The shapes in the image consist of six unit cubes. Which of the following 3D objects can be filled up with the aforementioned shapes: a) a cube with side length $3$, from which one edge has been removed (i.e. three layers of the shape [img]https://i.imgur.com/vUqgHS2.png[/img] )? b) a rectangular prism of size $5 \times 4 \times 3$, from which two edges of length $3$ have been removed from one of the $5 \times 3$ sides (i.e. three layers of the shape [img]https://imgur.com/W4pfEfz.png[/img] )? We can use each of shapes at most once, no two shapes can overlap, nor protrude from the 3D object and every unit cube of the 3D object must be covered by a unit cube of one of the constituent shapes. [center][img]https://imgur.com/evAmuep.png[/img][/center] [i]Proposed by Ilija Jovčeski[/i]

A set $S$ is called perfect if it has the following two properties: a) $S$ has exactly four elements b) for every element $x$ of $S$, at least one of the numbers $x - 1$ or $x+1$ belongs to $S$. Find the number of all perfect subsets of the set $\{1,2,... ,n\}$

Hamza and Majid play a game on a horizontal $3 \times 2015$ white board. They alternate turns, with Hamza going first. A legal move for Hamza consists of painting three unit squares forming a horizontal $1 \times 3$ rectangle. A legal move for Majid consists of painting three unit squares forming a vertical $3\times 1$ rectangle. No one of the two players is allowed to repaint already painted squares. The last player to make a legal move wins. Which of the two players, Hamza or Majid, can guarantee a win no matter what strategy his opponent chooses and what is his strategy to guarantee a win? Lê Anh Vinh

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]

Rakesh is flipping a fair coin repeatedly. If $T$ denotes the event where the coin lands on tails and $H$ denotes the event where the coin lands on heads, what is the probability Rakesh flips the sequence $HHH$ before the sequence $THH$?

[hide=D stands for Dedekind, Z stands for Zermelo]they had two problem sets under those two names[/hide] [b]D1.[/b] The product of two positive integers is $5$. What is their sum? [b]D2.[/b] Gavin is $4$ feet tall. He walks $5$ feet before falling forward onto a cushion. How many feet is the top of Gavin’s head from his starting point? [b]D3.[/b] How many times must Nathan roll a fair $6$-sided die until he can guarantee that the sum of his rolls is greater than $6$? [b]D4 / Z1.[/b] What percent of the first $20$ positive integers are divisible by $3$? [b]D5.[/b] Let $a$ be a positive integer such that $a^2 + 2a + 1 = 36$. Find $a$. [b]D6 / Z2.[/b] It is said that a sheet of printer paper can only be folded in half $7$ times. A sheet of paper is $8.5$ inches by $11$ inches. What is the ratio of the paper’s area after it has been folded in half $7$ times to its original area? [b]D7 / Z3.[/b] Boba has an integer. They multiply the number by $8$, which results in a two digit integer. Bubbles multiplies the same original number by 9 and gets a three digit integer. What was the original number? [b]D8.[/b] The average number of letters in the first names of students in your class of $24$ is $7$. If your teacher, whose first name is Blair, is also included, what is the new class average? [b]D9 / Z4.[/b] For how many integers $x$ is $9x^2$ greater than $x^4$? [b]D10 / Z5.[/b] How many two digit numbers are the product of two distinct prime numbers ending in the same digit? [b]D11 / Z6.[/b] A triangle’s area is twice its perimeter. Each side length of the triangle is doubled,and the new triangle has area $60$. What is the perimeter of the new triangle? [b]D12 / Z7.[/b] Let $F$ be a point inside regular pentagon $ABCDE$ such that $\vartriangle FDC$ is equilateral. Find $\angle BEF$. [b]D13 / Z8.[/b] Carl, Max, Zach, and Amelia sit in a row with $5$ seats. If Amelia insists on sitting next to the empty seat, how many ways can they be seated? [b]D14 / Z9.[/b] The numbers $1, 2, ..., 29, 30$ are written on a whiteboard. Gumbo circles a bunch of numbers such that for any two numbers he circles, the greatest common divisor of the two numbers is the same as the greatest common divisor of all the numbers he circled. Gabi then does the same. After this, what is the least possible number of uncircled numbers? [b]D15 / Z10.[/b] Via has a bag of veggie straws, which come in three colors: yellow, orange, and green. The bag contains $8$ veggie straws of each color. If she eats $22$ veggie straws without considering their color, what is the probability she eats all of the yellow veggie straws? [b]Z11.[/b] We call a string of letters [i]purple[/i] if it is in the form $CVCCCV$ , where $C$s are placeholders for (not necessarily distinct) consonants and $V$s are placeholders for (not necessarily distinct) vowels. If $n$ is the number of purple strings, what is the remainder when $n$ is divided by $35$? The letter $y$ is counted as a vowel. [b]Z12.[/b] Let $a, b, c$, and d be integers such that $a+b+c+d = 0$ and $(a+b)(c+d)(ab+cd) = 28$. Find $abcd$. [b]Z13.[/b] Griffith is playing cards. A $13$-card hand with Aces of all $4$ suits is known as a godhand. If Griffith and $3$ other players are dealt $13$-card hands from a standard $52$-card deck, then the probability that Griffith is dealt a godhand can be expressed in simplest form as $\frac{a}{b}$. Find $a$. [b]Z14.[/b] For some positive integer $m$, the quadratic $x^2 + 202200x + 2022m$ has two (not necessarily distinct) integer roots. How many possible values of $m$ are there? [b]Z15.[/b] Triangle $ABC$ with altitudes of length $5$, $6$, and $7$ is similar to triangle $DEF$. If $\vartriangle DEF$ has integer side lengths, find the least possible value of its perimeter. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

In a hidden friend, suppose no one takes oneself. We say that the hidden friend has "marmalade" if there are two people $A$ and $ B$ such that A took $B$ and $ B $ took $A$. For each positive integer n, let $f (n)$ be the number of hidden friends with n people where there is no “marmalade”, i.e. $f (n)$ is equal to the number of permutations $\sigma$ of {$1, 2,. . . , n$} such that: *$\sigma (i) \neq i$ for all $i=1,2,...,n$ * there are no $ 1 \leq i <j \leq n $ such that $ \sigma (i) = j$ and $\sigma (j) = i. $ Determine the limit $\lim_{n \to + \infty} \frac{f(n)}{n!}$

We call a $6\times 6$ table consisting of zeros and ones [i]right[/i] if the sum of the numbers in each row and each column is equal to $3$. Two right tables are called [i]similar[/i] if one can get from one to the other by successive interchanges of rows and columns. Find the largest possible size of a set of pairwise similar right tables.

Let $ n > 1, n \in \mathbb{Z}$ and $ B \equal{}\{1,2,\ldots, 2^n\}.$ A subset $ A$ of $ B$ is called weird if it contains exactly one of the distinct elements $ x,y \in B$ such that the sum of $ x$ and $ y$ is a power of two. How many weird subsets does $ B$ have?

We call a graph with n vertices $k-flowing-chromatic$ if: 1. we can place a chess on each vertex and any two neighboring (connected by an edge) chesses have different colors. 2. we can choose a hamilton cycle $v_1,v_2,\cdots , v_n$, and move the chess on $v_i$ to $v_{i+1}$ with $i=1,2,\cdots ,n$ and $v_{n+1}=v_1$, such that any two neighboring chess also have different colors. 3. after some action of step 2 we can make all the chess reach each of the n vertices. Let T(G) denote the least number k such that G is k-flowing-chromatic. If such k does not exist, denote T(G)=0. denote $\chi (G)$ the chromatic number of G. Find all the positive number m such that there is a graph G with $\chi (G)\le m$ and $T(G)\ge 2^m$ without a cycle of length small than 2017.

Is there a six-link broken line in space that passes through all the vertices of a given cube?

The number 7 is written on a board. Alice and Bob in turn (Alice begins) write an additional digit in the number on the board: it is allowed to write the digit at the beginning (provided the digit is nonzero), between any two digits or at the end. If after someone’s turn the number on the board is a perfect square then this person wins. Is it possible for a player to guarantee the win? [i]Alexandr Gribalko[/i]

Let $S$ a finite subset of $\mathbb{N}$. For every positive integer $i$, let $A_{i}$ the number of partitions of $i$ with all parts in $ \mathbb{N}-S$. Prove that there exists $M\in \mathbb{N}$ such that $A_{i+1}>A_{i}$ for all $i>M$. ($ \mathbb{N}$ is the set of positive integers)

A design $X$ is an array of the digits $1,2,..., 9$ in the shape of an $X$, for example, [img]https://cdn.artofproblemsolving.com/attachments/8/e/d371a2cd442cb7a8784e1cc7635344df722e20.png[/img] We will say that a design $X$ is [i]balanced [/i] if the sum of the numbers of each of the diagonals match. Determine the number of designs $X$ that are balanced.

For $n>1$ consider an $n\times n$ chessboard and place identical pieces at the centers of different squares. [list=i] [*] Show that no matter how $2n$ identical pieces are placed on the board, that one can always find $4$ pieces among them that are the vertices of a parallelogram. [*] Show that there is a way to place $(2n-1)$ identical chess pieces so that no $4$ of them are the vertices of a parallelogram. [/list]

Four unit squares form a $2 \times 2$ grid. Each of the $12$ unit line segments forming the sides of the squares is colored either red or blue in such way that each square has $2$ red sides and blue sides. One example is shown below (red is solid, blue is dashed). Find the number of such colorings. [asy] size(4cm); defaultpen(linewidth(1.2)); draw((0, 0) -- (2, 0) -- (2, 1)); draw((0, 1) -- (1, 1) -- (1, 2) -- (2,2)); draw((0, 0) -- (0, 1), dotted); draw((1, 0) -- (1, 1) -- (2, 1) -- (2, 2), dotted); draw((0, 1) -- (0, 2) -- (1, 2), dotted); [/asy]

Several napkins of equal size and of shape of a unit disc were placed on a table (with overlappings). Is it always possible to hammer several point-sized nails so that all the napkins will be thus attached to the table with the same number of nails? (The nails cannot be hammered into the borders of the discs). Vladimir Dolnikov, Pavel Kozhevnikov

Adamand Topher are playing a game in which each of them starts with $2$ pickles. Each turn, they flip a fair coin: if it lands heads, Topher takes $1$ pickle from Adam; if it lands tails, Adam takes $2$ pickles from Topher. (If Topher has only $1$ pickle left, Adam will just take it.) What’s the probability that Topher will have all $4$ pickles before Adam does?

Let $N=\overline{abc}$ be a three-digit number. It is known that we can construct an isoceles triangle with $a,b$ and $c$ as the length of sides. Determine how many possible three-digit number $N$ there are. ($N=\overline{abc}$ means that $a,b$ and $c$ are digits of $N$, and not $N=a\times b\times c$.)

In preparation for a game of Fish, Carl must deal $48$ cards to $6$ players. For each card that he deals, he runs through the entirety of the following process: $1$. He gives a card to a random player. $2$. A player $Z$ is randomly chosen from the set of players who have at least as many cards as every other player (i.e. $Z$ has the most cards or is tied for having the most cards). $3$. A player $D$ is randomly chosen from the set of players other than $Z$ who have at most as many cards as every other player (i.e. $D$ has the fewest cards or is tied for having the fewest cards). $4$. $Z$ gives one card to $D$. He repeats steps $1-4$ for each card dealt, including the last card. After all the cards have been dealt, what is the probability that each player has exactly $8$ cards?

There are $ n$ websites $ 1,2,\ldots,n$ ($ n \geq 2$). If there is a link from website $ i$ to $ j$, we can use this link so we can move website $ i$ to $ j$. For all $ i \in \left\{1,2,\ldots,n - 1 \right\}$, there is a link from website $ i$ to $ i+1$. Prove that we can add less or equal than $ 3(n - 1)\log_{2}(\log_{2} n)$ links so that for all integers $ 1 \leq i < j \leq n$, starting with website $ i$, and using at most three links to website $ j$. (If we use a link, website's number should increase. For example, No.7 to 4 is impossible). Sorry for my bad English.

Let $N$ be the set of positive integers. If $A,B,C \ne \emptyset$, $A \cap B = B \cap C = C \cap A = \emptyset$ and $A \cup B \cup C = N$, we say that $A,B,C$ are partitions of $N$. Prove that there are no partitions of $N, A,B,C$, that satisfy the following: (i) $\forall a \in A, b \in B$, we have $a + b + 1 \in C$ (ii) $\forall b \in B, c \in C$, we have $b + c + 1 \in A$ (iii) $\forall c \in C, a \in A$, we have $c + a + 1 \in B$

For any two positive integers $n$ and $p$, prove that there are exactly ${{(p+1)}^{n+1}}-{{p}^{n+1}}$ functions $f:\left\{ 1,2,...,n \right\}\to \left\{ -p,-p+1,-p+2,....,p-1,p \right\}$ such that $\left| f(i)-f(j) \right|\le p$ for all $i,j\in \left\{ 1,2,...,n \right\}$.

A polygonal line of the length $1001$ is given in a unit square. Prove that there exists a line parallel to one of the sides of the square that meets the polygonal line in at least $500$ points.

Students $ A $ and $ B $ play according to the following rules: student $ A $ selects a vector $ \overrightarrow{a_1} $ of length 1 in the plane, then student $ B $ gives the number $ s_1 $, equal to $ 1 $ or $ - $1; then the student $ A $ chooses a vector $ \overrightarrow{a_1} $ of length $ 1 $, and in turn the student $ B $ gives a number $ s_2 $ equal to $ 1 $ or $ -1 $ etc. $ B $ wins if for a certain $ n $ vector $ \sum_{j=1}^n \varepsilon_j \overrightarrow{a_j} $ has a length greater than the number $ R $ determined before the start of the game. Prove that student $B$ can achieve a win in no more than $R^2 + 1$ steps regardless of partner $A$'s actions.