There are exactly 7 possible tetrominos (groups of 4 connected squares in a grid): [img]https://cdn.discordapp.com/attachments/813077401265242143/816189385859006474/tetris.png[/img] Daniel has a $2 \times 20210$ rectangle and wants to tile the interior with tetrominos without overlaps, pieces sticking out, or extra pieces left over. Note that you are allowed to rotate tetrominos but not reflect them. For how many multisets of tetrominos (ie. an ordered tuple of how many of each tile he has) is it possible to exactly tile his $2\times20210$ rectangle? [i]Proposed by Dilhan Salgado[/i]

Suppose $\frac{1}{2} < s < 1$ . An insect flying on $[0,1]$ . If it is on point $a$ , it jump into point $ a\times s$ or $(a-1) \times s +1$ . For every real number $0 \le c \le 1$, Prove that insect can jump that after some jumps , it has a distance less than $\frac {1}{1402}$ from point $c$. [i]Proposed by Navid Safaei [/i]

Consider the following one-person game played on the real line. During the game disks are piled at some of the integer points on the line. To perform a move in the game, the player chooses a point $ j$ at which at least two disks are piled and then takes two disks from the point $ j$ and places one of them at $ j\minus{}1$ and one at $ j\plus{}1$. Initially, $ 2n\plus{}1$ disks are placed at point $ 0$. The player proceeds to perform moves as long as possible. Prove that after $ \frac{1}{6} n(n\plus{}1)(2n\plus{}1)$ moves no further moves will be possible and that at this stage, one disks remains at each of the positions $ \minus{}n,\minus{}n\plus{}1,...,0,...n$.

The numbers $$\frac{1}{i+j-1} \,\,\,\,\,\,\, (i = 1,2,...,n; j = 1,2,...,n)$$ are written in an $n$ by $n$ table: the number $1/(i + j - 1)$ stands at the intersection of the $i$-th row and $j$-th column. Chose any $n$ squares of the table so that no two of them stand in the same row and no two of them stand in the same column. Prove that the sum of the numbers in these $n$ squares is not less than $1$. (Sergey Ivanov, St Petersburg)

Is it possible to cover an $n \times n$ chessboard which has its center square cut out with tiles shown in the picture (each tile covers exactly $4$ squares, tiles can be rotated and turned around) if a) $n = 5$, b) $n = 2003$? [img]https://cdn.artofproblemsolving.com/attachments/6/5/8fddeefc226ee0c02353a1fc11e48ce42d8436.png[/img]

Determine for which $m$ there exist exactly $2^{15}$ subsets $X$ of $\{1,2,...,47\}$ with the following property: $m$ is the smallest element of $X$, and for every $x \in X$, either $x+m \in X$ or $x+m > 47$.

Given a positive integer $ n\geq 2$, let $ B_{1}$, $ B_{2}$, ..., $ B_{n}$ denote $ n$ subsets of a set $ X$ such that each $ B_{i}$ contains exactly two elements. Find the minimum value of $ \left|X\right|$ such that for any such choice of subsets $ B_{1}$, $ B_{2}$, ..., $ B_{n}$, there exists a subset $ Y$ of $ X$ such that: (1) $ \left|Y\right| \equal{} n$; (2) $ \left|Y \cap B_{i}\right|\leq 1$ for every $ i\in\left\{1,2,...,n\right\}$.

Prove that among any $51$ vertices of the $101$-regular polygon there are three that are the vertices of an isosceles triangle.

Consider a $n$x$n$ table such that the unit squares are colored arbitrary in black and white, such that exactly three of the squares placed in the corners of the table are white, and the other one is black. Prove that there exists a $2$x$2$ square which contains an odd number of unit squares white colored.

Let $p_n$ denote the probability that, in $n$ tosses, a fair coin shows the head up $100$ consecutive times. Prove that the sequence $(p_n)$ converges and determine its limit.

For a positive integer $n(\geq 5)$, there are $n$ white stones and $n$ black stones (total $2n$ stones) lined up in a row. The first $n$ stones from the left are white, and the next $n$ stones are black. $$\underbrace{\Circle \Circle \cdots \Circle}_n \underbrace{\CIRCLE \CIRCLE \cdots \CIRCLE}_n $$ You can swap the stones by repeating the following operation. [b](Operation)[/b] Choose a positive integer $k (\leq 2n - 5)$, and swap $k$-th stone and $(k+5)$-th stone from the left. Find all positive integers $n$ such that we can make first $n$ stones to be black and the next $n$ stones to be white in finite number of operations.

In a certain college containing $1000$ students, students may choose to major in exactly one of math, computer science, finance, or English. The [i]diversity ratio[/i] $d(s)$ of a student $s$ is the defined as number of students in a different major from $s$ divided by the number of students in the same major as $s$ (including $s$). The [i]diversity[/i] $D$ of the college is the sum of all the diversity ratios $d(s)$. Determine all possible values of $D$.

Each vertex of a regular $17$-gon is colored red, blue, or green in such a way that no two adjacent vertices have the same color. Call a triangle “multicolored” if its vertices are colored red, blue, and green, in some order. Prove that the $17$-gon can be cut along nonintersecting diagonals to form at least two multicolored triangles. (A diagonal of a polygon is a a line segment connecting two nonadjacent vertices. Diagonals are called nonintersecting if each pair of them either intersect in a vertex or do not intersect at all.)

[hide=C stands for Cantor, G stands for Gauss]they had two problem sets under those two names[/hide] [u]Set 1[/u] [b]C.1 / G.1[/b] Daniel is exactly one year younger than his friend David. If David was born in the year $2008$, in what year was Daniel born? [b]C.2 / G.3[/b] Mr. Pham flips three coins. What is the probability that no two coins show the same side? [b]C.3 / G.2[/b] John has a sheet of white paper which is $3$ cm in height and $4$ cm in width. He wants to paint the sky blue and the ground green so the entire paper is painted. If the ground takes up a third of the page, how much space (in cm$^2$) does the sky take up? [b]C.4 / G.5[/b] Jihang and Eric are busy fidget spinning. While Jihang spins his fidget spinner at $15$ revolutions per second, Eric only manages $10$ revolutions per second. How many total revolutions will the two have made after $5$ continuous seconds of spinning? [b]C.5 / G.4[/b] Find the last digit of $1333337777 \cdot 209347802 \cdot 3940704 \cdot 2309476091$. [u]Set 2[/u] [b]C.6[/b] Evan, Chloe, Rachel, and Joe are splitting a cake. Evan takes $\frac13$ of the cake, Chloe takes $\frac14$, Rachel takes $\frac15$, and Joe takes $\frac16$. There is $\frac{1}{x}$ of the original cake left. What is $x$? [b]C.7[/b] Pacman is a $330^o$ sector of a circle of radius $4$. Pacman has an eye of radius $1$, located entirely inside Pacman. Find the area of Pacman, not including the eye. [b]C.8[/b] The sum of two prime numbers $a$ and $b$ is also a prime number. If $a < b$, find $a$. [b]C.9[/b] A bus has $54$ seats for passengers. On the first stop, $36$ people get onto an empty bus. Every subsequent stop, $1$ person gets off and $3$ people get on. After the last stop, the bus is full. How many stops are there? [b]C.10[/b] In a game, jumps are worth $1$ point, punches are worth $2$ points, and kicks are worth $3$ points. The player must perform a sequence of $1$ jump, $1$ punch, and $1$ kick. To compute the player’s score, we multiply the 1st action’s point value by $1$, the $2$nd action’s point value by $2$, the 3rd action’s point value by $3$, and then take the sum. For example, if we performed a punch, kick, jump, in that order, our score would be $1 \times 2 + 2 \times 3 + 3 \times 1 = 11$. What is the maximal score the player can get? [u]Set 3[/u] [b]C.11[/b] $6$ students are sitting around a circle, and each one randomly picks either the number $1$ or $2$. What is the probability that there will be two people sitting next to each other who pick the same number? [b]C.12 / G. 8[/b] You can buy a single piece of chocolate for $60$ cents. You can also buy a packet with two pieces of chocolate for $\$1.00$. Additionally, if you buy four single pieces of chocolate, the fifth one is free. What is the lowest amount of money you have to pay for $44$ pieces of chocolate? Express your answer in dollars and cents (ex. $\$3.70$). [b]C.13 / G.12[/b] For how many integers $k$ is there an integer solution $x$ to the linear equation $kx + 2 = 14$? [b]C.14 / G.9[/b] Ten teams face off in a swim meet. The boys teams and girls teams are ranked independently, each team receiving some number of positive integer points, and the final results are obtained by adding the points for the boys and the points for the girls. If Blair’s boys got $7$th place while the girls got $5$th place (no ties), what is the best possible total rank for Blair? [b]C.15 / G.11[/b] Arlene has a square of side length $1$, an equilateral triangle with side length $1$, and two circles with radius $1/6$. She wants to pack her four shapes in a rectangle without items piling on top of each other. What is the minimum possible area of the rectangle? PS. You should use hide for answers. C16-30/G10-15, G25-30 have been posted [url=https://artofproblemsolving.com/community/c3h2790676p24540145]here[/url] and G16-25 [url=https://artofproblemsolving.com/community/c3h2790679p24540159]here [/url] . Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Given is a sequence of real numbers $\{a_n\}^{\infty}_{n=1}$ such that $a_n \ne a_m$ for $n\ne m,$ given is a natural number $k$. Construct an injective map $P:\{1,2,\ldots,20k\}\to\mathbb Z^+$ such that the following inequalities hold: $$a_{p(1)}<a_{p(2)}<...<a_{p(10)}$$ $$ a_{p(10)}>a_{p(11)}>...>a_{p(20)}$$ $$a_{p(20)}<a_{p(21)}<...<a_{p(30)}$$ $$...$$ $$a_{p(20k-10)}>a_{p(20k-9)}>...>a_{p(20k)}$$ $$a_{p(10)}>a_{p(30)}>...>a_{p((20k-10))} $$ $$a_{p(1)}<a_{p(20)}<...<a_{p(20k)},$$

Call a positive integer [i]almost square[/i] if it is not a perfect square, but all of its digits are perfect squares. For example, both $149$ and $904$ are almost square, but $144$ and $936$ are not. Find the number of positive integers less than $1000$ that are not almost square.

Some of the vertices of a convex $n$-gon are connected by segments, such that any two of them have no common interior point. Prove that, for any $n$ points in general position, there exists a one-to-one correspondence between the points and the vertices of the $n$ gon, such that any two segments between the points, corresponding to the respective segments from the $n$ gon, have no common interior point.

Given $12$ segments, it is known that they can be divided into $4$ groups of $3$ segments each in such a way that a triangle can be formed from the segments of each triplet. Is it always possible to divide these $12$ segments into $3$ groups of $4$ segments each in such a way that a quadrilateral can be formed from the segments of each quartet? [i]Proposed by Mykhailo Shtandenko[/i]

Decide whether the integers $1,2,\ldots,100$ can be arranged in the cells $C(i, j)$ of a $10\times10$ matrix (where $1\le i,j\le 10$), such that the following conditions are fullfiled: i) In every row, the entries add up to the same sum $S$. ii) In every column, the entries also add up to this sum $S$. iii) For every $k = 1, 2, \ldots, 10$ the ten entries $C(i, j)$ with $i-j\equiv k\bmod{10}$ add up to $S$. [i](Proposed by Gerhard Woeginger, Austria)[/i]

In a lecture hall some chairs are placed in rows and columns, forming a rectangle. In each row there are $6$ boys sitting and in each column there are $8$ girls sitting, whereas $15$ places are not taken. What can be said about the number of rows and that of columns?

Given a $m \times n$ rectangle where $m,n\geq 2023$. The square in the $i$-th row and $j$-th column is filled with the number $i+j$ for $1\leq i \leq m, 1\leq j \leq n$. In each move, Alice can pick a $2023 \times 2023$ subrectangle and add $1$ to each number in it. Alice wins if all the numbers are multiples of $2023$ after a finite number of moves. For which pairs $(m,n)$ can Alice win? [i]Proposed by Boon Qing Hong[/i]

A group of $n{}$ $(n>1)$ people each visited $k{}$ $(k>1)$ citites. Each person makes a list of these $k$ cities in the order they want to visit them. A permutation $(a_1,a_2,\ldots,a_k)$ is called $m-prefered$ $(m\in\mathbb{N})$, if for every $i=1,2,\ldots,k$ there are at least $m$ people that would prefer to visit the city $a_i$ before the city $a_{i+1}$, $(a_{k+1}=a_1)$. Prove that there exists an m-prefered permutation if and only if $km\leq n(k-1)$.

Can we partition the positive integers in two sets such that none of the sets contains an infinite arithmetic progression of nonzero ratio ?

Maisy and Jeff are playing a game with a deck of cards with $4$ $0$’s, $4$ $1$’s, $4$ $2$’s, all the way up to $4$ $9$’s. You cannot tell apart cards of the same number. After shuffling the deck, Maisy and Jeff each take $4$ cards, make the largest $4$-digit integer they can, and then compare. The person with the larger $4$-digit integer wins. Jeff goes first and draws the cards $2,0,2,1$ from the deck. Find the number of hands Maisy can draw to beat that, if the order in which she draws the cards matters. [i]Proposed by Richard Chen[/i]

Each segment whose endpoints are the vertices of a given regular $100$-gon is colored red, if the number of vertices between its endpoints is even, and blue otherwise. (For example, all sides of the $100$-gon are red.) A number is placed in every vertex so that the sum of their squares is equal to $1$. On each segment the product of the numbers at its endpoints is written. The sum of the numbers on the blue segments is subtracted from the sum of the numbers on the red segments. What is the greatest possible result? (Ilya Bogdanov)