For an ${n\times n}$ matrix $A$, let $X_{i}$ be the set of entries in row $i$, and $Y_{j}$ the set of entries in column $j$, ${1\leq i,j\leq n}$. We say that $A$ is [i]golden[/i] if ${X_{1},\dots ,X_{n},Y_{1},\dots ,Y_{n}}$ are distinct sets. Find the least integer $n$ such that there exists a ${2004\times 2004}$ golden matrix with entries in the set ${\{1,2,\dots ,n\}}$.

The following figure shows a $3^2 \times 3^2$ grid divided into $3^2$ subgrids of size $3 \times 3$. This grid has $81$ cells, $9$ in each subgrid. [asy] draw((0,0)--(9,0)--(9,9)--(0,9)--cycle, linewidth(2)); draw((0,1)--(9,1)); draw((0,2)--(9,2)); draw((0,3)--(9,3), linewidth(2)); draw((0,4)--(9,4)); draw((0,5)--(9,5)); draw((0,6)--(9,6), linewidth(2)); draw((0,7)--(9,7)); draw((0,8)--(9,8)); draw((1,0)--(1,9)); draw((2,0)--(2,9)); draw((3,0)--(3,9), linewidth(2)); draw((4,0)--(4,9)); draw((5,0)--(5,9)); draw((6,0)--(6,9), linewidth(2)); draw((7,0)--(7,9)); draw((8,0)--(8,9)); [/asy] Now consider an $n^2 \times n^2$ grid divided into $n^2$ subgrids of size $n \times n$. Find the number of ways in which you can select $n^2$ cells from this grid such that there is exactly one cell coming from each subgrid, one from each row and one from each column.

Five people are standing in a straight line, and the distance between any two people is a unique positive integer number of units. Find the least possible distance between the leftmost and rightmost people in the line in units.

suppose that $\mathcal F\subseteq X^{(k)}$ and $|X|=n$. we know that for every three distinct elements of $\mathcal F$ like $A,B,C$, at most one of $A\cap B$,$B\cap C$ and $C\cap A$ is $\phi$. for $k\le \frac{n}{2}$ prove that: a) $|\mathcal F|\le max(1,4-\frac{n}{k})\times \dbinom{n-1}{k-1}$.(15 points) b) find all cases of equality in a) for $k\le \frac{n}{3}$.(5 points)

Uri must paint some integers from $1$ to $2022$ (inclusive) in red, such that none of the differences between two red numbers is a prime number. Determine the maximum number of numbers Uri can paint red. [b]Note 1:[/b] The [i]difference [/i]between two distinct numbers is the subtraction of the larger minus the smaller. [b]Note 2:[/b] $1$ is not a prime number.

[u]Round 1[/u] [b]1.1.[/b] What is the area of a circle with diameter $2$? [b]1.2.[/b] What is the slope of the line through $(2, 1)$ and $(3, 4)$? [b]1.3.[/b] What is the units digit of $2^2 \cdot 4^4 \cdot 6^6$ ? [u]Round 2[/u] [b]2.1.[/b] Find the sum of the roots of $x^2 - 5x + 6$. [b]2. 2.[/b] Find the sum of the solutions to $|2 - x| = 1$. [b]2.3.[/b] On April $1$, $2018$, Mr. Dospinescu, Mr. Phaovibul and Mr. Pohoata all go swimming at the same pool. From then on, Mr. Dospinescu returns to the pool every 4th day, Mr. Phaovibul returns every $7$th day and Mr. Pohoata returns every $13$th day. What day will all three meet each other at the pool again? Give both the month and the day. [u]Round 3[/u] [b]3. 1.[/b] Kendall and Kylie are each selling t-shirts separately. Initially, they both sell t-shirts for $\$ 33$ each. A week later, Kendall marks up her t-shirt price by $30 \%$, but after seeing a drop in sales, she discounts her price by $30\%$ the following week. If Kim wants to buy $360$ t-shirts, how much money would she save by buying from Kendall instead of Kylie? Write your answer in dollars and cents. [b]3.2.[/b] Richard has English, Math, Science, Spanish, History, and Lunch. Each class is to be scheduled into one distinct block during the day. There are six blocks in a day. How many ways could he schedule his classes such that his lunch block is either the $3$rd or $4$th block of the day? [b]3.3.[/b] How many lattice points does $y = 1 + \frac{13}{17}x$ pass through for $x \in [-100, 100]$ ? (A lattice point is a point where both coordinates are integers.) [u]Round 4[/u] [b]4. 1.[/b] Unsurprisingly, Aaron is having trouble getting a girlfriend. Whenever he asks a girl out, there is an eighty percent chance she bursts out laughing in his face and walks away, and a twenty percent chance that she feels bad enough for him to go with him. However, Aaron is also a player, and continues asking girls out regardless of whether or not previous ones said yes. What is the minimum number of girls Aaron must ask out for there to be at least a fifty percent chance he gets at least one girl to say yes? [b]4.2.[/b] Nithin and Aaron are two waiters who are working at the local restaurant. On any given day, they may be fired for poor service. Since Aaron is a veteran who has learned his profession well, the chance of him being fired is only $\frac{2}{25}$ every day. On the other hand, Nithin (who never paid attention during job training) is very lazy and finds himself constantly making mistakes, and therefore the chance of him being fired is $\frac{2}{5}$. Given that after 1 day at least one of the waiters was fired, find the probability Nithin was fired. [b]4.3.[/b] In a right triangle, with both legs $4$, what is the sum of the areas of the smallest and largest squares that can be inscribed? An inscribed square is one whose four vertices are all on the sides of the triangle. PS. You should use hide for answers. Rounds 5-8 have been posted [url=https://artofproblemsolving.com/community/c3h2784569p24468582]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Does there exist a set of $100$ triangles in which not one of the triangles can be covered by the other $99$?

Suppose $A\subseteq \{0,1,\dots,29\}$. It satisfies that for any integer $k$ and any two members $a,b\in A$($a,b$ is allowed to be same), $a+b+30k$ is always not the product of two consecutive integers. Please find $A$ with largest possible cardinality.

Let $\,S\,$ be a finite set of points in three-dimensional space. Let $\,S_{x},\,S_{y},\,S_{z}\,$ be the sets consisting of the orthogonal projections of the points of $\,S\,$ onto the $yz$-plane, $zx$-plane, $xy$-plane, respectively. Prove that \[ \vert S\vert^{2}\leq \vert S_{x} \vert \cdot \vert S_{y} \vert \cdot \vert S_{z} \vert, \] where $\vert A \vert$ denotes the number of elements in the finite set $A$. [hide="Note"] Note: The orthogonal projection of a point onto a plane is the foot of the perpendicular from that point to the plane. [/hide]

Let $n$ be a positive integer. There are $2018n+1$ cities in the Kingdom of Sellke Arabia. King Mark wants to build two-way roads that connect certain pairs of cities such that for each city $C$ and integer $1\le i\le 2018,$ there are exactly $n$ cities that are a distance $i$ away from $C.$ (The [i]distance[/i] between two cities is the least number of roads on any path between the two cities.) For which $n$ is it possible for Mark to achieve this? [i]Proposed by Michael Ren[/i]

Let $ A_i,i=1,2,\dots,t$ be distinct subsets of the base set $\{1,2,\dots,n\}$ complying to the following condition $$ \displaystyle A_ {i} \cap A_ {k} \subseteq A_ {j}$$for any $1 \leq i <j <k \leq t.$ Find the maximum value of $t.$ Thanks @dgrozev

Let $G$ be a finite graph. Prove that one can partition $G$ into two graphs $A \cup B=G$ such that if we erase all edges conecting a vertex from $A$ to a vertex from $B$, each vertex of the new graph has even degree.

Geetha wants to cut a cube of size $4 \times 4\times 4$ into $64$ unit cubes (of size $1\times 1\times 1$). Every cut must be straight, and parallel to a face of the big cube. What is the minimum number of cuts that Geetha needs? Note: After every cut, she can rearrange the pieces before cutting again. At every cut, she can cut more than one pieces as long as the pieces are on a straight line.

$101$ wise men stand in a circle. Each of them either thinks that the Earth orbits Jupiter or that Jupiter orbits the Earth. Once a minute, all the wise men express their opinion at the same time. Right after that, every wise man who stands between two people with a different opinion from him changes his opinion himself. The rest do not change. Prove that at one point they will all stop changing opinions.

Find all pairs of positive integers \((a,b)\) with the following property: there exists an integer \(N\) such that for any integers \(m\ge N\) and \(n\ge N\), every \(m\times n\) grid of unit squares may be partitioned into \(a\times b\) rectangles and fewer than \(ab\) unit squares. [i]Proposed by Holden Mui[/i]

Let $n \geqslant 2$ be an integer. We consider a square grid of size $2n \times 2n$ divided into $4n^2$ unit squares. The grid is called [i]balanced[/i] if: [list] [*]Each cell contains a number equal to $-1$, $0$ or $1$. [*]The absolute value of the sum of the numbers in the grid does not exceed $4n$. [/list] Determine, as a function of $n$, the smallest integer $k \geqslant 1$ such that any balanced grid always contains an $n \times n$ square whose absolute sum of the $n^2$ cells is less than or equal to $k$.

A mason has bricks with dimensions $2\times5\times8$ and other bricks with dimensions $2\times3\times7$. She also has a box with dimensions $10\times11\times14$. The bricks and the box are all rectangular parallelepipeds. The mason wants to pack bricks into the box filling its entire volume and with no bricks sticking out. Find all possible values of the total number of bricks that she can pack.

Is it is possible to choose 24 distinct points in the space such that no three of them lie on the same line and choose 2019 distinct planes in a way that each plane passes through at least 3 of the chosen points and each triple belongs to one of the chosen planes?

Let $n$ be a positive integer divisible by $4$. We consider the permutations $(a_1, a_2,...,a_n)$ of $(1,2,..., n)$ having the following property: for each j we have $a_i + j = n + 1$ where $i = a_j$ . Prove that there are exactly $\frac{ (\frac12 n)!}{(\frac14 n)!}$ such permutations.

How many ways are there to divide a $2\times n$ rectangle into rectangles having integral sides, where $n$ is a positive integer?

In a social network with a fixed finite setback of users, each user had a fixed set of [i]followers[/i] among the other users. Each user has an initial positive integer rating (not necessarily the same for all users). Every midnight, the rating of every user increases by the sum of the ratings that his followers had just before midnight. Let $m$ be a positive integer. A hacker, who is not a user of the social network, wants all the users to have ratings divisible by $m$. Every day, he can either choose a user and increase his rating by 1, or do nothing. Prove that the hacker can achieve his goal after some number of days. [i]Vladislav Novikov[/i]

How many ways are there to arrange the five letters P,U,M,A,C, such that the two vowels are not adjacent?

Let $n$ be a positive integer greater than $1$ and let us call an arbitrary set of cells in a $n\times n$ square $\textit{good}$ if they are the intersection cells of several rows and several columns, such that none of those cells lie on the main diagonal. What is the minimum number of pairwise disjoint $\textit{good}$ sets required to cover the entire table without the main diagonal?

A positive integer is called extra-even if all of its digits are even. Compute the number of positive integers $n$ less than or equal to $2022$ such that both $n$ and $2n$ are both extra-even.

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 $2016$ good partitions. PS. [url=https://artofproblemsolving.com/community/c6h1268855p6622233]2015 ISL C3 [/url] has 2015 instead of 2016