Tag Archives: RMO mathematics

References for RMO, INMO and IMO

Below is a list of basic references for RMO, INMO and IMO:

  • Problem Primer for the Olympiad by C R Pranesachar, B J Venkatachala et al.
  • Higher Algebra by Hall and Knight
  • Higher Algebra by Bernard and Child
  • Plane Trigonometry Part I by S L Loney
  • A School Geometry by Hall and Stevens
  • An Introduction to Number Theory by Niven and Zuckermann
  • Elementary Number Theory by David M Burton
  • Problem Solving Strategies by Arthur Engel
  • Problems in Plane Geometry by Sharygin, MIR Publishers.
  • Combinatorics — Schaum Series
  • Mathematical Circles by Fomin et al.
  • An Excursion in Mathematics by M R Modak
  • Selected Problems and Theorems in Elementary Mathematics by Shklyarsky et al.
  • International Mathematical Olympiads, 1959-77, S. L. Greitzer.
  • International Mathematical Olympiads, 1978-85, M. S. Klamkin
  • USA Mathematical Olympiads, 1972-85, M. S. Klamkin
  • Mathematical Challenges for Olympiads, 2nd edition, C R Pranesachar et al.
  • An Excursion in Mathematics — M. R. Modak,
  • College Geometry — Howard Eve.
  • 1000 Mathematical Challenges — J N Kapur

Start cracking …

Nalin Pithwa

Polynomials — quartics

Let us continue our exploration of polynomials. Just as in the previous blog, let me present to you an outline of some method(s) to solve Quartics. As I said earlier, “filling up the gaps” will kindle your intellect. Above all, the main aim of all teaching is teaching “to think on one’s own”.

I) The Quartic Equation. Descartes’s Method (1637).

(a) Argue that any quartic equation can be solved once one has a method to handle quartic equations of  the form:

t^{4}+pt^{2}+qt+r=0

(b) Show that the quartic polynomial in (a) can be written as the product of two factors

(t^{2}+ut+v)(t^{2}-ut+w)

where u, v, w satisfy the simultaneous system

v + w - u^{2}=p

u(w-v)=q

vw=r

Eliminate v and w to obtain a cubic equation in u^{2}.

(c) Show how any solution u obtained in (b) can be used to find all the roots of the quartic equation.

(d) Use Descartes’s Method to solve the following:

t^{4}+t^{2}+4t-3=0

t^{4}-2t^{2}+8t-3=0

II) The Quartic Equation. Ferrari’s Method:

(a) Let a quartic equation be presented in the form:

t^{4}+2pt^{3}+qt^{2}+2rt+s=0

The strategy is to complete the square on the left side in such a way as to incorporate the cubic term. Show that the equation can be rewritten in the form

(t^{2}+pt+u)^{2}=(p^{2}-q+2u)t^{2}+2(pu-r)t+(u^{2}-s)

where u is indeterminate.

(b) Show that the right side of the transformed equation in (a) is the square of  a linear polynomial if u satisfies a certain cubic equation. Explain how such a value of u can be used to completely solve the quartic.

(c) Use Ferrari’s Method to solve the following:

t^{4}+t^{2}+4t-3=0

t^{4}-2t^{3}-5t^{2}+10t-3=0.

Happy problem-solving 🙂

Do leave your comments, questions, suggestions…!

More later…

Nalin