Maths item of the month
4 points, 2 lengths
In how many ways can you arrange 4 distinct points in the plane so there are exactly two different distances amongst the 6 pairs?
e.g. if the four points are at the corners of a square then the four sides are the same length and the two diagonals are the same length.
1, 2, 3, 4
Find two quadratic functions f(x), g(x) so the equation f(g(x)) = 0 has the four roots x = 1, 2, 3, 4.
Is it possible to find three quadratic functions f(x), g(x), h(x) so the equation f(g(h(x))) = 0 has the eight roots x = 1, 2, 3, 4, 5, 6, 7, 8?
Happy 2015: A Triple of Triples
2015 is the product of 3 distinct primes: 5×13×31
2014 and 2013 are also the product 3 distinct primes.
Can you find a smaller triple (n, n+1, n+2) where n, n+1 and n+2 are all the product of 3 distinct primes?
Are there any quadruples (n, n+1, n+2, n+3) where n, n+1, n+2 and n+3 are all the product of 3 distinct primes?