Consider a cube with a fly standing at each of its vertices. When a whistle blows, each fly moves to a vertex in the same face as the previous one but diagonally opposite to it. After the whistle blows, in how many ways can the flies change position so that there is no vertex with 2 or more flies?

Every cell of a $2005 \times 2005$ square board is colored white or black so that every $2 \times 2$ subsquare contains an odd number of black cells. Show that among the corner cells there is an even number of black ones. How many ways are there to color the board in this manner?

Cary has six distinct coins in a jar. Occasionally, he takes out three of the coins and adds a dot to each of them. Determine the number of orders in which Cary can choose the coins so that, eventually, for each number $i \in \{0, 1, . . . , 5\}$, some coin has exactly $i$ dots on it.

On a board with $a$ rows and $b$ columns, each square has a switch and an unlit light bulb. By pressing the switch of a house, the lamp in that house changes state, along with the lamps in the same row and those in the same column (those that are on go out and the that are off light up). For what values of $a$ and $b$ is it possible to have just one lamp on, by pressing a series of switches?

Let $f$ be a function defined on the positive integers with $f(n) \ge 0$ and $f(n) \le f(n+1)$ for all $n$. Prove that if \[\sum_{n = 1}^{\infty} \frac{f(n)}{n^2}\] diverges, there exists a sequence $a_1, a_2, \dots$ such that the sequence $\tfrac{a_n}{n}$ hits every natural number, while \[a_{n+m} \le a_n + a_m + f(n+m)\] holds for every pair $n$, $m$.

Given convex $1000$-gon. Inside this polygon, $1020$ points are chosen so that no $3$ of the $2020$ points do not lie on one line. Polygon is cut into triangles so that these triangles have vertices only those specified $2020$ points and each of these points is the vertex of at least one of cutting triangles. How many such triangles were formed?

Consider $2018$ pairwise crossing circles no three of which are concurrent. These circles subdivide the plane into regions bounded by circular $edges$ that meet at $vertices$. Notice that there are an even number of vertices on each circle. Given the circle, alternately colour the vertices on that circle red and blue. In doing so for each circle, every vertex is coloured twice- once for each of the two circle that cross at that point. If the two colours agree at a vertex, then it is assigned that colour; otherwise, it becomes yellow. Show that, if some circle contains at least $2061$ yellow points, then the vertices of some region are all yellow. Proposed by [i]India[/i]

In a galaxy far, far away there are $225$ inhabited planets. Between some pairs of inhabited planets there is a bidirectional space connection, and it is possible to reach any planet from any other (possibly with several transfers). The [i]influence[/i] of a planet is the number of other planets with which this planet has a direct connection. It is known that if two planets are not connected by a direct space flight, they have different influences. What is the smallest number of connections possible under these conditions? [i]Proposed by Arsenii Nikolaev, Bogdan Rublov[/i]

There are $100$ members in a ladies' club.Each lady has had tea (in private) with exactly $56$ of her lady friends.The Board,consisting of the $50$ most distinguished ladies,have all had tea with one another.Prove that the entire club may be split into two groups in such a way that,with in each group,any lady has had tea with any other.

$N$ lines lie on a plane, no two of which are parallel and no three of which are concurrent. Prove that there exists a non-self-intersecting broken line $A_0A_1A_2A_3...A_N$ with $N$ parts, such that on each of the $N$ lines lies exactly one of the $N$ segments of the line.

Is it possible to arrange $n \times n$ in the cells of a square table the numbers $0$,$ 1$ or $2$ so that the sums of the numbers in rows and columns took on all different values from $1$ to $2n$? Consider two cases: a) $n$ is an odd number; b) $n$ is an even number.

Ninety-one students in a school are distributed in three classes. Each student took part in a competition. It is known that among any six students of the same sex some two got the same number of points. Show that here are four students of the same sex who are in the same class and who got the same number of points.

Let $n$ be a natural number. The leader of the math team invites $n$ girls for winter training, and each leaves her two gloves in a common box upon entry. The mischievous little brother randomly pairs the gloves into pairs, where each pair consists of one left glove and one right glove. A pairing is called [i]weak[/i] if there is a set of $k < \frac{n}{2}$ pairs containing gloves of exactly $k$ girls. Find the probability that the pairing is not weak.

Prove that for every $n \in N$, $n > 6$, every equilateral triangle can be divided into $n$ pieces, which are also equilateral triangles.

The integers $1 , 2,..., n$ are rearranged in such a way that if the integer $k, 1 \le k\le n$, is not the first term, then one of the integers $k + 1$ or $k-1$ occurs to the left of $k$ . How many arrangements of the integers $1 , 2,..., n$ satisfy this condition? (A. Andjans, Riga)

Let $S$ be a set of size $n$, and $k$ be a positive integer. For each $1 \le i \le kn$, there is a subset $S_i \subset S$ such that $|S_i| = 2$. Furthermore, for each $e \in S$, there are exactly $2k$ values of $i$ such that $e \in S_i$. Show that it is possible to choose one element from $S_i$ for each $1 \le i \le kn$ such that every element of $S$ is chosen exactly $k$ times.

In a closed rectangular neighborhood there are: $S$ streets (these are straight roads of maximum length), $V$ four-arm intersections ( [img]https://cdn.artofproblemsolving.com/attachments/e/4/6a5974a30dc182b59a519a8ef4eb4f1412e05e.png[/img]), $H$ city blocks (these are rectangular areas bounded by four streets, which are no be intersected by another street) and $T$ represents the number of $T$-intersections ([img]https://cdn.artofproblemsolving.com/attachments/0/a/b390a30a0b27d83db681f70f633bdeed697163.png[/img] ). For example, in the neighborhood below, there are $15$ streets, $8$ four-arm intersections, $20$ city blocks and $22$ $T$-intersections. [img]https://cdn.artofproblemsolving.com/attachments/a/2/c1a5e463d0fb5671ac0702c91cfc2272d4e2c3.png[/img] Prove that in each district $S + V = H + 3$.

A numerist has $n$ eurodollars and distributes them to two bank accounts $A, B$ in Germany and Switzerland, whereby the Eurodollars cannot be split into smaller monetary units due to the lack of a suitable name. In order to hide all money from the tax authorities if necessary, he would like to be able to move all of his money to account $B$. Due to the immense bureaucracy, money is only allowed to be moved between two accounts if the deposited amount in one account is double. Of course, he can carry out several such transfers in a row. Show that the number of ways to initially distribute the money appropriately is a power of two.

Given an equilateral triangle, cut it into four polygonal shapes so that, reassembled appropriately, these figures form a square.

In a regular 100-gon, 41 vertices are colored black and the remaining 59 vertices are colored white. Prove that there exist 24 convex quadrilaterals $Q_{1}, \ldots, Q_{24}$ whose corners are vertices of the 100-gon, so that [list] [*] the quadrilaterals $Q_{1}, \ldots, Q_{24}$ are pairwise disjoint, and [*] every quadrilateral $Q_{i}$ has three corners of one color and one corner of the other color. [/list]

Baron Munchausen claims that he has drawn a polygon and chosen a point inside the polygon in such a way that any line passing through the chosen point divides the polygon into three polygons. Could the Baron’s claim be correct?

[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].

Will and Lucas are playing a game. Will claims that he has a polynomial $f$ with integer coefficients in mind, but Lucas doesn’t believe him. To see if Will is lying, Lucas asks him on minute $i$ for the value of $f(i)$, starting from minute $ 1$. If Will is telling the truth, he will report $f(i)$. Otherwise, he will randomly and uniformly pick a positive integer from the range $[1,(i+1)!]$. Now, Lucas is able to tell whether or not the values that Will has given are possible immediately, and will call out Will if this occurs. If Will is lying, say the probability that Will makes it to round $20$ is $a/b$. If the prime factorization of $b$ is $p_1^{e_1}... p_k^{e_k}$ , determine the sum $\sum_{i=1}^{k} e_i$.

Let $n\ge 4$ be a positive integer and let $M$ be a set of $n$ points in the plane, where no three points are collinear and not all of the $n$ points being concyclic. Find all real functions $f:M\to\mathbb{R}$ such that for any circle $\mathcal{C}$ containing at least three points from $M$, the following equality holds: \[\sum_{P\in\mathcal{C}\cap M} f(P)=0\] [i]Dorel Mihet[/i]

A tournament has $5$ players and is in round-robin format (each player plays each other exactly once). Each game has a $\frac13$ chance of player $A$ winning, a $\frac13$ chance of player $B$ winning, and a$ \frac13$ chance of ending in a draw. The probability that at least one player draws all of their games can be written in simplest form as $\frac{m}{3^n}$ where $m, n$ are positive integers. Find $m + n$.