Tessellate S.T.E.M.S. 2018

Chennai Mathematical Institute (CMI), a premier center for research and education in mathematical sciences, is organizing a nationwide (India) mathematics, physics, and computer science contest, the S.T.E.M.S. (Scholastic Test of Excellence in Mathematical Sciences), as a part of the college fest Tessellate.

Visit this wiki for S.T.E.M.S. 2019.

By participating in S.T.E.M.S., you get to show off your mathematics/physics/computer science skills at a national level and have the chance to attend a science camp at CMI, where you'd meet some of the finest mathematicians and physicists in the country, not to mention the gifts.

The camp will feature some of the best mathematicians, physicists, and computer scientists from some of the best research institutes in India such as CMI, ISI Bangalore, IMSc, IISc, the IIT's, etc. (The complete list of speakers shall be uploaded soon.) CMI shall provide travel fare, food, and accommodations for the selected participants. Students from all age groups will have fair representation. It promises to be a great opportunity for the students to interact with CMI-ites and get an insight into college mathematics and college life (probably in that order, though). The camp shall also feature student talks from students of the aforementioned institutes. They shall also receive certificates signed by some of the best mathematicians of the country, books, and other prizes. The top 100 participants shall receive certificates of participation.

The following are the rules of the contest:

• The entire contest will occur online in two parts for school and college students. The dates for the contest are the $13^\text{th}$ and $14^\text{th}$ of January, 2018.
• Both parts will take place according to the following schedule: On the $13^\text{th},$ the Computer Science exam shall start at 12:00 PM and submissions will be accepted till 3:00 PM. After an hour's break, the Physics exam shall begin at 4:00 PM and submissions must be handed in before 7:00 PM. The Mathematics exam is to be held on the $14^\text{th}.$ We shall be uploading the question papers at 12:00 PM and the submission of answers must be made online by 6:00 PM.
• For students who are writing MTRP: For students of class 9 and 11 who are appearing for MTRP, conducted by ISI, we understand that you might want to appear for both the examinations, and to avoid a clash with STEMS (school mathematics), we have decided that we can give those people a different time slot for writing STEMS. Please contact us in the given email IDs if you are interested in writing both exams, and we will give you a revised time slot for STEMS Mathematics. Note that this will only be done for students who are writing MTRP and STEMS Math together, upon showing appropriate evidence.
• The dates and times are subject to change. We will inform the participants in case such a change takes place.
• The event is divided into two sections: Section A is for students from $8^\text{th}$ to $12^\text{th}$ grade. Section B is for undergraduate and postgraduate students. The question paper, needless to say, shall be different for the two sections.
• The question paper shall consist of 20 multiple choice questions and 6 subjective questions. The question papers shall be mailed to the registered participants exactly at the starting time.
• Students are allowed to use books and electronic resources to solve the problems. However, your solutions must be strictly original. There might be an interview after the selection is made. In case of any discrepancies, the answer sheet of the student in question will be invalidated.
• The solutions have to be submitted in the form of scanned copies or clear photographs of the answer sheets. Any electronic formats will also be accepted. They have to be mailed to tessellate.cmi@gmail.com from your registered email ID strictly before the ending time. Any submissions made after the deadline will not be accepted.
• In the registration portal, please enter your name, gender, email ID, phone number, and permanent address in the mentioned format. Fill in the name of your school/college with its city in the 'College/City' tab.
• In case of any issues regarding registration, send us a mail at tessellate.cmi@gmail.com.
• Registration is hassle-free. Just click on our website http://tessellate.cmi.ac.in/#stems and register online. The fee is Rs. 100 per subject. For any queries, contact us again at tessellate.cmi@gmail.com or the following:
  • Ankita Sarkar: 8428532216 (ankita_s@cmi.ac.in)
  • Soham Chakraborty: 9884232190 (sochak@cmi.ac.in)
  • Srijan Ghosh: 9433777622 (srijang@cmi.ac.in)
  • Sarvesh Bandhaokar: 9405956066 (bandhaokar@cmi.ac.in)

• Tessellate S.T.E.M.S 2018 - Mathematics - College
• Tessellate S.T.E.M.S 2018 - Mathematics - School
• Tessellate S.T.E.M.S 2018 - Physics - College
• Tessellate S.T.E.M.S 2018 - Physics - School
• Tessellate S.T.E.M.S 2018 - Computer Science - College
• Tessellate S.T.E.M.S 2018 - Computer Science - School

• Tessellate S.T.E.M.S - Mathematics - School - Set 1
• Tessellate S.T.E.M.S - Mathematics - College - Set 1
• Tessellate S.T.E.M.S - Mathematics - School - Set 2
• Tessellate S.T.E.M.S - Mathematics - College - Set 2
• Tessellate S.T.E.M.S - Mathematics - School - Set 3
• Tessellate S.T.E.M.S - Mathematics - College - Set 3

• Tessellate S.T.E.M.S. - Physics - School - Set 1
• Tessellate S.T.E.M.S. - Physics - School - Set 2
• Tessellate S.T.E.M.S. - Physics - College - Set 1
• Tessellate S.T.E.M.S. - Physics - College - Set 2
• Tessellate S.T.E.M.S. - Physics - College - Set 3

• Tessellate S.T.E.M.S. - Computer Science - School - Set 1
• Tessellate S.T.E.M.S - Computer Science - School - Set 2
• Tessellate S.T.E.M.S - Computer Science - School - Set 3
• Tessellate S.T.E.M.S. - Computer Science - College - Set 1
• Tessellate S.T.E.M.S. - Computer Science - College - Set 2

School

• Combinatorics

  • Basic Counting (Rule of Sum, Rule of Product, Combinations, Permutations, etc.)
  • Pigeonhole Principle
  • Elementary Recurrence Relations
  • Binomial Theorem

• Algebra

  • Linear Equations, Quadratic Equations
  • Polynomials
  • Classical Inequalities
  • Exponents and Logarithms
  • Complex Numbers (Basics)
  • Arithmetic Progressions, Geometric Progression, etc.

• Geometry

  • Euclidean Geometry
  • Coordinate Geometry (Distance Formula, Equations of Straight Lines, Equation of Circles, etc.)
  • Conic Sections (Basics)

• Number Theory

  • Divisibility
  • Modular Congruences (Euler's Theorem, Fermat's Little Theorem, etc. may be helpful.)

• Trigonometry

  • Trigonometry (Basics)

• Set Theory

  • Basics of Set Theory
  • Relations
  • Functions

• Probability

  • Basics of Probability

College

• Advanced knowledge of all concepts mentioned in the high school syllabus

• Linear Algebra

  • Matrices
  • Linear Transformations
  • Eigenvalues and Eigenvectors
  • Diagonalization
  • Jordan Normal Form
  • Dual Spaces
  • Elementary knowledge of Forms (Bilinear Forms, Skew Symmetric Forms, etc.)

• Calculus and Real Analysis

  • Relations and Functions
  • Sequences and Series
  • Limits
  • Continuity
  • Uniform Continuity
  • Derivatives
  • Mean Value Theorem
  • L'Hopital's Rule
  • Taylor's Theorem
  • Riemann Integration
  • Fundamental Theorem of Calculus
  • Fubini's Theorem
  • Multivariable Calculus $\big($Functions from $\mathbb{R}^n \to \mathbb{R}^m$, their derivatives, and inverse function theorem (not mandatory) might be useful.$\big)$

• Abstract Algebra

  • Group Theory $($Matrix Groups, Cauchy and Sylow Theorems, Cayley's Theorems, Permutations, Finite Abelian Groups (not mandatory), Isomorphism Theorems$)$
  • Ring Theory (Basics)
  • Field Theory (Basics)

• Discrete Mathematics

  • Advanced Combinatorial Concepts
  • Graph Theory

• Probability Theory

  • Probability Density Function
  • Probability Distribution Function (Bernoulli Distribution, Binomial Distribution, Poisson Distribution, Normal Distribution, Uniform Distribution, etc.)
  • Mean and Variance
  • Joint Probability Distribution

School

• Mechanics

  • Kinematics in 1 and 2 dimensions
  • Newton's Laws of Motion
  • Friction (Static and Dynamic)
  • Kinetic and Potential Energy
  • Work and Power
  • Conservation of Energy
  • Conservation of Momentum
  • Elastic and Inelastic Collisions
  • Gravitation
  • Dynamics of Rigid Bodies
  • Linear and Angular Harmonic Motions
  • Pressure in Fluids, Pascal's Law
  • Surface Energy and Surface Tension
  • Streamline Flow
  • Equations of Continuity
  • Bernoulli's Theorems and its applications
  • Wave Motion
  • Vibration of Strings and Air Columns
  • Doppler Effect (Sound)

• Electrodynamics

  • Coulomb's Law
  • Electric Fields and Electric Potential
  • Gauss's Law and its application in simple cases
  • Capacitance
  • Electric Current, Ohm's Law, Series and Parallel arrangements of Resistors and Cells, Kirchoff's Laws (and simple applications)
  • Heating Effect of Current
  • Biot-Savart's Law and Ampere's Law
  • Lorentz Force
  • Magnetic Moment of a Current Loop
  • Electromagnetic Induction: Faraday's Law, Lenz's Law, RC, LC, and RL Circuits

• Thermal Physics

  • Thermal Expansion of Solids, Liquids, and Gases
  • Latent Heat
  • Conduction in 1-dimension, Elementary concepts of Convection and Radiation
  • Newton's Law of Cooling
  • Ideal Gas Laws
  • Specific Heats
  • Isothermal and Adiabatic Processes
  • First Law of Thermodynamics
  • Black Body Radiation (absorptive and emissive powers): Kirchoff's Law, Wein's Displacement Law, Stefan Law

• Optics

  • Rectilinear Propagation of Light
  • Reflection and Refraction
  • Thin Lenses
  • Wave Nature of Light: Huygens Principle, Interference

• Modern Physics

  • Law of Radioactive Decay, Decay Constant, Half-life and Mean Life, Binding Energy and its calculation, Fission and Fusion Processes
  • Photoelectric Effect
  • Bohr's Theory of Hydrogen-like Atoms
  • de Broglie Wavelength of Matter Waves

College

• Mechanics

  • Newtonian Mechanics, Lagrangian Mechanics, Hamiltonian Mechanics
  • Rigid Body Dynamics
  • Simple Harmonic Oscillator
  • Central Forces
  • Special Relativity (Time Dilation, Length Contraction, Lorentz Transformation)
  • Noether's Theorem
  • Elementary Topics in Fluid Dynamics

• Electrodynamics

  • Gauss's Law, Coulomb's Law, Application of Gauss's Law in the presence of Symmetries
  • Currents and AC and DC Circuits
  • Solution of Laplace's Equations in Cartesian, Spherical, and Cylindrical Coordinates
  • Multipole Expansion
  • Ampere's Law
  • Faraday's Law
  • Continuity Equation
  • Electromagnetic Waves and Poynting's Theorem

• Quantum Mechanics

  • Heisenberg's Formulation, Schrodinger's Formulation
  • Linear Algebra
  • Spin $\frac{1}{2}$ systems
  • Angular Momentum Quantization and Addition
  • Perturbation Theory (Basics)
  • Fourier Transforms
  • Quantum Harmonic Oscillator

• Optics

  • Wave Properties
  • Superposition, Diffraction
  • Geometric Optics
  • Polarization
  • Doppler Effect

• Thermal Physics

  • Thermodynamic Processes, Equations of State
  • Ideal Gases, Kinetic Theory
  • Ensembles
  • Statistical concepts and calculation of Thermodynamic Quantities
  • Heat Transfer
  • Thermal Expansion

• Modern Physics

  • Bohr's Model
  • Energy Quantization
  • Black Body Radiation
  • X-Rays
  • Atoms in Electric and Magnetic Fields

School

The objective of the exam is to test the student on their computational, algorithmic, and logical thinking abilities. Specific details about hardware architecture, operating systems, software systems, web technologies, programming languages, etc. will not be asked. To find the answer to a problem, one would not require programming.

The main focus will be on the following aspects:

1. Systematically following, simulating and reasoning about sets of instructions, protocols, structures, etc.
2. Understanding correctness of algorithms
3. Assessing performance of algorithms
4. Reasoning about discrete structures
5. Reasoning about combinatorial games
6. Understanding implications of logical statements

We do not expect any pre-requisite formal training of the candidates in any of these areas. Any relevant definitions and/or hints that are necessary for the understanding of the problem shall be provided. High school mathematical knowledge (and an inquisitive and computational mind!) should be enough to get started on any of the problems in this category.

Besides the sample problems and papers, these resources on Brilliant might be helpful:

• Algorithms
  • Introduction to Algorithms - Concept Quiz
  • Algorithms - Wiki
  • Runtime of Algorithms
  • Computer Science Fundamentals - Course (Premium Content)
  • Computer Science Algorithms - Course (Premium Content)
• Discrete Mathematics
  • Discrete Mathematics Warmup - Concept and Challenge Quizzes
  • Graph Theory - Concept and Challenge Quizzes
  • Graph Theory - Wiki
  • Discrete Probability - Warmup Quiz
  • Probability - Course (Premium Content)
  • Probability - Wiki
• Logic and Games
  • Combinatorial Games - Concept Quiz
  • Combinatorial Games - Wiki
  • Logic Puzzles - Quiz
  • Logic - Course (Premium Content)

College

The objective of the exam is to test the student on the theoretical aspects of computation. Specific details about hardware architecture, operating systems, web technologies, etc. will not be asked. To find the answer to a problem, one would not require programming.

The main focus will be on the following aspects:

1. Comprehensive understanding of algorithms and algorithmic paradigms such as greedy algorithms, dynamic programming, divide and conquer, and introductory graph algorithms. A preliminary knowledge of analysis of these algorithms is essential.
2. Understanding of data structures and various discrete structures such as graphs, trees, heaps, stacks, and queues
3. An understanding of finite state machines, pushdown systems, and turing machines, along with their properties and representations including grammars and computation models
4. An understanding of computation in terms of complexity and decidability

Besides the sample problems and papers, these resources on Brilliant might be helpful:

• Algorithms

  • Algorithms - Wiki
  • Greedy Algorithms - Wiki
  • Divide and Conquer - Wiki
  • Dynamic Programming - Wiki
  • Graphs - Wiki
  • Big O Notation - Wiki

• Data Structures

  • Abstract Data Types
  • Binary Search Trees
  • Stacks
  • Queues

• Models of Computation

  • Finite State Machines
  • Push Down Automata
  • Turing Machines

• Decidability and Complexity

  • Halting Problem
  • Complexity Classes
  • P vs NP

• Logic

  • Propositional Logic
  • Logic Gates
  • Predicate Logic

You may refer to the material recommended for the school section for topics in discrete mathematics.