Category: Math Olympiad

An Invitation to Math Olympiad 1 – Plato’s Dianoa and the identity of Sophie Germain
Once upon a time there was a beautiful problem. Is there a number $n$ such that $n^4 + 4^n$ is a prime number?

Round robin tournament
Problem : Suppose there are teams playing a round robin tournament; that is, each team plays against all the other teams and no game ends in a draw.Suppose the team loses games and wins games.Show that = Solution : Each team plays exactly one match against each other team. Consider the expression Since each team…

Orthogonality
Let ABC be a triangle and D be the midpoint of BC. Suppose the angle bisector of \[\angle ADC \] is tangent to the circumcircle of triangle ABD at D. Prove that \[\angle A = 90^o \] . (Regional Mathematics Olympiad, India, 2016)

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…

Number Theory in Math Olympiad – Beginner’s Toolbox
This article is aimed at entry level Math Olympiad (AMC and AIME in U.S. , SMO Junior in Singapore, RMO in India). We have complied some of the most useful results and tricks in elementary number theory that helps in problem solving at this level. Note that only with a lot of practice and conceptual…

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.

Arithmetic of Remainders
Consider the two number: 37 and 52 What is the remainder when we divide 37 by 7? 2 of course. And 52 produces remainder 3 when divided by 7. Suppose we want to know the remainder when the product of 37 and 52 is divided by 7.

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…

Chinese Remainder Theorem
I want to discuss the ‘ideai behind the famous Chinese Remainder Theorem. Let us leave the jargon and start our exploration by a problem.Find a number that leaves remainder 1, when divided by 5, 7 and 13. Clearly such a number can be found by trial. A stupid method is to check out all the…