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Teachers for Tomorrow

Bose Olympiad Senior – Resources

Bose Olympiad Senior is suitable for kids in Grade 8 and above. There are two levels of this olympiad:

  • Prelims
  • Mains

Curriculum

  • Number Theory
  • Combinatorics
  • Algebra
    • Polynomials
    • Complex Numbers
    • Inequality
  • Geometry

Number Theory

The following topics in number theory are useful for the Senior round:

  • Bezout’s Theorem and Euclidean Algorithm
  • Theory of congruence
  • Number Theoretic Functions
  • Theorems of Fermat, Euler, and Wilson
  • Pythagorean TriplesChinese Remainder Theorem

Here is an example of a Number Theory problem that may appear in Seinor Bose Olympiad:

Suppose $a, b, c$ are the side lengths of an integer sided right-angled triangle such that $GCD(a, b, c) = 1$. If $c$ is the length of the hypotenuse, then what is the largest value of the $GCD (b, c)$?

Key idea: Pythagorean Triples

Geometry

The following topics in geometry are useful for the Senior Bose Olympiad round:

  • Synthetic geometry of triangles, circles
  • Barycentric Coordinates
  • Miquel Point Configuration
  • Translation
  • Rotation
  • Screw Similarity

Here is an example of a geometry problem that may appear in the Senior Bose Olympiad:

Suppose the river Basumoti is 25 meters wide and its banks are parallel straight lines. Sudip’s house 10 meters away from the bank of Basumoti. Apu’s house is on the other side of the river, 15 meter away from the bank. If you are allowed to construct a bridge perpendicular to the banks of Basumoti, what is the shortest distance from Sudip to Apu’s house.

Key idea: Reflection

Algebra

The following topics in Algebra are useful for Intermediate Bose Olympiad:

  • Screw similarity, Cyclotomic Polynomials using Complex Numbers
  • AM, GM, and Cauchy Schwarz Inequality
  • Rational Root Theorem, Remainder Theorem
  • Roots of a polynomial

Here is an example of an algebra problem that may appear in Senior Bose Olympiad:

The following sum is greater than which integer: $$ \frac{2}{3} + \frac{3}{4} \cdots + \frac{2019}{2020} + \frac{2020}{2} $$

(A) $2019$ (B) $2020$ (C) $2021$ (D) $2022$

Key idea: inequality

Reference Books

  • Elementary Number Theory by David Burton
  • Principles and Techniques in Combinatorics by Chen Chuan Chong and Koh Khee Meng
  • Polynomials by Barbeau
  • Secrets in Inequalities by Pham Kim Hung
  • Complex Numbers from A to Z by Titu Andreescu
  • Challenges and Thrills of Pre College Mathematics
  • Lines and Curves by Vasiliyev (something else) 
  • Geometric Transformation by Yaglom
  • Notes by Yufei Zhao
  • Trigonometric Delights by El Maor
  • Trigonometry by S.L. Loney
  • 101 Problems in Trigonometry by Titu Andreescu

By Dr. Ashani Dasgupta

Ph.D. in Mathematics, University of Wisconsin, Milwaukee, United States.

Research Interest: Geometric Group Theory, Relatively Hyperbolic Groups.

Founder, Cheenta

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