Tessellate S.T.E.M.S.
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.
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 CMIites 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.
Contents
Rules
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 hasslefree. 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)
Mathematics Sample Problems
 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
Physics Sample Problems
Computer Science Sample Problems
Mathematics Syllabus
School
Combinatorics
 Basic Counting (Rule of Sum, Rule of Product, Combinations, Permutations, etc.)
 Pigeonhole Principle
 Elementary Recurrence Relations
 Binomial Theorem
Algebra
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

 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
Physics Syllabus
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
 BiotSavart'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 1dimension, 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, Halflife and Mean Life, Binding Energy and its calculation, Fission and Fusion Processes
 Photoelectric Effect
 Bohr's Theory of Hydrogenlike 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
 XRays
 Atoms in Electric and Magnetic Fields
Computer Science Syllabus
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:
 Systematically following, simulating and reasoning about sets of instructions, protocols, structures, etc.
 Understanding correctness of algorithms
 Assessing performance of algorithms
 Reasoning about discrete structures
 Reasoning about combinatorial games
 Understanding implications of logical statements
We do not expect any prerequisite 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
 Logic and Games
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:
 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.
 Understanding of data structures and various discrete structures such as graphs, trees, heaps, stacks, and queues
 An understanding of finite state machines, pushdown systems, and turing machines, along with their properties and representations including grammars and computation models
 An understanding of computation in terms of complexity and decidability
Besides the sample problems and papers, these resources on Brilliant might be helpful:
Algorithms
Data Structures
Models of Computation
Decidability and Complexity
Logic
You may refer to the material recommended for the school section for topics in discrete mathematics.