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Books
  • History of Mathematics
    • Leonhard Euler
      • The life of Leonhard Euler
      • The language that Euler left us
      • The Basel problem
      • The number $e$ and compound interest
      • Euler's formula and the "most beautiful equation"
      • The spring and differential equations
      • The Königsberg bridges and the birth of graphs
      • $V - A + F = 2$ and the soccer ball
      • Infinitely many prime numbers from a sum
      • Fermat's number that was not prime
      • Euler's theorem and cryptography
      • Other contributions and legacy
      • Image credits
    • René Descartes
      • The life of René Descartes
      • Coordinates: geometry becomes algebra
      • Square root with ruler and compass
      • The rule of signs
      • The angles of polyhedra and Euler's formula
      • The kissing circles theorem
      • Why does the rainbow appear at 42°
      • Descartes and the idea of ​​function
      • A reading of the Cogito: listening and speaking
      • Descartes and the problem of cannons
      • The word "function"
      • *Physis*: deciphering the traits of nature
      • Infinity and division by zero
      • Other contributions and legacy
      • Image credits
    • Carl Friedrich Gauss
      • The life of Carl Friedrich Gauss
      • The sum of 1 to 100
      • The 17 sided polygon
      • Clock arithmetic
      • The fundamental theorem of algebra
      • Counting prime numbers
      • Ceres and the least squares method
      • The normal curve, Gauss's bell
      • The Egregium Theorem and curvature
      • Systems of equations: the determinant and Gaussian elimination
      • Gaussian quadrature: integrate with two points
      • Other contributions and legacy
      • Image credits
    • Isaac Newton
      • The life of Isaac Newton
      • The generalized binomial: beyond whole-number powers
      • Fluxions: how to measure an instantaneous change
      • Areas and integration: the inverse of fluxions
      • Newton's method: approximating roots with tangents
      • The laws of motion: forces and changes in velocity
      • Gravitation: from the apple to the Moon
      • Light and colours: the prism does not make the rainbow
      • Newton's legacy
      • Image credits
    • Gottfried Wilhelm Leibniz
      • The life of Gottfried Wilhelm Leibniz
      • Leibniz's notation: a language for change
      • The product rule: two changes in one calculation
      • The ∫ sign: from sum to accumulation
      • Binary arithmetic: writing numbers with 0 and 1
      • Leibniz's machine: turning sums into motion
      • The Leibniz series for π: a simple, slow sum
      • Calculemus: the idea of calculating with reasoning
      • Leibniz's legacy
      • Image credits
    • Pierre de Fermat
      • The life of Pierre de Fermat
      • Fermat and analytic geometry: a curve in an equation
      • Maxima and minima: the rectangle with the largest area
      • Fermat's little theorem: powers and remainders
      • Fermat numbers: a conjecture and a counterexample
      • Fermat and Pascal: fairly dividing an interrupted game
      • Fermat's principle: light takes the fastest path
      • Infinite descent: a smaller solution leads to a contradiction
      • Fermat's last theorem: from the margin to Wiles's proof
      • Fermat's legacy
      • Image credits
    • Blaise Pascal
      • The life of Blaise Pascal
      • Pascal's triangle: build it by adding
      • Combinations: counting without listing
      • The Chevalier de Méré's dice and the fair split
      • The Pascaline: a sum made of gears
      • The mystic hexagram: six points on a conic
      • The cycloid: the toothache curve
      • Pascal's principle: multiplying forces with water
      • The weight of air: the Puy de Dôme experiment
      • Pascal's wager
      • Pascal's legacy
      • Image credits
    • Joseph-Louis Lagrange
      • The life of Joseph-Louis Lagrange
      • The calculus of variations
      • The Analytical Mechanics
      • The Lagrange points
      • Four squares are enough
      • Shuffling the roots
      • The derivative and the mean value
      • Lagrange multipliers
      • Lagrange's legacy
      • Image credits
    • Sophie Germain
      • The life of Sophie Germain
      • The letters to Gauss
      • The theorem on Fermat
      • Sophie Germain primes
      • The Sophie Germain identity
      • Vibrating plates
      • Mean curvature
      • Sophie Germain's legacy
      • Image credits
    • James Bernoulli
      • The life of James Bernoulli
      • Interest and the number e
      • The harmonic series
      • The spiral and the lemniscate
      • The art of conjecturing
      • The law of large numbers
      • The Bernoulli numbers
      • The legacy of James Bernoulli
      • Image credits
    • Albert Einstein
      • The life of Albert Einstein
      • The photoelectric effect
      • Brownian motion
      • Special relativity
      • E = mc²
      • General relativity
      • The legacy of Albert Einstein
      • Image credits
    • Johann Bernoulli
      • The life of Johann Bernoulli
      • The catenary
      • L'Hôpital's rule
      • The brachistochrone
      • The exponential calculus
      • Virtual work
      • The legacy of Johann Bernoulli
      • Image credits
    • Euclid of Alexandria
      • Euclid: the author without a face
      • The Elements
      • The equilateral triangle
      • Pythagoras's theorem
      • Geometric algebra
      • Euclid's algorithm
      • The primes never run out
      • The fifth postulate
      • The legacy of Euclid
      • Image credits
    • James Clerk Maxwell
      • The life of James Clerk Maxwell
      • Colour
      • Saturn's rings
      • The speeds of molecules
      • Maxwell's equations
      • The displacement current
      • Light is a wave
      • The legacy of James Clerk Maxwell
      • Image credits
    • John Nash
      • The life of John Nash
      • The game of Hex
      • The Nash equilibrium
      • Mixed strategies
      • The Nash bargaining solution
      • The embedding theorem
      • The differential equations
      • The legacy of John Nash
      • Image credits
    • Niels Henrik Abel
      • The life of Niels Henrik Abel
      • The fifth-degree equation
      • The binomial series
      • Elliptic functions
      • The lemniscate
      • Abelian groups
      • The legacy of Niels Henrik Abel
      • Image credits
    • Évariste Galois
      • The life of Évariste Galois
      • The group of an equation
      • What a group is
      • Solving by radicals
      • Finite fields
      • The impossible constructions
      • The legacy of Évariste Galois
      • Image credits
    • John Napier
      • The life of John Napier
      • Multiplying by adding
      • Napier's two points
      • Briggs and base 10
      • Napier's bones
      • The slide rule
      • The legacy of John Napier
      • Image credits
    • Alan Turing
      • The life of Alan Turing
      • The Turing machine
      • The halting problem
      • Computable numbers
      • The universal machine
      • The Enigma
      • Banburismus
      • The imitation game
      • Morphogenesis
      • The legacy of Alan Turing
      • Image credits
    • John von Neumann
      • The life of John von Neumann
      • Ordinals
      • The minimax theorem
      • Quantum mechanics
      • The von Neumann architecture
      • Merge sort
      • The Monte Carlo method
      • The biased coin
      • Self-reproducing automata
      • The legacy of John von Neumann
      • Image credits
    • William Rowan Hamilton
      • The life of William Rowan Hamilton
      • Optics and conical refraction
      • Complex numbers as couples
      • Hamilton's mechanics
      • Quaternions
      • Rotations with quaternions
      • Vectors and scalars
      • Cayley–Hamilton
      • The Icosian game
      • The legacy of William Rowan Hamilton
      • Image credits
    • Pierre-Simon Laplace
      • The life of Pierre-Simon Laplace
      • Jupiter, Saturn and the stability of the Solar System
      • The nebular hypothesis
      • The dark stars
      • Laplace's equation
      • Laplace's expansion of determinants
      • Classical probability and the Gaussian integral
      • The central limit theorem
      • Will the Sun rise tomorrow? The rule of succession
      • The Laplace transform
      • The speed of sound
      • Laplace's demon
      • The legacy of Pierre-Simon Laplace
      • Image credits
    • Maria Gaetana Agnesi
      • The life of Maria Gaetana Agnesi
      • The curve of Agnesi
      • The Instituzioni analitiche
      • Maxima, minima and points of inflection
      • The area under the curve and the number π
      • The inverse method of tangents
      • The witch in statistics
      • A woman at the university
      • The legacy of Maria Gaetana Agnesi
      • Image credits
    • Archimedes
      • The life of Archimedes
      • Eureka: the king's crown
      • The law of the lever
      • The measurement of the circle and the number π
      • The quadrature of the parabola
      • The sphere and the cylinder
      • The Method and the palimpsest
      • The spiral of Archimedes
      • The Sand Reckoner
      • The screw, the claw and the mirrors
      • The legacy of Archimedes
      • Image credits
    • Bernhard Riemann
      • The life of Bernhard Riemann
      • The Riemann integral
      • The rearrangement theorem
      • Riemann surfaces
      • The hypotheses on which geometry is founded
      • The zeta function
      • Counting the prime numbers
      • The Riemann hypothesis
      • The legacy of Bernhard Riemann
      • Image credits
    • Augustin-Louis Cauchy
      • The life of Augustin-Louis Cauchy
      • The polygonal number theorem
      • Determinants and permutations
      • Limits and continuity
      • The convergence of series
      • Cauchy's inequality
      • The derivative and the integral
      • The integral in the complex plane
      • Stress in elastic solids
      • The legacy of Augustin-Louis Cauchy
      • Image credits
  • Mathematics for Children
    • Natural numbers
      • Elementary I (1st to 5th year)
        • Introduction
        • The number zero
        • Sum
        • Subtraction
        • Multiplication
        • Division
        • Even and odd numbers
        • Introduction to Geometry
        • Quadrilateral
      • Elementary II (6th to 9th year)
        • Prime numbers
        • Greatest Common Divisor
        • Least Common Multiple
        • Geometry with sequences of numbers
        • Triangles
      • High School
        • Introduction to Arithmetic Progression
        • Arithmetic Progression
      • Solved exercises (all levels)
        • Natural Number Subtraction Exercises
        • Exercises for Adding Natural Numbers
    • Whole Numbers
      • Elementary I (1st to 5th year)
        • Above and below zero
        • Measure
      • Elementary II (6th to 9th year)
        • Introduction to Integers
        • Straight and operations
        • Signal rule
        • Potentiation of integers
        • Angles and Geometry
          • Angles in basic planes
          • Property of Angles in Planes
      • High School (1st to 3rd grade)
        • Pascal's triangle
        • Function Basics
          • Function Discovery
          • Cartesian Plane
          • Classification in mathematics
          • Graphs for integers
    • Rational Numbers
      • Elementary I (1st to 5th year)
        • Equal parts and fractions
        • Decimals in everyday life
      • Elementary II (6th to 9th year)
        • Introduction to Rational Numbers
        • Operations with Rational Numbers
        • Percentage Basics
        • Proportions
        • Rule of 3
        • Fees
        • Quantity Analysis
        • Proportions and Geometry
        • Algebraic Expressions
        • First Degree Equations
      • High School (1st to 3rd grade)
        • Power with Negative Exponent
        • Nth Root
        • Models and restrictions
    • Irrational Numbers
      • Elementary I (1st to 5th year)
        • Measure and discover
      • Elementary II (6th to 9th year)
        • Operation with Radicals
          • Introduction to Irrational Numbers
          • Introduction to Radiciation
        • Approaches on the straight
        • Geometry - Irrational Numbers
          • Circumferences
          • Pi Number
          • Circle Area
          • Angles and Circumferences
          • Triangles and Circles
          • Pythagorean Suits
      • High School (1st to 3rd grade)
        • Arguments and operations
        • Logarithm
          • Introduction to Logarithm Table
          • Logarithm Properties
        • Rectangle Triangles
        • Basic Finance
          • Increases and Discounts
          • Simple Interest
          • Compound interest and Euler's number
    • Real Numbers
      • Elementary I (1st to 5th year)
        • Measures between integers
        • Everyday data
      • Elementary II (6th to 9th year)
        • Introduction to Real Numbers
        • Rational, irrational and approximations
        • Inequalities of real numbers
        • 2° degree equation
        • Bhaskara
        • Notions of Probability
          • Introduction to Probability
          • Fundamentals
          • Equiprobable Events
      • High School (1st to 3rd grade)
        • Notions of Sets
          • Introduction to Sets
          • Equivalence between Sets
          • Venn-Euler Diagrams
          • Diagram Operations
        • Intervals and infinity
          • Notion of Infinity
          • Ranges in real numbers
          • Interval Operations
        • Functions
          • Introduction to Function Theory
          • Understanding Relationships
          • Correspondence law
          • Function Concept
          • First Degree Functions
          • Domain and Image
        • Statistics Notions
          • Introduction to Statistics
          • Basic Concepts of Statistics
          • Descriptive Statistics
            • Position Measurements
            • Dispersion Measures
    • Plane Geometry
      • Elementary I (1st to 5th year)
        • Flat shapes
        • Lines, parallel and perpendicular
        • Angles: turns and corners
        • Polygons
        • Symmetry
        • Perimeter and area in the mesh
        • Enlargement and reduction
      • Elementary II (6th to 9th year)
        • 1. Primitive notions and propositions
          • A little history
          • I. Primitive notions
          • II. Primitive propositions
          • III. Competing straights
          • Exercises
        • 2. straight segment
          • I. The notion of "being between"
          • II. Segment and ray
          • III. Positions of two segments
          • IV. Segment congruence
          • V. Midpoint
          • VI. Measurement of a segment
          • VII. Distance between two points
          • Segments and straight lines in everyday life
          • Exercises
        • 3. Angles
          • I. Introduction: convexity and half-planes
          • II. Angle: definitions
          • III. Congruence and comparison
          • IV. Right, acute and obtuse angles — measurement
          • Exercises
        • 4. Triangles
          • I. Concept, elements and classification
          • II. Congruence of triangles
          • III. Inequalities in triangles
          • Exercises
          • Reading: Euclid and deductive geometry
        • 5. Parallelism
          • I. Parallel lines
          • II. Existence and uniqueness of parallel
          • III. Consequences in the triangle
          • Exercises
          • Reading: The fifth postulate
        • 6. Perpendicularity
          • I. Definitions — right angle
          • II. Existence and uniqueness of the perpendicular
          • III. Projections and distance
          • Exercises
        • Constructions with ruler and compass
        • 7. Notable quadrilaterals
          • I. Quadrilateral — definition and elements
          • II. Notable quadrilaterals — definitions
          • III. Properties of trapezoids
          • IV. Properties of parallelograms
          • V. Rectangle, rhombus and square
          • VI. Consequences — medium bases
          • Exercises
        • 9. Polygons
          • I. Definitions and elements
          • II. Diagonals, interior angles and exterior angles
          • Exercises
        • Areas of flat figures
        • 10. Circumference and circle
          • I. Definitions and elements
          • II. Relative positions of straight line and circle
          • III. Relative positions of two circles
          • IV. Tangent segments — circumscribable quadrilaterals
          • Exercises
        • Circumference length
        • Pythagoras Theorem
      • High School
        • 8. Notable points of the triangle
          • I. centroid — medians
          • II. incenter — angle bisectors internal
          • III. circumcenter — perpendicular bisectors
          • IV. orthocenter — heights
          • Exercises
          • Reading: Papus: the epilogue of Greek geometry
        • 11. Angles on the circumference
          • I. Congruence, addition and inequality of arcs
          • II. Central angle
          • III. Inscribed angle
          • IV. Segment angle (semi-inscribed)
          • V. Eccentric angles
          • Exercises
        • 12. Thales' theorem
          • I. Thales' Theorem
          • II. angle bisectors Theorems
          • Exercises
          • Reading: Legendre: towards a rigorous and didactic geometry
        • 13. Similarity of triangles and power of point
          • I. Similarity of triangles
          • II. Cases (criteria) of similarity
          • III. Point power
          • Exercises
        • 14. Right triangles
          • I. Metric relationships
          • II. Applications of the Pythagorean theorem
          • Exercises
        • 15. Any triangles
          • I. Sines theorem
          • II. Metric relations and cosine theorem
          • III. Calculation of notable lines
          • Exercises
        • Regular Polygons
        • Circumference ratio in regular polygons
    • Trigonometry
      • Elementary I (1st to 5th year)
        • 01. Whole turns, half turns and quarters
        • 02. Corners, spokes and paths on the wheel
        • 03. Triangles that grow without changing shape
        • 12. Triangulation in the field
      • Elementary II (6th to 9th year)
        • 01. Arcs, chords, rays and angles
        • 02. Measurement of angles in degrees, minutes and seconds
        • 03. Inscribed polygons and the sine of a central half-angle
        • 04. Demonstrations of the values ​​of 30, 45 and 60 degrees
        • 05. Pythagoras, height and projections in the right triangle
        • 06. Trigonometric ratios and solving right triangles
        • I. Arcs and angles
          • I. Arcs and measurements
          • II. Angles and their measurements
          • III. Trigonometric cycle
          • Exercises
          • Reading: Bolzano
        • V. Notable arcs
          • I. Theorem and applications
          • Exercises
        • IX. Right triangles
          • I. Metric elements and relationships
          • II. Trigonometric ratios
          • III. Solving right-angled triangles
          • Exercises
        • 01. Arcs and angles
        • 05. Notable angles
        • 09. Right triangles
      • High School
        • 01. Radians, arc length and trigonometric cycle
        • 02. Circular functions and periodicity
        • 03. Sine function: definition, signs and graph
        • 04. Cosine function: horizontal coordinate and graph
        • 05. Tangent function: slope, domain and graph
        • 06. Cotangent, secant and cosecant
        • 07. Fundamental relationships demonstrated
        • 08. Identities and three demonstration paths
        • 09. Reduction of the second, third and fourth quadrants
        • 10. Complements, reduction identities, and parity
        • 11. Arc addition and subtraction formulas
        • 12. Sine, cosine and tangent of double and triple arc
        • 13. Arc half division and tangent formulas
        • 14. Transform sums into products and products into sums
        • 15. Fundamental equations of sine, cosine and tangent
        • 16. Intervals and classical trigonometric equations
        • 17. Arcsine, arccosine and arctangent
        • 18. Sine, cosine and tangent inequalities
        • 19. Laws of sines, cosines, projections and areas
        • 20. Heights, Heron, Medians, angle bisectors and Radii
        • 21. Solving triangles and discussing the ambiguous case
        • II. Circular functions
          • I. General notions and periodicity
          • II. Sine function
          • III. Cosine function
          • IV. Tangent function
          • V. Cotangent, secant and cosecant
          • Exercises
          • Reading: Fourier
        • III. Fundamental relationships
          • I. Fundamental relationships
          • II. Identities
          • Exercises
        • IV. Reduction to the 1st quadrant
          • I. Reduction by quadrants
          • II. Complementary arches and board
          • III. Identities and parity
          • Exercises
        • VI. Transformations
          • I. Addition formulas
          • II. Multiplication, division and half arc
          • III. Transformation into product
          • Exercises
          • Reading: Poncelet
        • VII. Equations
          • I. Fundamental equations
          • II. Intervals and classical equations
          • III. Inverse circular functions
          • Exercises
          • Reading: Euler
        • VIII. Inequalities
          • I. Sine inequalities
          • II. Cosine inequalities
          • III. Tangent inequalities
          • Exercises
        • X. Any triangles
          • I. Laws of cosines and sines
          • II. Projections, areas and law of tangents
          • III. Heights, medians, angle bisectors and radii
          • IV. Solving any triangles
          • Exercises
        • Entrance exams
          • Circular functions
          • Fundamental identities
          • Transformations
          • Equations
          • Inverse functions
          • Inequalities
          • Triangles
        • 02. Circular functions
        • 03. Fundamental relationships
        • 04. Symmetries
        • 06. Transformations
        • 07. Equations
        • 08. Inequalities
        • 10. Any triangles
        • Final review
      • Map of texts and formulas
  • Mathematical programming
    • Project Euler
      • Level Problems 1
  • Pure Mathematics
    • Elements of Mathematics
      • Mathematical Logic
      • Truth Table
      • Quantifier
      • Axioms and Postulates
      • Theorems
      • Demo
    • Set Theory
      • Primitive Notions
      • Introduction to Set Theory
      • Operations with Sets
      • Set Family
      • Cartesian Product of Sets
    • Relations
      • Introduction to Relationships
      • Relationship Properties
      • Equivalence Relations
      • Set Partition
    • Natural Numbers
      • Peano axioms
  • OBMEP Exercises
    • 2018
      • OBMEP 2018
      • OBMEP 2018 - Second Phase
    • 2019
      • OBMEP 2019 - Second Phase
    • 2021
      • OBMEP 2021 - 1 Phase
      • OBMEP 2021 - 2 Phase
    • 2022
      • OBMEP 2022 - 2 Phase - N1
      • OBMEP 2022 - 1 Phase
    • 2023
      • 1st phase – Level 1
      • 1st phase – Level 2
      • 1st phase – Level 3
      • 2nd phase – Level 1
      • 2nd phase – Level 2
      • 2nd phase – Level 3
  • Entrance Exam Exercises
    • ENEM
      • ENEM 2025
        • Questions 136 to 150
        • Questions 151 to 165
        • Questions 166 to 180
      • ENEM 2024
        • Questions 136 to 150
        • Questions 151 to 165
        • Questions 166 to 180
      • ENEM 2023
        • Questions 136 to 150
        • Questions 151 to 165
        • Questions 166 to 180
      • ENEM 2022
        • Questions 136 to 150
        • Questions 151 to 165
      • ENEM 2021
        • Questions 136 to 150
        • Questions 151 to 165
        • Questions 166 to 180
    • Fuvest
      • FUVEST 2026
        • 1st phase
        • 2nd phase
      • FUVEST 2025
        • 1st phase
        • 2nd phase
      • FUVEST 2024
        • 1st phase
        • 2nd phase
      • FUVEST 2023
        • 1st phase
        • 2nd phase
      • FUVEST 2022
        • 1st phase
        • 2nd phase
      • FUVEST 2021 – 2nd phase
    • ITA
      • ITA 2026
        • 1st phase
        • 2nd phase
      • ITA 2025
        • 1st phase
        • 2nd phase
      • ITA 2024
        • 1st phase
        • 2nd phase
      • ITA 2023
        • 1st phase
        • 2nd phase
      • ITA 2022
        • 1st phase
        • 2nd phase
  • The Semiotics of Mathematical Notation
    • 1. From word to symbol
      • Radix quadrati: mathematics has a script
      • Zero and place value
      • Euclid: the figure and the proof
      • Sine, leg and hypotenuse
      • From words to letters
      • +, −, × and =: the history of the signs
      • Napier and the logarithm
      • Leibniz against the dots
      • From exhaustion to the integral
      • Euler and functions
      • Cauchy and the limit
      • Galois and Cayley
      • Hamilton, Gibbs and Heaviside
      • Differential equations
      • Boole, Frege and Peano
      • References and credits
    • 2. The mathematical sign
      • What is a sign?
      • Peirce: icon, index and symbol
      • The object and its representations
      • The grammar of formulas
      • When place is the algorithm
      • The equals sign is a balance
      • Square root: the side of the square
      • Formula, table and graph
      • How to read a notation
      • References and credits
    • 3. Variables, unknowns and functions
      • One letter, five roles
      • The unknown
      • Letters that generalise
      • The variable
      • Parameters
      • Naming a function
      • Dummy letters
      • When the letter misleads
      • Guide and glossary
      • References and credits
    • 4. Equality, equivalence and proof
      • One sign, many equals
      • Frege and the morning star
      • Equation, identity, definition
      • Equivalence
      • ⇒ and ⇔
      • The chain of equalities
      • The words of a proof
      • Examples do not prove
      • Guide and glossary
      • References and credits
    • 5. Writing calculus
      • Four ways to write the derivative
      • From Δ to d
      • Is dy/dx a fraction?
      • The D that factorises
      • The integral sign by sign
      • The fundamental theorem
      • Wallis's ∞
      • Sums that never end
      • Guide and glossary
      • References and credits
    • 6. Diagrams and graphs
      • Euler's and Venn's circles
      • From Oresme to the graph
      • Charts that saved lives
      • The scale of a chart
      • Argand's plane
      • Graphs and maps
      • Proofs without words
      • The fourth dimension
      • Guide and glossary
      • References and credits
    • 7. Reading and writing mathematics
      • Counting, saying and writing
      • Read before you calculate
      • Two languages in one sentence
      • The captain's age
      • The hidden conventions
      • Errors that make sense
      • Writing to think
      • Reading with hands and ears
      • Guide and glossary
      • References and credits
    • 8. Notation and culture
      • Other scripts
      • The metre and decimal time
      • Tribal conventions
      • A tool of thought
      • The equals sign that gives orders
      • The names on theorems
      • Street and school mathematics
      • Notation for machines
      • Guide and glossary
      • References and end of the book
  1. Books
  2. OBMEP Exercises
  3. Year 2022
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Euler's Eyes

2020-06-19

OBMEP Exercises Page 7 of 13

OBMEP 2022 - First Phase

  • Mathematics for children
  • Children
  • Obmep
  • 1 phase
  • Year 2022

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