with numbers from $1$ to $6$ with $1$ and $2$ appearing on opposite faces. Further, each face is colored either red or yellow with opposite faces always of the same color. Two dice are considered to have the same design if one of them can be rotated to obtain a dice that has the same numbers and colors on the corresponding faces as the other one. Find the number of distinct dice that can be designed.

How many possible arrangements of bishops are there on a $8 \times 8$ chessboard such that no bishop threatens a square on which another lies and the maximum number of bishops are used? (Note that a bishop threatens any square along a diagonal containing its square.)

In a certain language words are formed using an alphabet of three letters. Some words of two or more letters are not allowed, and any two such distinct words are of different lengths. Prove that one can form a word of arbitrary length that does not contain any non-allowed word.

In a video game, there is a board divided into squares, with $27$ rows and $27$ columns. The squares are painted alternately in black, gray and white as follows: $\bullet$ in the first row, the first square is black, the next is gray, the next is white, the next is black, and so on; $\bullet$ in the second row, the first is white, the next is black, the next is gray, the next is white, and so on; $\bullet$ in the third row, the order would be gray-white-black-gray and so on; $\bullet$ the fourth row is painted the same as the first, the fifth the same as the second, $\bullet$ the sixth the same as the third, and so on. In the box in row $i$ and column $j$, there are $ij$ coins. For example, in the box in row $15$ and column $20$ there are $(15) (20) = 300$ coins. Verify that in total there are, in the black squares, $9^2 (13^2 + 14^2 + 15^2)$ coins.

Let $n_1, n_2$ be positive integers. Consider in a plane $E$ two disjoint sets of points $M_1$ and $M_2$ consisting of $2n_1$ and $2n_2$ points, respectively, and such that no three points of the union $M_1 \cup M_2$ are collinear. Prove that there exists a straightline $g$ with the following property: Each of the two half-planes determined by $g$ on $E$ ($g$ not being included in either) contains exactly half of the points of $M_1$ and exactly half of the points of $M_2.$

Write either $1$ or $-1$ in each of the cells of a $(2n) \times (2n)$-table, in such a way that there are exactly $2n^2$ entries of each kind. Let the minimum of the absolute values of all row sums and all column sums be $M$. Determine the largest possible value of $M$.

Pete puts 100 stones in a row: black one, white one, black one, white one, ..., black one, white one. In a single move either Pete chooses two black stones with only white stones between them, and repaints all these white stones in black, or Pete chooses two white stones with only black stones between them, and repaints all these black stones in white. Can Pete with a sequence of moves described above obtain a row of 50 black stones followed by 50 white stones? Egor Bakaev

Let $ N \equal{} \{1,2 \ldots, n\}, n \geq 2.$ A collection $ F \equal{} \{A_1, \ldots, A_t\}$ of subsets $ A_i \subseteq N,$ $ i \equal{} 1, \ldots, t,$ is said to be separating, if for every pair $ \{x,y\} \subseteq N,$ there is a set $ A_i \in F$ so that $ A_i \cap \{x,y\}$ contains just one element. $ F$ is said to be covering, if every element of $ N$ is contained in at least one set $ A_i \in F.$ What is the smallest value $ f(n)$ of $ t,$ so there is a set $ F \equal{} \{A_1, \ldots, A_t\}$ which is simultaneously separating and covering?

A checkered polygon $A$ is drawn on the checkered plane. We call a cell of $A$ [i]internal[/i] if all $8$ of its adjacent cells belong to $A$. All other (non-internal) cells of $A$ we call [i]boundary[/i]. It is known that $1)$ each boundary cell has exactly two common sides with no boundary cells; and 2) the union of all boundary cells can be divided into isosceles trapezoid of area $2$ with vertices at the grid nodes (and acute angles of the trapezoids are equal $45^\circ$). Prove that the area of the polygon $A$ is congruent to $1$ modulo $4$.

A square with a side 1 is colored in 3 colors. What’s the greatest real number $a$ such that there can always be found 2 points of the same color at a distance $a$?

Let $n$ be a positive integer, where $n \geq 3$ and let $a_1, a_2, ..., a_n$ be the lengths of sides of some $n$-gon. Prove that $$a_1 + a_2 + ... + a_n \geq \sqrt{2 \cdot (a_1^2 + a_2^2 + ... + a_n^2)} $$

Consider a $4 \times 4$ grid of squares. We place coins in some of the grid squares so that no two coins are orthogonally adjacent, and each $2 \times 2$ square in the grid has at least one coin. How many ways are there to place the coins?

All the divisors of a) $8\cdot 10^6$ and b) $360^{10}$ are written on a board. At a move, we can take two numbers, neither of which is divisible by the other, and replace them with their greatest common divisor and lowest common multiple. At some point, we will no longer be able to perform new operations. How many different numbers will be on the board at this moment? [i]Proposed by V. Bragin[/i]

[b]7.1[/b] A rectangle that is not a square is inscribed in a square. Prove that its semi-perimeter is equal to the diagonal of the square. [b]7.2[/b] Find five numbers whose pairwise sums are 0, 2, 4,5, 7, 9, 10, 12, 14, 17. [b]7.3 [/b] In a $1000$-digit number, all but one digit is a five. Prove that this number is not a perfect square. [b]7.4 / 6.5[/b] Several teams took part in the volleyball tournament. Team A is considered stronger than team B if either A beat B or there is a team C such that A beat C, and C beat B. Prove that if team T is the winner of the tournament, then it is the strongest the rest of the teams. [b]7.5[/b] In a pentagon $ABCDE$, $K$ is the midpoint of $AB$, $L$ is the midpoint of $BC$, $M$ is the midpoint of $CD$, $N$ is the midpoint of $DE$, $P$ is the midpoint of $KM$, $Q$ is the midpoint of $LN$. Prove that the segment $ PQ$ is parallel to side $AE$ and is equal to its quarter. [img]https://cdn.artofproblemsolving.com/attachments/2/5/be8e9b0692d98115dbad04f960e8a856dc593f.png[/img] [b]7.6 / 8.4[/b] Several circles are arbitrarily placed in a circle of radius $3$, the sum of their radii is $25$. Prove that there is a straight line that intersects at least $9$ of these circles. PS. You should use hide for answers.Collected [url=https://artofproblemsolving.com/community/c3988084_1968_leningrad_math_olympiad]here[/url].

Konstantin moves a knight on a $n \times n$- chess board from the lower left corner to the lower right corner with the minimal number of moves. Then Isabelle takes the knight and moves it from the lower left corner to the upper right corner with the minimal number of moves. For which values of $n$ do they need the same number of moves?

You are given some equilateral triangles and squares, all with side length 1, and asked to form convex $n$ sided polygons using these pieces. If both types must be used, what are the possible values of $n$, assuming that there is sufficient supply of the pieces?