## Problems

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

Find the smallest positive integer N such that N/2 is a perfect
cube, N/3 is a perfect fifth power and N/5 is a perfect seventh
power.

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

a) A four digit number (in base 10) aabb is a perfect square. Discuss ways of systematically finding this number.
(b) Prove that 11^{10}-1 is divisible by 100.

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

In how many ways can the number 1 000 000 be expressed as the
product of three positive integers?

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

A tetrahedron has two identical equilateral triangles faces, of side length 1 unit. The other two faces are right angled isosceles triangles. Find the exact volume of the tetrahedron.

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

ABCD is a rectangle and P, Q, R and S are moveable points on the
edges dividing the edges in certain ratios. Strangely PQRS is
always a cyclic quadrilateral and you can find the angles.

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