A cube is partitioned into finitely many rectangular parallelepipeds with the edges parallel to the edges of the cube. Prove that if the sum of the volumes of the circumspheres of these parallelepipeds equals the volume of the circumscribed sphere of the cube, then all the parallelepipeds are cubes.

For any subset $S \subseteq \mathbb{Z}^+$, a function $f : S \to S$ is called [i]interesting[/i] if the following two conditions hold: [b]1.[/b] There is no element $a \in S$ such that $f(a) = a$. [b]2.[/b] For every $a \in S$, we have $f^{f(a) + 1}(a) = a$ (where $f^{k}$ denotes the $k$-th iteration of $f$). Prove that: [b]a) [/b]There exist infinitely many interesting functions $f : \mathbb{Z}^+ \to \mathbb{Z}^+$. [b]b) [/b]There exist infinitely many positive integers $n$ for which there is no interesting function $$ f : \{1, 2, \ldots, n\} \to \{1, 2, \ldots, n\}. $$ [i]Proposed by Giorgi Kekenadze, Georgia[/i]

A circle is divided into $n$ sectors ($n \geq 3$). Each sector can be filled in with either $1$ or $0$. Choose any sector $\mathcal{C}$ occupied by $0$, change it into a $1$ and simultaneously change the symbols $x, y$ in the two sectors adjacent to $\mathcal{C}$ to their complements $1-x$, $1-y$. We repeat this process as long as there exists a zero in some sector. In the initial configuration there is a $0$ in one sector and $1$s elsewhere. For which values of $n$ can we end this process?

There are three colleges in a town. Each college has $n$ students. Any student of any college knows $n+1$ students of the other two colleges. Prove that it is possible to choose a student from each of the three colleges so that all three students would know each other.

Let $\mathcal{C}$ be a family of subsets of $A=\{1,2,\dots,100\}$ satisfying the following two conditions: 1) Every $99$ element subset of $A$ is in $\mathcal{C}.$ 2) For any non empty subset $C\in\mathcal{C}$ there is $c\in C$ such that $C\setminus\{c\}\in \mathcal{C}.$ What is the least possible value of $|\mathcal{C}|$?

Let $m$ be a positive integer and let $A$, respectively $B$, be two alphabets with $m$, respectively $2m$ letters. Let also $n$ be an even integer which is at least $2m$. Let $a_n$ be the number of words of length $n$, formed with letters from $A$, in which appear all the letters from $A$, each an even number of times. Let $b_n$ be the number of words of length $n$, formed with letters from $B$, in which appear all the letters from $B$, each an odd number of times. Compute $\frac{b_n}{a_n}$.

On the circle with center 0 and radius 1 the point $A_0$ is fixed and points $A_1, A_2, \ldots, A_{999}, A_{1000}$ are distributed in such a way that the angle $\angle A_00A_k = k$ (in radians). Cut the circle at points $A_0, A_1, \ldots, A_{1000}.$ How many arcs with different lengths are obtained. ?

A rectangle $\mathcal{R}$ with odd integer side lengths is divided into small rectangles with integer side lengths. Prove that there is at least one among the small rectangles whose distances from the four sides of $\mathcal{R}$ are either all odd or all even. [i]Proposed by Jeck Lim, Singapore[/i]

A finite number of coins are placed on an infinite row of squares. A sequence of moves is performed as follows: at each stage a square containing more than one coin is chosen. Two coins are taken from this square; one of them is placed on the square immediately to the left while the other is placed on the square immediately to the right of the chosen square. The sequence terminates if at some point there is at most one coin on each square. Given some initial configuration, show that any legal sequence of moves will terminate after the same number of steps and with the same final configuration.

Paul and his pet octahedron like to play games together. For this game, the octahedron randomly draws an arrow on each of its faces pointing to one of its three edges. Paul then randomly chooses a face and progresses from face to adjacent face, as determined by the arrows on each face, and he wins if he reaches every face of the octahedron. What is the probability that Paul wins?

The integers $1, 2, \cdots, n^2$ are placed on the fields of an $n \times n$ chessboard $(n > 2)$ in such a way that any two fields that have a common edge or a vertex are assigned numbers differing by at most $n + 1$. What is the total number of such placements?

There are $N$ red cards and $N$ blue cards. Each card has a positive integer between $1$ and $N$ (inclusive) written on it. Prove that we can choose a (non-empty) subset of the red cards and a (non-empty) subset of the blue cards, so that the sum of the numbers on the chosen red cards equals the sum of the numbers on the chosen blue cards.

[hide=C stands for Cantor, G stands for Gauss]they had two problem sets under those two names[/hide] [u]Set 4[/u] [b]G.16[/b] A number $k$ is the product of exactly three distinct primes (in other words, it is of the form $pqr$, where $p, q, r$ are distinct primes). If the average of its factors is $66$, find $k$. [b]G.17[/b] Find the number of lattice points contained on or within the graph of $\frac{x^2}{3} +\frac{y^2}{2}= 12$. Lattice points are coordinate points $(x, y)$ where $x$ and $y$ are integers. [b]G.18 / C.23[/b] How many triangles can be made from the vertices and center of a regular hexagon? Two congruent triangles with different orientations are considered distinct. [b]G.19[/b] Cindy has a cone with height $15$ inches and diameter $16$ inches. She paints one-inch thick bands of paint in circles around the cone, alternating between red and blue bands, until the whole cone is covered with paint. If she starts from the bottom of the cone with a blue strip, what is the ratio of the area of the cone covered by red paint to the area of the cone covered by blue paint? [b]G.20 / C.25[/b] An even positive integer $n$ has an odd factorization if the largest odd divisor of $n$ is also the smallest odd divisor of n greater than 1. Compute the number of even integers $n$ less than $50$ with an odd factorization. [u] Set 5[/u] [b]G.21[/b] In the magical tree of numbers, $n$ is directly connected to $2n$ and $2n + 1$ for all nonnegative integers n. A frog on the magical tree of numbers can move from a number $n$ to a number connected to it in $1$ hop. What is the least number of hops that the frog can take to move from $1000$ to $2018$? [b]G.22[/b] Stan makes a deal with Jeff. Stan is given 1 dollar, and every day for $10$ days he must either double his money or burn a perfect square amount of money. At first Stan thinks he has made an easy $1024$ dollars, but then he learns the catch - after $10$ days, the amount of money he has must be a multiple of $11$ or he loses all his money. What is the largest amount of money Stan can have after the $10$ days are up? [b]G.23[/b] Let $\Gamma_1$ be a circle with diameter $2$ and center $O_1$ and let $\Gamma_2$ be a congruent circle centered at a point $O_2 \in \Gamma_1$. Suppose $\Gamma_1$ and $\Gamma_2$ intersect at $A$ and $B$. Let $\Omega$ be a circle centered at $A$ passing through $B$. Let $P$ be the intersection of $\Omega$ and $\Gamma_1$ other than $B$ and let $Q$ be the intersection of $\Omega$ and ray $\overrightarrow{AO_1}$. Define $R$ to be the intersection of $PQ$ with $\Gamma_1$. Compute the length of $O_2R$. [b]G.24[/b] $8$ people are at a party. Each person gives one present to one other person such that everybody gets a present and no two people exchange presents with each other. How many ways is this possible? [b]G.25[/b] Let $S$ be the set of points $(x, y)$ such that $y = x^3 - 5x$ and $x = y^3 - 5y$. There exist four points in $S$ that are the vertices of a rectangle. Find the area of this rectangle. PS. You should use hide for answers. C1-15/ G1-10 have been posted [url=https://artofproblemsolving.com/community/c3h2790674p24540132]here [/url] and C16-30/G10-15, G25-30 [url=https://artofproblemsolving.com/community/c3h2790676p24540145]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url]

In some company, consisting of $n$ people, any two have at most $k \geq 2$ common friends. Lets call group of people working in the company unsocial if everyone in the group has at most one friend from the group. Prove that there exists an unsocial group consisting of at least $\sqrt{\frac{2n}{k}}$ people [i]M. Zorka[/i]

Let $n> 1$ be an odd integer. On a square surface have been placed $n^2 - 1$ white slabs and a black slab on the center. Two workers $A$ and $B$ take turns removing them, betting that whoever removes black will lose. First $A$ picks a slab; if it has row number $i \ge (n + 1) / 2$, then it will remove all tiles from rows with number greater than or equal to$ i$, while if $i <(n + 1) / 2$, then it will remove all tiles from the rows with lesser number or equal to $i$. Proceed in a similar way with columns. Then $B$ chooses one of the remaining tiles and repeats the process. Determine who has a winning strategy and describe it. Note: Row and column numbering is ascending from top to bottom and from left to right.

Given a positive integer $n$, find the proportion of the subsets of $\{1,2, \ldots, 2n\}$ such that their smallest element is odd.

There are $ n$ football teams in a round-robin competition where every 2 teams meet once. The winner of each match receives 3 points while the loser receives 0 points. In the case of a draw, both teams receive 1 point each. Let $ k$ be as follows: $ 2 \leq k \leq n \minus{} 1$. At least how many points must a certain team get in the competition so as to ensure that there are at most $ k \minus{} 1$ teams whose scores are not less than that particular team's score?

Three distinct positive integers are chosen at random from $\{1,2,3...,12\}$. The probability that no two elements of the set have an absolute difference less than or equal to $2$ can be written as $\frac{a}{b}$ where $a$ and $b$ are relatively prime positive integers. Find $a +b$.

One hundred balls labelled $1$ to $100$ are to be put into two identical boxes so that each box contains at least one ball and the greatest common divisor of the product of the labels of all the balls in one box and the product of the labels of all the balls in the other box is $1$. Determine the number of ways that this can be done.

Given a sequence of $19$ positive (not necessarily distinct) integers not greater than $93$, and a set of $93$ positive (not necessarily distinct) integers not greater than $19$. Show that we can find non-empty subsequences of the two sequences with equal sum.

An angle of size $n \times m$, where $m, n \ge 2$, is called a figure, resulting from a rectangle of size $n \times m$ cells by removing the rectangle size $(n - 1) \times (m - 1)$ cells. Two players take turns making moves consisting in painting in a corner an arbitrary non-zero number of cells forming a rectangle or square.

Alice and Bob are playing a game on a graph with $n\ge3$ vertices. At each moment, Alice needs to choose two vertices so that the graph is connected even if one of them (along with the edges incident to it) is removed. Each turn, Bob removes one edge in the graph, and upon the removal, Alice needs to re-select the two vertices if necessary. However, Bob has to guarantee that after each removal, any two vertices in the graph are still connected via at most $k$ intermediate vertices. Here $0\le k\le n-2$ is some given integer. Suppose that Bob always knows which two vertices Alice chooses, and that initially, the graph is a complete graph. Alice's objective is to change her choice of the two vertices as few times as possible, and Bob's objective is to make Alive re-select as many times as possible. If both Alice and Bob are sufficiently smart, how many times will Alice change her choice of the two vertices? (usjl)

A sequence of $X$s and $O$s is given, such that no three consecutive characters in the sequence are all the same, and let $N$ be the number of characters in this sequence. Maia may swap two consecutive characters in the sequence. After each swap, any consecutive block of three or more of the same character will be erased (if there are multiple consecutive blocks of three or more characters after a swap, then they will be erased at the same time), until there are no more consecutive blocks of three or more of the same character. For example, if the original sequence were $XXOOXOXO$ and Maia swaps the fifth and sixth character, the end result will be $$XXOOOXXO \to XXXXO \to O.$$ Find the maximum value $N$ for which Maia can’t necessarily erase all the characters after a series of swaps. Partial credit will be awarded for correct proofs of lower and upper bounds on $N$.

A special kind of chess knight is in the origin of an infinite grid. It can make one of twelve different moves: it can move directly up, down, left, or right one unit square, or it can move $1$ units in one direction and $3$ units in an orthogonal direction. How many different squares can it be on after $2$ moves?

There are 2008 congruent circles on a plane such that no two are tangent to each other and each circle intersects at least three other circles. Let $ N$ be the total number of intersection points of these circles. Determine the smallest possible values of $ N$.