Key Formulas and Concepts to Remember for IMO and NSO Level 2

• Prime Numbers: A prime number is a number greater than 1 that has no divisors other than 1 and itself (e.g., 2, 3, 5, 7, 11).
• Even and Odd Numbers:
  Even numbers: Divisible by 2 (e.g., 2, 4, 6, 8).
  Odd numbers: Not divisible by 2 (e.g., 1, 3, 5, 7).

• Divisibility Rules:
  Divisible by 2: Last digit is even.
  Divisible by 3: Sum of digits is divisible by 3.
  Divisible by 5: Last digit is 0 or 5.
  Divisible by 10: Last digit is 0.

• GCD (Greatest Common Divisor)
  Use Euclidean Algorithm: GCD(a,b) = GCD(b,a mod b).
• LCM (Least Common Multiple)
  LCM(a,b) = LCM (a,b) = (a x b)/GCD(a,b)

• Addition and Subtraction: Understanding properties like commutativity and associativity:
  Commutative Property: a + b = b + a
  Associative Property: (a + b) + c = a + (b + c)
• Multiplication:
  Multiplying by 10, 100, etc.: Multiply the number and add zeroes accordingly.
  Multiplication properties:
  Commutative: a × b = b × a
  Distributive: a × (b + c) = a × b + a × c

• Division: Long division for dividing large numbers.

• Fractions:
  Fraction representation: , where a is the numerator and b is the denominator.
  Simplifying fractions: Divide numerator and denominator by their GCD.
  Equivalent fractions:
  Addition/Subtraction of fractions: Same denominator.

• Decimals:
  Converting fractions to decimals and vice versa.
  Adding and subtracting decimals.

• 2D Shapes:
  Perimeter and Area of simple shapes:
  Rectangle: Perimeter = 2 × (l + b); Area = l × b
  Square: Perimeter = 4 × a, Area = a²
  Circle: Circumference = , Area = πr²
• Angles:
  Right angle: 90°
  Acute angle: Less than 90°
  Obtuse angle: Greater than 90°
• Symmetry:
  Identifying lines of symmetry in shapes like squares, circles, and triangles.

• Arithmetic Sequences: A sequence in which the difference between consecutive terms is constant (e.g., 2, 4, 6, 8, 10…).
  Formula for n^th term: , where a is the first term and d is the common difference.

• Geometric Sequences: A sequence where each term is multiplied by the same number (e.g., 3, 6, 12, 24…).
  Formula for n^th term:, where a​ is the first term and r is the common ratio.

• Modeling Word Problems: Translating word problems into mathematical expressions.

Example: “If there are 5 bags with 6 apples in each, how many apples in total?” (Multiplication: 5 × 6 = 30).

• Length: Conversion between units (cm, m, km).
  1 m = 100 cm, 1 km = 1000 m

• Weight: Conversion between units (g, kg).
  1 kg = 1000 g

• Time: Conversion between units (seconds, minutes, hours).
  1 hr = 60 minutes, 1 minute = 60 seconds, 1 hr = 3600 seconds

• Volume: Understanding the units for volume (liters, milliliters).
  1 ltr = 1000 ml

• Understanding simple probability as the likelihood of an event happening.
  Probability formula: P(A) = Number of favorable outcomes / Total number of possible outcomes

• Mean: Average of a set of numbers:Mean = Sum of numbers / Number of items

• Median: The middle value when numbers are arranged in order.
• Mode: The most frequent number in a set of data.

• Clock: Time in hours, minutes, and seconds, including both analog and digital.
• Calendar: Calculating the number of days between two dates, leap years.

• Area of triangle: Area = 1/2 x Base x Height
• Volume of Rectangular prism: Volume = Length x Width x Height
• Pythagoras theorem: For right angled triangle: a² + b² + c² (where c is the hypotenuse).

• Proportions Shortcut: a/b = c/d ⇒ ad = bc
• Percentage calculation: percentage = part/whole x 100
• Percentage increase/Decrease: New value =Orignal value x (1+percentagechange/100)

• Speed Formula: Speed = Distance/time
• Time Formula: Time = Distance/speed
• Distance Formula: Distance = Speed x Time

• work done (W = F ⋅ d  ⋅ cos θ)
• kinetic energy (KE = ½mv²)
• potential energy (PE = mgh)
• conservation of energy

• Momentum (p = mv)
• Impulse-momentum theorem (FΔt = Δp)
• Conservation of momentum

• Torque (T = rF sin θ)
• Angular momentum (L = Iω)
• Moment of inertia
• Rotational kinetic energy

• Coulomb’s Law (F = k q₁q₂/ _r2)
• Electric field (E = F/q)
• Electric potential (V = k q/r)

• Ohm’s Law (V = IR)
• Power (P = IV)
• Kirchhoff’s Laws

• Biot-Savart Law, Ampere’s Law
• Lorentz Force (F = q(v × B))
• Electromagnetic Induction (Faraday’s Law, Lenz’s Law)

• Protons, neutrons, electrons
• Atomic number, mass number, isotopes
• Electron configuration, orbitals, quantum numbers

• Trends: atomic radius, ionization energy, electronegativity, electron affinity
• Groups (alkali metals, halogens, noble gases) and periods

• Types: synthesis, decomposition, single replacement, double replacement, combustion
• Redox reactions, oxidation states
• Endothermic vs. exothermic reactions

• Heat, work, enthalpy, entropy, Gibbs free energy
• Calorimetry, Hess’s law

• Structural isomers, stereoisomers (geometric, optical)
• Chirality, enantiomers, diastereomers

• IUPAC naming rules
• Prefixes, suffixes for functional groups