2007 xIMC keystage2-EMIC individual

Three-digit numbers $\overline{abc},\overline{cba}$ have product $396396$, with $a\gt c$. Find $\overline{abc}$.

Right triangle $ACD$ has shaded triangle $BDE$ of area $10$ cm$^2$, $AD=5$ cm, $AB=BC$, and $DE=EC$, as in the original figure. Find $AB$ in cm.

A $4\times5\times6$ cm wooden prism is painted red outside and cut into unit cubes. Find the ratio of cubes with exactly two red faces to cubes with exactly three red faces.

Eve’s two-digit age reverses to her mother’s current age. Her mother’s birthday is tomorrow, when she will be twice Eve’s age. Their birthdays differ. How old is Eve?

How many three-digit numbers leave remainders $1,2,3,4,7$ when divided by $2,3,4,5,8$ respectively?

A fenced right-triangular habitat has perpendicular sides $12$ m and $16$ m. A giraffe can reach grass outside at maximum distance $2$ m from the fence. Find the maximum outside area accessible, using $\pi=3.14$.

Mary and Peter start together at the same point on a $400$ m circular track and run in opposite directions. Mary’s speed is $3/5$ of Peter’s. After $200$ seconds they have met four times. How much faster, in m/s, does Peter run?

Evaluate $2^{2007}-(2^{2006}+2^{2005}+\cdots+2^2+2+1)$.

Stamp collector $A$ has $18$ more stamps than $B$. $B:C=7:5$, and $(B+C):A=6:5$. How many stamps does $C$ have?

What is the smallest number of factors to remove from $1\times2\times\cdots\times27$ so the remaining product is a perfect square?

Two trains travel towards each other between towns $1320$ km apart at constant speeds. After meeting, train $A$ takes $5$ hours to finish, train $B$ takes $7.2$ hours. Find $A$’s speed in km/h.

A container has eight colours of balls, ninety balls of each colour. What minimum number drawn guarantees at least nine balls in each of four different colours?

In regular hexagon $ABCDEF$, diagonals $FC$ and $BD$ meet at $G$. Find the ratio of the area of triangle $BCG$ to quadrilateral $FEDG$.

Three primes have squares summing to $5070$. Find their product.

$ABCDEF$ is a regular hexagon with centre $O$. $M,N$ are the midpoints of $DE,OB$ respectively. If triangles $FNO$ and $FME$ have combined area $3$ cm$^2$, find the hexagon area.