## Problems

##### Age 11 to 16 Challenge Level:

Jo has three numbers which she adds together in pairs. When she
does this she has three different totals: 11, 17 and 22 What are
the three numbers Jo had to start with?”

##### Age 14 to 16 Challenge Level:

Triangle ABC is right angled at A and semi circles are drawn on all three sides producing two 'crescents'. Show that the sum of the areas of the two crescents equals the area of triangle ABC.

##### Age 14 to 16 Challenge Level:

A right-angled isosceles triangle is rotated about the centre point
of a square. What can you say about the area of the part of the
square covered by the triangle as it rotates?

##### Age 11 to 18 Challenge Level:

Given the nets of 4 cubes with the faces coloured in 4 colours, build a tower so that on each vertical wall no colour is repeated, that is all 4 colours appear.

##### Age 16 to 18 Challenge Level:

A polite number can be written as the sum of two or more
consecutive positive integers. Find the consecutive sums giving the
polite numbers 544 and 424. What characterizes impolite numbers?

##### Age 16 to 18 Challenge Level:

Prove that in every tetrahedron there is a vertex such that the
three edges meeting there have lengths which could be the sides of
a triangle.

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