2006 xIMC keystage2-EMIC team

Four different natural numbers, each greater than $3$, are placed in increasing order in four boxes with sum $27$. How many ways are possible?

For $22$, the sum of its digits equals their product. Find the smallest eight-digit natural number with this property.

A four-digit number $X$ has no zero digits. Reversing them gives $Y$. Their sum is $14773$ and their difference is $3177$. Find the larger number.

$ABCD$ is a parallelogram. Points $P,Q,R,S$ lie respectively on sides $AB,BC,CD,DA$, with $AP=DR$. If $ABCD$ has area $16$ cm$^2$, find the area of quadrilateral $PQRS$.

Adi needed $100$ in his final exam to obtain an overall average of $90$. He scored only $75$, giving an overall average of $85$. How many exams did he take altogether?

Annisa makes a rectangular prism from $120$ unit cubes and paints all six outer faces. After disassembling it, $24$ cubes have no painted face. Find the prism’s surface area in square units.

Unit cubes form the tower-like solid shown, with a hole passing left to right, top to bottom, and front to back. How many unit cubes remain?

$31513$ and $34369$ leave the same remainder when divided by the same three-digit number. What is that remainder?

Place any four digits from $1$ to $5$ in a $2\times2$ square. In each row the left digit exceeds the right; in each column the top digit exceeds the bottom. How many arrangements are there? Two examples are shown in the original figure.

A robot remote has three buttons moving it forward $1$, $2$, or $3$ steps. How many different sequences of button presses move it exactly $8$ steps?