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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.

Minus One Two Three

Stage: 4 Challenge Level: Challenge Level:2 Challenge Level:2

Look at the two brackets for the factors that make the product. You are evaluating it for $x$ values of -1, -2, and -3 . Think what you could change systematically about those factors and see how it affects the size of the product for each of the three $x$ values.