Problem 1:

An arrow is formed in a 2 2 square by joining the bottom corners to the midpoint of the top edge and the centre of the square.

Find the area of the arrow.

Problem 2:Two ladders are placed on opposite diagonals in an alley such that one ladder reaches *a* units up one wall, the other ladder reaches *b* units up the opposite wall and they intersect *h* units above the ground.

Prove the following result.

1 a | + | 1 b | = | 1 h |

Problem 3:

How many digits does the number 2

^{1000}contain?

Solution to 1:

Consider the two diagrams below.

The area of the square is 4, so the area of the large triangle is 2 (half of the square) and the area of the small triangle is 1 (quarter of the square).

Hence the area of the arrow is 2 1 = 1 square unit.

Solution to 2:Consider the following diagram:

By similar triangles:

x+ya= yhand x+yb= xh

Adding equations:

x+ya+ x+yb= x+yh

Dividing by (*x* + *y*):

1 a | + | 1 b | = | 1 h |

Solution to 3:

As 2^{1000} is not a multiple of 10, it follows that,

10^{m} 2^{1000} 10^{m + 1}, where 10^{m} contains *m* + 1 digits.

Solving 2^{1000} = 10^{k}, where *m* *k* *m* + 1

*k* = *log* 2^{1000} = 1000 *log* 2 301.02999... , so *m* = [1000 *log* 2] = 301.

Hence 2^{1000} contains 302 digits.

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