One day, Professor Umzar Azum decided to fry dumplings for dinner. He took out a frying pan, opened a pack of dumplings, but suddenly thought about the question: how many dumplings could he fit in his frying pan? Measuring the sizes of the frying pan and dumplings, the professor came to the conclusion that the dumplings have the shape of a semicircle, the diameter of which is four times smaller than the diameter of the frying pan. Show how on the frying pan it is possible to place (without overlap): a) $20$ pieces of dumplings; b) $24$ pieces of dumplings; . (The problem boils down to placing, without overlapping, the appropriate number of identical semicircles inside a circle with a diameter four times larger.) [i]Note: We (the authors of the problem) do not know the answer to the question whether it is possible to place 25 semicircles in a circle with a diameter four times smaller, and even more so we do not know what the largest number of such semicircles is. We will welcome any progress in solving the problem and evaluate it accordingly. [/i]

Can one paint four vertices of a cube red and the other four points black so that any plane passing through three points of the same colour contains a vertex of the other colour? (Mebius, Sharygin)

$7$ boys each went to a shop $3$ times. Each pair met at the shop. Show that $3$ must have been in the shop at the same time.

Let $ABCD$ be a square with side $4$. Find, with proof, the biggest $k$ such that no matter how we place $k$ points into $ABCD$, such that they are on the interior but not on the sides, we always have a square with sidr length $1$, which is inside the square $ABCD$, such that it contains no points in its interior(they can be on the sides).

Theseus starts at the point $(0, 0)$ in the plane. If Theseus is standing at the point $(x, y)$ in the plane, he can step one unit to the north to point $(x, y+1)$, one unit to the west to point $(x-1, y)$, one unit to the south to point $(x, y-1)$, or one unit to the east to point $(x+1, y)$. After a sequence of more than two such moves, starting with a step one unit to the south (to point $(0, -1)$), Theseus finds himself back at the point $(0, 0)$. He never visited any point other than $(0, 0)$ more than once, and never visited the point $(0, 0)$ except at the start and end of this sequence of moves. Let $X$ be the number of times that Theseus took a step one unit to the north, and then a step one unit to the west immediately afterward. Let $Y$ be the number of times that Theseus took a step one unit to the west, and then a step one unit to the north immediately afterward. Prove that $|X - Y| = 1$. [i]Mitchell Lee[/i]

A "[size=100][i]walking sequence[/i][/size]" is a sequence of integers with $a_{i+1} = a_i \pm 1$ for every $i$ .Show that there exists a sequence $b_1, b_2, . . . , b_{2016}$ such that for every walking sequence $a_1, a_2, . . . , a_{2016}$ where $1 \leq a_i \leq1010$, there is for some $j$ for which $a_j = b_j$ .

Let $N$ be a positive integer. Two persons play the following game. The first player writes a list of positive integers not greater than $25$, not necessarily different, such that their sum is at least $200$. The second player wins if he can select some of these numbers so that their sum $S$ satisfies the condition $200-N\le S\le 200+N$. What is the smallest value of $N$ for which the second player has a winning strategy?

Path graph $U$ is given, and a blindfolded player is standing on one of its vertices. The vertices of $U$ are labeled with positive integers between 1 and $n$, not necessarily in the natural order. In each step of the game, the game master tells the player whether he is in a vertex with degree 1 or with degree 2. If he is in a vertex with degree 1, he has to move to its only neighbour, if he is in a vertex with degree 2, he can decide whether he wants to move to the adjacent vertex with the lower or with the higher number. All the information the player has during the game after $k$ steps are the $k$ degrees of the vertices he visited and his choice for each step. Is there a strategy for the player with which he can decide in finitely many steps how many vertices the path has?

A convex $N\text{-gon}$ is divided by diagonals into triangles so that no two diagonals intersect inside the polygon. The triangles are painted in black and white so that any two triangles are painted in black and white so that any two triangles with a common side are painted in different colors. For each $N$ find the maximal difference between the numbers of black and white triangles.

[b]p1.[/b] A canoeist is paddling upstream in a river when she passes a log floating downstream,, She continues upstream for awhile, paddling at a constant rate. She then turns around and goes downstream and paddles twice as fast. She catches up to the same log two hours after she passed it. How long did she paddle upstream? [b]p2.[/b] Let $g(x) =1-\frac{1}{x}$ and define $g_1(x) = g(x)$ and $g_{n+1}(x) = g(g_n(x))$ for $n = 1,2,3, ...$. Evaluate $g_3(3)$ and $g_{1982}(l982)$. [b]p3.[/b] Let $Q$ denote quadrilateral $ABCD$ where diagonals $AC$ and $BD$ intersect. If each diagonal bisects the area of $Q$ prove that $Q$ must be a parallelogram. [b]p4.[/b] Given that: $a_1, a_2, ..., a_7$ and $b_1, b_2, ..., b_7$ are two arrangements of the same seven integers, prove that the product $(a_1-b_1)(a_2-b_2)...(a_7-b_7)$ is always even. [b]p5.[/b] In analyzing the pecking order in a finite flock of chickens we observe that for any two chickens exactly one pecks the other. We decide to call chicken $K$ a king provided that for any other chicken $X, K$ necks $X$ or $K$ pecks a third chicken $Y$ who in turn pecks $X$. Prove that every such flock of chickens has at least one king. Must the king be unique? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Prove that it is possible to place $2n(2n + 1)$ parallelepipedic (rectangular) pieces of soap of dimensions $1 \times 2 \times (n + 1)$ in a cubic box with edge $2n + 1$ if and only if $n$ is even or $n = 1$. [i]Remark[/i]. It is assumed that the edges of the pieces of soap are parallel to the edges of the box.

In a $m\times n$ square grid, with top-left corner is $A$, there is route along the edges of the grid starting from $A$ and visits all lattice points (called "nodes") exactly once and ending also at $A$. a. Prove that this route exists if and only if at least one of $m,\ n$ is odd. b. If such a route exists, then what is the least possible of turning points? *A turning point is a node that is different from $A$ and if two edges on the route intersect at the node are perpendicular.

For a finite set $A$ of positive integers, a partition of $A$ into two disjoint nonempty subsets $A_1$ and $A_2$ is $\textit{good}$ if the least common multiple of the elements in $A_1$ is equal to the greatest common divisor of the elements in $A_2$. Determine the minimum value of $n$ such that there exists a set of $n$ positive integers with exactly $2015$ good partitions.

Ward and Gabrielle are playing a game on a large sheet of paper. At the start of the game, there are $999$ ones on the sheet of paper. Ward and Gabrielle each take turns alternatingly, and Ward has the first turn. During their turn, a player must pick two numbers a and b on the sheet such that $gcd(a, b) = 1$, erase these numbers from the sheet, and write the number $a + b$ on the sheet. The first player who is not able to do so, loses. Determine which player can always win this game.

A $100\times \sqrt{3}$ rectangular table is given. What is the minimum number of disk-shaped napkins of radius $1$ required to cover the table completely? [i]Remark:[/i] The napkins are allowed to overlap and protrude the table's edges.

Graph $G$ has $n\geq 2$ vertices. Find the largest $m$ such that one of the following is true for always: 1. There exists a cycle with $k\geq m$ vertices. 2. There exists an independent set with $m$ vertices.

Given a natural number $ n, $ prove that the set $ \{ -n+1,-n+2,\ldots ,-1,1,2,\ldots ,n-1,n\} $ can be partitioned into $ k $ subsets such that the sums of all elements of each of these subsets are equal, if and only if $ n $ is multiple of $ k. $

If the probability that the sum of three distinct integers between $16$ and $30$ (inclusive) is even can be written as $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers, find $m + n$.

In an attempt to beat the Belgian handshake record come on $20/09/2009$ exactly $2009$ Belgians together in a large sports hall. Among them are Nathalie and thomas. During this event, everyone shakes hands with everyone exactly once other attendees. Afterwards, Nathalie says: “I have exactly $5$ times as many Flemish people shaken hands as people from Brussels.” Thomas replies with “I have exactly $3$ times as much Walloons and Brussels people shook hands”. From which region does Nathalie come and from which region comes Thomas?

At a party there were some couples attending. As they arrive each person gets to talk with all the other persons which are [i]already [/i] in the room. During the party, after all the guests arrive, groups of persons form, such that no two persons forming a couple belong to the same group, and for each two persons that do not form a couple, there is one and only one group to which both belong. Find the number of couples attending the party, knowing that there are less groups than persons at the party.

[b]p1.[/b] An Egyptian fraction has the form $1/n$, where $n$ is a positive integer. In ancient Egypt, these were the only fractions allowed. Other fractions between zero and one were always expressed as a sum of distinct Egyptian fractions. For example, $3/5$ was seen as $1/2 + 1/10$, or $1/3 + 1/4 + 1/60$. The preferred method of representing a fraction in Egypt used the "greedy" algorithm, which at each stage, uses the Egyptian fraction that eats up as much as possible of what is left of the original fraction. Thus the greedy fraction for $3/5$ would be $1/2 + 1/10$. a) Find the greedy Egyptian fraction representations for $2/13$. b) Find the greedy Egyptian fraction representations for $9/10$. c) Find the greedy Egyptian fraction representations for $2/(2k+1)$, where $k$ is a positive integer. d) Find the greedy Egyptian fraction representations for $3/(6k+1)$, where $k$ is a positive integer. [b]p2.[/b] a) The smaller of two concentric circles has radius one unit. The area of the larger circle is twice the area of the smaller circle. Find the difference in their radii. [img]https://cdn.artofproblemsolving.com/attachments/8/1/7c4d81ebfbd4445dc31fa038d9dc68baddb424.png[/img] b) The smaller of two identically oriented equilateral triangles has each side one unit long. The smaller triangle is centered within the larger triangle so that the perpendicular distance between parallel sides is always the same number $d$. The area of the larger triangle is twice the area of the smaller triangle. Find $d$. [img]https://cdn.artofproblemsolving.com/attachments/8/7/1f0d56d8e9e42574053c831fa129eb40c093d9.png[/img] [b]p3.[/b] Suppose that the domain of a function $f$ is the set of real numbers and that $f$ takes values in the set of real numbers. A real number $x_0$ is a fixed point of f if $f(x_0) = x_0$. a) Let $f(x) = m x + b$. For which $m$ does $f$ have a fixed point? b) Find the fixed point of f$(x) = m x + b$ in terms of m and b, when it exists. c) Consider the functions $f_c(x) = x^2 - c$. i. For which values of $c$ are there two different fixed points? ii. For which values of $c$ are there no fixed points? iii. In terms of $c$, find the value(s) of the fixed point(s). d) Find an example of a function that has exactly three fixed points. [b]p4.[/b] A square based pyramid is made out of rubber balls. There are $100$ balls on the bottom level, 81 on the next level, etc., up to $1$ ball on the top level. a) How many balls are there in the pyramid? b) If each ball has a radius of $1$ meter, how tall is the pyramid? c) What is the volume of the solid that you create if you place a plane against each of the four sides and the base of the balls? [b]p5.[/b] We wish to consider a general deck of cards specified by a number of suits, a sequence of denominations, and a number (possibly $0$) of jokers. The deck will consist of exactly one card of each denomination from each suit, plus the jokers, which are "wild" and can be counted as any possible card of any suit. For example, a standard deck of cards consists of $4$ suits, $13$ denominations, and $0$ jokers. a) For a deck with $3$ suits $\{a, b, c\}$ and $7$ denominations $\{1, 2, 3, 4, 5, 6, 7\}$, and $0$ jokers, find the probability that a 3-card hand will be a straight. (A straight consists of $3$ cards in sequence, e.g., $1 \heartsuit$ ,$2 \spadesuit$ , $3\clubsuit$ , $2\diamondsuit$ but not $6 \heartsuit$ ,$7 \spadesuit$ , $1\diamondsuit$). b) For a deck with $3$ suits, $7$ denominations, and $0$ jokers, find the probability that a $3$-card hand will consist of $3$ cards of the same suit (i.e., a flush). c) For a deck with $3$ suits, $7$ denominations, and $1$ joker, find the probability that a $3$-card hand dealt at random will be a straight and also the probability that a $3$-card hand will be a flush. d) Find a number of suits and the length of the denomination sequence that would be required if a deck is to contain $1$ joker and is to have identical probabilities for a straight and a flush when a $3$-card hand is dealt. The answer that you find must be an answer such that a flush and a straight are possible but not certain to occur. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $ \mathcal{P}$ be a square and let $ n$ be a nonzero positive integer for which we denote by $ f(n)$ the maximum number of elements of a partition of $ \mathcal{P}$ into rectangles such that each line which is parallel to some side of $ \mathcal{P}$ intersects at most $ n$ interiors (of rectangles). Prove that \[ 3 \cdot 2^{n\minus{}1} \minus{} 2 \le f(n) \le 3^n \minus{} 2.\]

A plane figure of area $A > n$ is given, where $n$ is a positive integer. Prove that this figure can be placed onto a Cartesian plane so that it covers at least $n+1$ points with integer coordinates.

Each of the players in a tennis tournament played one match against each of the others. If every player won at least one match, show that there is a group A; B; C of three players for which A beat B, B beat C and C beat A.

Prove that there exists a set $ S$ of $ n \minus{} 2$ points inside a convex polygon $ P$ with $ n$ sides, such that any triangle determined by $3$ vertices of $ P$ contains exactly one point from $ S$ inside or on the boundaries.