On New Year’s Eve, a group of kids gathered together to play a simple math game. The first kid writes $2022$ on a blackboard. The second kid and every kid thereafter replaces the number written by the previous kid with the product of the digits of that number plus $12$. What number does the $59$th kid write?
Fill up the infinite grid with the numbers $1,2,3,4,5,6,7,1,2,3,4,5,6,7,1,2,3,4,5,\ldots$ (numbers from $1$ to $7$, repeatedly) in a counterclockwise spiral-like manner starting at the shaded cell as shown in the diagram below. Which number is written in the cell that is located $2022$ cells below the shaded cell?
Select any positive integer $n$ and write down the integers from $0$ to $n$, inclusive, in some order with no spaces.
What is the minimum value of $n$ for which the resulting string can have the same value when read forwards and backwards? What is the resulting string?
A four-digit positive integer is called “good” if it consists of two pairs of same digits in some order but not all four digits are the same. For example, the numbers $2211$, $2424$ and $7007$ are good, while $5555$ and $3111$ are not. Find the total number of “good” four-digit positive integers which are divisible by $7$ or $101$, but not divisible by both.
In a group of five people, the ages of four of them are known to be $21$, $53$, $19$ and $60$. It is known that the average of the five people is an odd number. If we arrange the ages of these five people in an increasing order, then the middle age is a multiple of $3$. Find the sum of all possible values for the fifth age.
Five chess players, Andy, Boris, Clark, Dick and Eric played in a tournament, where every two participants played against each other exactly once. For each game, a player will get $2$ points for a win, $1$ point for a draw, and $0$ point for a loss. At the end of the tournament, one noticed that: Boris drew all his games, Clark won exactly two games, Dick is ahead of Andy by $1$ point, Eric is not in last place and he lost only to the only player who accumulated the lowest number of points. If their total scores are $a,b,c,d,e$, respectively, then what is the five-digit number $\overline{abcde}$?
A rectangular sheet of paper is cut into two pieces as shown in the diagram below, where one region is shaded. The eight sides of the shaded region have lengths of $1$ cm, $2$ cm, $3$ cm, $4$ cm, $5$ cm, $6$ cm, $7$ cm and $8$ cm, in some order. What are the maximum and the minimum possible areas, in cm$^2$, of the unshaded region?
The maximum possible area of the unshaded region is ______ cm$^2$.
The minimum possible area of the unshaded region is ______ cm$^2$.
In a $8\times8$ board, each cell has a bug. After a bell rings, all bugs jump to an adjacent cell in the same row or in the same column, and do not jump out of the board. Each cell can be empty, or contain more than one bug. What is the maximum number of empty cells after the bell rings?
We arrange the following seven $4$-digit numbers: $2021$, $2022$, $2023$, $2024$, $2025$, $2026$ and $2027$ in a row, in such a way that any two adjacent numbers are relatively prime. For example: $2027,2022,2023,2024,2025,2026,2021$ is one such arrangement.
