MathDB

2

Part of 2012 Iran MO (2nd Round)

Problems(2)

Numbers around a circle

Source: Iran 2nd round 2012-Day1-P2

4/30/2012
Suppose nn is a natural number. In how many ways can we place numbers 1,2,....,n1,2,....,n around a circle such that each number is a divisor of the sum of it's two adjacent numbers?
inductioncombinatorics proposedcombinatorics
fourth degree polynomail, 4 variable polynomial!

Source: Iran 2nd round 2012-Day2-P5

5/1/2012
Consider the second degree polynomial x2+ax+bx^2+ax+b with real coefficients. We know that the necessary and sufficient condition for this polynomial to have roots in real numbers is that its discriminant, a2−4ba^2-4b be greater than or equal to zero. Note that the discriminant is also a polynomial with variables aa and bb. Prove that the same story is not true for polynomials of degree 44: Prove that there does not exist a 44 variable polynomial P(a,b,c,d)P(a,b,c,d) such that:
The fourth degree polynomial x4+ax3+bx2+cx+dx^4+ax^3+bx^2+cx+d can be written as the product of four 11st degree polynomials if and only if P(a,b,c,d)≥0P(a,b,c,d)\ge 0. (All the coefficients are real numbers.)
Proposed by Sahand Seifnashri
algebrapolynomialVietacomplex numbersalgebra proposed