Category: Algebra

Construction of polynomials
The polynomial P(x) has the property that P(1), P(2), P(3), P(4), and P(5) are equal to 1, 2, 3, 4, 5 in some order. How many possibilities are there for the polynomial P, given that the degree of P is strictly less than 4? (Duke Math Meet 2013 Tiebreaker round) Discussion: Let \[P(x) = a…

Lifting the exponent and math olympiad number theory
In math olympiads around the world, number theory problems have many recurring themes. One such theme is the ‘LTE’ or lifting the exponent.

The Dreams of Pythagoras
Pythagoras is famous. Even those who do not like mathematics, have heard of Pythagoras’ Theorem (regarding right angled triangles). He lived about 2500 years ago and did about 2500 wonderful things (well may be a little less) but all that is not the subject matter of this note. We want to talk about the shattered…