MathDB

5

Part of 2013 Tuymaada Olympiad

Problems(2)

Representation of a degree 4 polynomial

Source: Tuymaada 2013, Day 2, Problem 5 Seniors

7/23/2013
Prove that every polynomial of fourth degree can be represented in the form P(Q(x))+R(S(x))P(Q(x))+R(S(x)), where P,Q,R,SP,Q,R,S are quadratic trinomials.
A. Golovanov
EDIT. It is confirmed that assuming the coefficients to be real, while solving the problem, earned a maximum score.
algebrapolynomialquadraticslinear algebramatrixparameterizationalgebra proposed
Colouring a cube of side 7

Source: Tuymaada 2013, Day 2, Problem 5 Juniors

7/22/2013
Each face of a 7×7×77 \times 7 \times 7 cube is divided into unit squares. What is the maximum number of squares that can be chosen so that no two chosen squares have a common point?
A. Chukhnov
geometry3D geometrycombinatorics proposedcombinatorics