2005 xIMC keystage2-EMIC team

In parallelogram $ABCD$, $BE=EC$. The shaded region has area $2$ cm$^2$. Find the area of $ABCD$ in cm$^2$.

The large square has side $4$ cm, the small square side $3$ cm. Find the shaded area in cm$^2$, using the original diagram.

A circle is divided into six equal sectors. Paint at least one sector black. How many patterns are there when patterns differing only by rotation are counted as one?

Let $n=9+99+999+\cdots+\underbrace{99\cdots9}_{2005\text{ digits}}$. How many digits $1$ occur in $n$?

Ten oil barrels labelled with the distinct digits $0$–$9$ form the triangular arrangement shown. Rearrange them so all three edge sums are equal and as small as possible.

Find a route from a top cell to a bottom cell giving total $175$. Passing through any cell adjacent to a zero resets the running total to zero. Use each cell at most once; use the original puzzle diagram.

Arrange the digits $1$–$9$ in a row of circles. The inclusive interval between $1,2$ sums to $9$; between $2,3$ to $19$; between $3,4$ to $45$; between $4,5$ to $18$.

A man has three children, all with birthdays today. Their ages multiply to $72$ and sum to his house number. The census taker sees that number but still cannot identify their ages. The man adds, “The oldest one likes ice cream,” and the ages are then determined. Find them.

In the multiplication shown, circles hide odd digits and squares hide even digits. Fill in the missing digits. Use the original aligned multiplication, including both partial products.

Donuts are sold in boxes of $7$, $13$, or $25$. What is the largest total that cannot be purchased by combining whole boxes?