Creating and manipulating expressions and formulae

problem
System speak

Age

16 to 18

Challenge level

Five equations... five unknowns... can you solve the system?
problem
Look before you leap

Age

16 to 18

Challenge level

Relate these algebraic expressions to geometrical diagrams.
problem
Sums of squares

Age

16 to 18

Challenge level

Can you prove that twice the sum of two squares always gives the sum of two squares?
problem
Calculus countdown

Age

16 to 18

Challenge level

Can you hit the target functions using a set of input functions and a little calculus and algebra?
problem
Polynomial interpolation

Age

16 to 18

Challenge level

Can you fit polynomials through these points?
problem
Ball bearings

Age

16 to 18

Challenge level

If a is the radius of the axle, b the radius of each ball-bearing, and c the radius of the hub, why does the number of ball bearings n determine the ratio c/a? Find a formula for c/a in terms of n.
problem
Particularly general

Age

16 to 18

Challenge level

By proving these particular identities, prove the existence of general cases.
problem
Operating machines

Age

16 to 18

Challenge level

What functions can you make using the function machines RECIPROCAL and PRODUCT and the operator machines DIFF and INT?
problem
Interpolating polynomials

Age

16 to 18

Challenge level

Given a set of points (x,y) with distinct x values, find a polynomial that goes through all of them, then prove some results about the existence and uniqueness of these polynomials.