A job at Hai Liang Education Park can be done by Alex alone in 6 hours and by Bob alone in 10 hours. Alex works on the job for one hour alone, then Bob continues to work on the job for one hour alone. If they repeat the pattern, in how many hours can the job be done? Express your answer as a common fraction.
In the diagram below, $E$ is a point on side $AD$ of rectangle $ABCD$. Points $F$, $G$, $H$ and $I$ are midpoints of $CE$, $BF$, $CG$ and $BH$ respectively.
If the area of triangle $BCI$ is 1 cm², find the area, in cm², of rectangle $ABCD$.
For any positive integer $n$, we define the function $f(n)$ to be the sum of the digits of $n$ and the number of digits of $n$. For example, $f(218)=2+1+8+3=14$. (Note: The first digit of $n$, reading from left to right, cannot be 0). What is the sum of maximum and minimum values of $n$ such that $f(n)=6$?
A rectangle is divided into 9 smaller portions as shown in the diagram below. The perimeter, in cm, of the 5 known portions are also given. Find the perimeter, in cm, of the original rectangle.
A cylindrical water tank, with diameter 2.8 m and height 4.2 m, is filled in by a pipe of diameter 7 cm, through which water flows at the rate of 4 m/sec. How many minutes will it take for the pipe to completely fill the tank? (Take $\pi=\frac{22}{7}$.)
We want to divide a square into obtuse triangles such that every two triangles meet at a common vertex or at a common edge or are disjoint. At least how many triangles can we have?
Two overlapped equilateral triangles are shown in the diagram on the right. The sides of each triangle are parallel to the sides of the other. The perimeter of the two triangles are 744 cm and 930 cm, respectively. What is the perimeter, in cm, of shaded hexagon?
Sunny got three boxes from his father, which contained some number of marbles. His father said that the number of marbles inside the first, second and third boxes are three consecutive integers in increasing order and that they are divisible by 5, 7 and 9 respectively. What is the minimum total number of marbles inside the three boxes?
Given the sequence 1, 1, 1, 3, 5, 9, 17, 31, …, where the $n$th term after the 3rd term is the sum of three previous terms. For example, the 4th term is $1+1+1=3$ and the 5th term is $1+1+3=5$. What is the remainder when the 2018th term is divided by 8?
If the eight-digit number $\overline{2018MN28}$ is divided by 7, the remainder is 5. If the same eight-digit number is divided by 11, the remainder is 9. Find the largest possible value of the two-digit number $\overline{MN}$.
Let $I$, $M$, $S$ and $O$ represent different digits and the sum of $IMSO$, $ISMO$, $OMSI$, $OSMI$, $MISO$, $MOSI$, $SIMO$ and $SOMI$ is equal to 60012. What is the value of $I+M+S+O$?
In the following $5\times5$ square grid, the numbers 1, 2, 3, 4 and 5 are filled in, such that each number appears only once in each row and only once in each column. Find the number filled in the shaded square.
We place 2018 distinct points inside a square, then divide the square into triangles, whose vertices are those 2018 points and the 4 vertices of the square. (that is, we have 2022 points in total, which are vertices of the triangles). For each triangle, each point is either one of its vertices or lies completely outside the triangle itself. In how many triangles has the square been divided?
In the diagram shown below, each empty white cell is filled with an integer from 1 to 9. Each number is used at most once. The gray cell is filled with a sign among $\times$, $\div$, $+$ or $-$. In each line with 3 numbers, the calculation is from left to right. In the last column, the calculation is from top to bottom. Show one possible solution.
In the grid below, each letter represents a different integer from 1 to 7. When comparing the sum of the numbers in any row or column to that of any other rows or columns, the two sums must be of the same parity (either both even or both odd) and differ by at most 2. Find the sum of all possible values of $a$.
