### Old Nuts

In turn 4 people throw away three nuts from a pile and hide a quarter of the remainder finally leaving a multiple of 4 nuts. How many nuts were at the start?

### Just Touching

Three semi-circles have a common diameter, each touches the other two and two lie inside the biggest one. What is the radius of the circle that touches all three semi-circles?

### Days and Dates

Investigate how you can work out what day of the week your birthday will be on next year, and the year after...

# Weekly Problem 52 - 2006

##### Stage: 4 and 5 Challenge Level:

1:3

Let the length of a short side of a rectangle be $x$ and the length of a long side be $y$. Then the whole square has side of length $(y+x)$, whilst the small square has side of length $y-x$. As the area of the whole square is four times the area of the small square, the length of the side of the whole square is twice the length of the side of the small square. Therefore $y+x=2(y-x)$, i.e., $y=3x$ so $x:y=$1:3.

This problem is taken from the UKMT Mathematical Challenges.

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