2009 xIMC keystage2-EMIC individual

Find the smallest positive integer whose product with $543$ ends in $2009$.

Linda turned ten on $13$ July $1991$, noticing $13\times7=91$. Her two younger brothers also have this day-times-month relationship on their tenth birthdays, in that century. Their birthdays are on consecutive calendar days. When was the youngest brother born?

Pair $1,2,\ldots,12$ into six pairs with distinct prime sums. Find the largest of those primes.

In the overlapping-square figure, $\frac34$ of the larger square and $\frac57$ of the smaller are shaded. Find the ratio of their shaded areas.

The sequence is $1,1,2,3,5,8,13,\ldots$, each later term the sum of the previous two. Find the remainder of term $2009$ on division by $8$.

Ampang Street has at most $15$ houses numbered $1,2,3,\ldots$. Mrs Lau does not live in the first. The product of all house numbers before hers equals the product after hers. How many houses are there?

Right triangle $ABC$ has $\angle B=90^\circ$, $BC=42$ cm, $AB=56$ cm. Draw a semicircle with diameter $AC$ and a quarter-circle centred at $B$ with radius $BC$. Find the shaded area shown, using $\pi=22/7$.

A three-digit number has distinct digits. The largest minus the smallest rearrangement of its digits equals the original number. Find it.

The last three digits of a perfect square are equal and nonzero. Find the smallest such square.

Lynn walks from $A$ to $B$ while Mike cycles from $B$ to $A$. They start together and meet after one hour. Mike turns at $A$ and catches Lynn $40$ minutes after that first meeting. He later turns at $B$. Find the ratio of the third meeting point\'s distances from $A$ and $B$.

The figure shows the net of a polyhedron. How many edges does the folded polyhedron have?

Five circles have radius $1$ cm, their centres at the triangle vertices shown. Find the total shaded area, using $\pi=22/7$.

Ten steps lead to a platform. Step $6$ cannot be stepped on, only jumped over. Michael moves up one or two steps at a time. How many ways can he reach step $10$?

Four positive integers $a\lt b\lt c\lt d$ form a friendly group if $(d-c)(c-b)(b-a)$ is divisible by $2009$. How many friendly groups lie in $1$ through $60$?

Colour the five circles in the given graph using five available colours. Joined circles must have different colours. How many colourings are there?