2007 xIMC keystage2-EMIC team

A highway from $A$ to $B$ has an uphill and a downhill section. A car goes $20$ km/h uphill, $35$ km/h downhill. It takes $9$ hours from $A$ to $B$, and $7\frac12$ hours back. Find the downhill distance from $A$ to $B$.

One side of a street has consecutive odd house numbers starting at $1$, the other consecutive even numbers starting at $2$. The even side uses $256$ digits altogether, the odd side $404$. Find the difference between the largest odd and even house numbers.

Parallelogram $ABCD$ has area $10$, $AB=3$, $BC=5$, $AE=BF=AG=2$, and $GH\parallel EF$, with points positioned as shown. Find the area of $EFHG$.

Find the two smallest integers differing by $3$, each with digit sum a multiple of $11$.

Join two different two-digit primes to form a four-digit number. Give one such number divisible by the average of those primes. For example $1317,1713$ do not work because they are not divisible by $15$.

How many prime factors does $2+2^2+2^3+\cdots+2^{16}$ have?

A pencil, eraser and notebook cost $100$ dollars altogether. The notebook costs more than two pencils, three pencils cost more than four erasers, and three erasers cost more than the notebook. Find each positive whole-dollar price.

Eight pairs of natural numbers have the product of their sum and difference equal to $1995$. Which pair has the greatest difference?

An already fenced $52$ m by $24$ m plot is subdivided into identical square sections with sides parallel to its boundary. $1172$ m of fencing is available for the subdivision. What is the maximum number of square sections?

Find the total number of ways to write $270$ as a sum of consecutive positive integers.