Odd Differences

The diagram illustrates the formula: 1 + 3 + 5 + ... + (2n - 1) = n² Use the diagram to show that any odd number is the difference of two squares.

Factorial

How many zeros are there at the end of the number which is the product of first hundred positive integers?

Rachel's Problem

Is it true that $99^n$ has 2n digits and $999^n$ has 3n digits? Investigate!

Fracmax

Age 14 to 16 Challenge Level:

Find the maximum of

$${1\over p} + {1\over q} + {1\over r}$$

where $p$, $q$ and $r$ are positive integers and

$${1\over p} + {1\over q} + {1\over r} < 1.$$

Prove that it is indeed a maximum.

Continued fractions. Integers. Creating and manipulating expressions and formulae. Inequalities. Properties of numbers. Fractions. Diophantine equations. Graphs. Mathematical reasoning & proof. Calculating with fractions.

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Odd Differences

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Rachel's Problem

Fracmax

Age 14 to 16 Challenge Level: