In the diagram, $ABCD$ is a square where $AB=24$ cm. Let point $Q$ be the midpoint of side $CD$, point $P$ be the intersection of diagonals $AC$ and $BD$, point $R$ be the intersection of line segments $AQ$ and $BD$. Find the sum of the area, in cm², of triangle $ARP$ and triangle $DRQ$.
A $9\times9\times9$ cube is formed by using 729 unit cubes, each of size $1\times1\times1$. How many different cubes of different sizes can we find in this $9\times9\times9$ cube? (Note: Two cubes having the same size but are located at different positions must be counted as two different cubes)
Peter is about to shade some cells in the $6\times6$ grid which is shown on the right. Any two cells that share a side must at most have one shaded cell. While on those un-shaded cells, Peter writes a number which is the total number of cells that shares a side with this cell that are shaded. Given that all the numbers written is at most 3, what is the maximum number of cells that Peter can shade? Fill the grid with one possible solution of shaded cells and filling number.
Find all possible three-digit numbers $\overline{abc}$ such that $\overline{abc}=a!+b!+c!$. (Note: $n!$ (read as $n$ factorial) means $n!=n\times(n-1)\times(n-2)\times\cdots\times3\times2\times1$. For example, $6!=6\times5\times4\times3\times2\times1=720$.)
In parallelogram $ABCD$, point $E$ is on $AD$ such that $AE=DE$, point $F$ is on $CD$ and point $H$ is on $AB$ such that $EF\perp DC$ and $CH\perp AB$. Let $AH:HB=3:2$ and point $G$ is on line $CH$. If the area of triangle $EFG$ is 296 cm², find the area, in cm², of parallelogram $ABCD$.
One may express a number in base-8. For example, 37 could be written as $(45)_8$ in base-8 because $(45)_8=4\times8+5=37$. On the other hand, $(49)_8$ does not exist because $4\times8+9=41$ should be expressed as $(51)_8$. Let $\overline{AB}$ be a two-digit prime number in base-10, where $A,B<8$. If the value of $(\overline{AB})_8$ is also a prime number in base-10, how many possible values of $(\overline{AB})_8$ can we obtain?
Colour each edge of the cube below by one of the 3 colours: blue, red or yellow so any 2 edges that share the same vertex should have different colours. How many ways can we colour the cube? (Note: Colouring is considered the same if when rotating or reflecting, we see it the same way.)
Using digits 1, 2, 3, 4, 5 and 6 to form a six-digit number $\overline{abcdef}$, where different letters represent different digits. If $2\mid\overline{ab}$, $3\mid\overline{abc}$, $4\mid\overline{abcd}$, $5\mid\overline{abcde}$ and $6\mid\overline{abcdef}$, find all possible six-digit numbers $\overline{abcdef}$.
A regular octahedron is shown in the diagram on the right. There are 6 vertices altogether connected by a total of 12 edges. An ant starts from vertex $A$ and moves along any edge but cannot go through any edge more than once. If the ant visits each vertex exactly once before it returns to vertex $A$, how many different routes are there?
Let $A$ be a positive integer consisting of different digits, and $B$ be a positive integer whose digits are a rearrangement of the digits of $A$. Suppose that $A-B=\underbrace{11\ldots1}_{m}$ ($m$ copies of 1’s). Find the maximum value of $m$ that satisfy the given requirements.
Let $n$ be a positive integer such that all of its digits are distinct and that its digit sum is 17. If $n<2018$, how many different possible values of $n$ are there?
Using 16 black pawns and 16 white pawns, how many ways can we place 16 pawns into the $4\times4$ chessboard, where each pawn is occupying 1 cell, such that in each row and in each column the number of white pawns is even?
