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Converse

Clearly if a, b and c are the lengths of the sides of a triangle and the triangle is equilateral then a^2 + b^2 + c^2 = ab + bc + ca. Is the converse true, and if so can you prove it? That is if a^2 + b^2 + c^2 = ab + bc + ca is the triangle with side lengths a, b and c necessarily equilateral?

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Consecutive Squares

The squares of any 8 consecutive numbers can be arranged into two sets of four numbers with the same sum. True of false?

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Parabolic Patterns

The illustration shows the graphs of fifteen functions. Two of them have equations y=x^2 and y=-(x-4)^2. Find the equations of all the other graphs.

Geometric Parabola

Stage: 4 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3
$$y=ax^2+2bx+c$$

Start with $a=1, b=2, c=4$ - what does the graph look like? What key points does it pass through?

Then try $a=2, b=4, c=8$, $a=4, b=8, c=16$ and so on. Look for similarities and differences between the graphs.