407 Polynomials & Trigonometry Review Problems

1. Let AC be a line segment in the plane and B a point between A and C. Construct isosceles triangles P AB and QBC on one side of the segment AC such that ∠AP B = ∠BQC = 120 • and an isosceles triangle RAC on the otherside of AC such that ∠ARC = 120 •. Show that P QR is an equilateral triangle. Solution: We give here 2 different solutions.

1994

2020

In this paper, by using trigonometric functions and generating functions, identities and relations associated with special numbers and polynomials are derived. Relations among the combinatorial numbers, the Bernoulli polynomials, the Euler numbers, the Stirling numbers and others special numbers and polynomials are given.

2011

Mathematics Magazine, 2005

International Mathematical Olympiad (1961) Problems and Solutions Day 2, 2020

This is a series of papers centralized around International Mathematical Olympiad (IMO). The context includes problems ranging from elementary algebra and other pre-calculus subjects to other elds occasionally not covered under pre-university curriculum. IMO being one of the highest pre-university maths competition requires a certain level of mastery. Mathematical maturity paired with the ability to connect ideas between dierent branches identify an adept math olympian. Usually problems cover pre-undergraduate mathematical texts, although they may require advanced undergraduate ideas, sometimes even connecting to unsolved problems in the research level.

PROBLEMS IN ELEMENTARY MATHEMATICS, 1982

A collection of 928 problems in arithmetic, algebra, geometry and trigonometry (with answers) prepared for home study by correspondence students and others studying or brushing supplementary mathematics without a teacher. Problems with similar solutions are grouped together with a detailed example of the solution of the first problem in the group. Will be found a useful resource of questions for revision, tests, and examination papers. Has had 17 large editions in Russian.

2005

We consider the problem of nding the trigonometric polynomial

Journal of Classical Analysis, 2014

We establish necessary and sufficient conditions for an arbitrary polynomial of degree n, especially with only real roots, to be trivial, i.e. to have the form a(x − b) n. To do this, we derive new properties of polynomials and their roots. In particular, it concerns new bounds and genetic sum representations of the Abel-Goncharov interpolation polynomials. Moreover, we prove the Sz.-Nagy type identities, the Laguerre and Obreshkov-Chebotarev type inequalities for roots of polynomials and their derivatives. As applications these results are associated with the known problem, conjectured by Casas-Alvero in 2001, which says, that any complex univariate polynomial, having a common root with each of its non-constant derivative must be a power of a linear polynomial. We investigate particular cases of the problem, when the conjecture holds true or, possibly, is false.