Something Maths!

Ok, well given that i'm a Maths Student I thought it was about time I added some maths problems with some solutions, the first problem is the easiest and the last problem is the most difficult, solutions are available below the problems.

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 2¹⁰⁰⁰ 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 + y a

=

y h

and

x + y b

=

x h

Adding equations:

x + y a

+

x + y b

=

x + y h

Dividing by (x + y):

1 a

+

1 b

=

1 h

Solution to 3:

As 2¹⁰⁰⁰ is not a multiple of 10, it follows that,
10^m 2¹⁰⁰⁰ 10^{m + 1}, where 10^m contains m + 1 digits.

Solving 2¹⁰⁰⁰ = 10^k, where m k m + 1

k = log 2¹⁰⁰⁰ = 1000 log 2 301.02999... , so m = [1000 log 2] = 301.

Hence 2¹⁰⁰⁰ contains 302 digits.

The problems and solutions are all taken from http://www.mathschallenge.net