Congratulations to Fok Chi Kwong, age 16 from Yuen Long Merchants Association Secondary School, Hong Kong on this solution.

We may find the required polynomial by starting from the expression :

x = 1 + √2 + √3. Squaring both sides and simplifying, we get

x − 1 = √2+ √3

x2 − 2x + 1 = 5 + 2√6

x2 − 2x − 4 = 2√6

(x2 − 2x − 4)2 = 24

x4 − 4x3 + 4x2 − 8x2 + 16x + 16 = 24

x4 − 4x3 − 4x2 + 16x − 8 = 0

Thus p(x) = x⁴ - 4x³ - 4x² + 16x - 8 is the required polynomial.

Tony Cardell, age 15, State College Area High School, PA, USA, also sent in a good solution.