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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
Categories
Teachers for Tomorrow

Bose Olympiad Intermediate – Resources

Bose Olympiad Intermediate is suitable for kids in Grade 5, 6, and 7. There are two levels of this olympiad:

  • Prelims
  • Mains

Curriculum

  • Elementary Number Theory
  • Counting Principles
  • Algebra
  • Geometry

Number Theory

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

  • Primes and Composites
  • Arithmetic of Remainders
  • Divisibility
  • Number Theoretic Functions

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

How many positive integer solutions are there of the equation $x^3 – y^3 = 121$ ?

Key idea: Primes

Geometry

The following topics in geometry are useful for the Intermediate round:

  • Locus problems
  • Geometry of lines (angles, parallels)
  • Geometry of triangles (centroid, circumcenter, orthocenter)
  • Geometry of circles (tangents, chords, cyclic quadrilaterals)
  • Conic sections (ellipse, parabola, hyperbola).
  • Triangular Inequality

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

There are two trees A and B on a field such that distance between A and B is 5 meter. Ayesha is continuously running on the field such that sum of her distances from A and B is always 5 meters. How many times does she visit the midpoint of A and B?

Key idea: Locus

Algebra

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

  • Factorization
  • Linear equations
  • Quadratic Equations
  • Inequality

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

Consider all rectangles of perimeter 40 cm. What is the largest area that can be enclosed by any such rectangle?

Key idea: inequality

Reference Books

  • Mathematical Circles by Fomin
  • Lines and Curves by Vasiliyev
  • Challenges and Thrills of Pre College Mathematics
Categories
Teachers for Tomorrow

Bose Olympiad Junior – Resources

Bose Olympiad Junior is suitable for kids in Grade 1, 2, 3 and 4. There are two levels of this olympiad:

  • Prelims
  • Mains

Curriculum

  • Arithmetic
  • Geometry
  • Mathematical Puzzles

Arithmetic

Basic skills of addition, subtraction and multiplication and division will be sufficient for attending arithmetic problems. Fundamental ideas about place-value system and ratios could be useful for Mains level.

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

Suppose Ajit has 35 cheese sticks. Ajit makes Red Packs containing 3 sticks in each packet. Then Ajit makes Green packs containing 3 Red Packs each. Finally he makes Blue packs, each containing 3 Green Packs. How many unpacked sticks are there at the end of this process?

Key idea: Place Value System

Geometry

A basic understanding is of shapes like triangle, circle, square is sufficient for prelims. Locus (path traced out by a moving point) is another key geometry topic that may appear. At the Mains level, the student may need notions of Area and Perimeter.

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

Ayesha is running on a field such that his distances from two trees A and B are always equal. That is the distance of the position of Manoj from tree A is equal to the distance of the position of Manoj from tree B at any point of time. Then what is the shape of the path along which Ayesha is running?

Key idea: Locus

Mathematical Puzzles

Mathematical puzzles may involve parallel channels, back tracking, greedy algorithm and recursive logic.

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

2019248 teams are playing in a knockout galactic football tournament. In this tournament no match ends in a draw and if you lose a match then you are out of the tournament. In the first round of the tournament the teams are paired up. In each subsequent round if even number of teams remain then they are again paired up, if odd number of teams remain then the highest scoring team is allowed to rest and directly go to the next round. How many matches are played in this tournament?

Key idea: one on one correspondence

Reference Books

  • Mathematics can be fun by Perelman
  • Mathematical Circles for 3 to 8
  • Lines and Curves by Vasiliyev
  • Puzzles by Martin Gardner