Combinatorial Analysis

ENEM 2021 – Solved mathematics: questions 136 to 150

Source: INEP – proof and template of the 2nd day, Notebook 5 – Yellow (1st application). The ENEM Mathematics and Technologies test 2021 was administered on the 2nd day and will from question 136 to 180. This post addresses questions 136 to 150. The numbering used here is that of the Notebook 5 – Yellow; us In other notebooks the same questions appear in a different order. Template Question Subject Template 136 Roman numeral system D 137 Addition in binary base D 138 Area and subdivision units C 139 Combinatorial analysis: combinations A 140 Cost per user (comparison) B 141 Percentage: heart rate D 142 Fewer number of washes (inequality) C 143 Discounts: lower expense C 144 Spatial geometry: faces of the structure A 145 Octahedron flattening E 146 Perimeter of equilateral triangle D 147 Volume of the cone frustum C 148 Areas of flat figures and cost E 149 Dimensional analysis B 150 Linear system: baggage mass B Question 136: Year of foundation in Roman numerals Statement The Roman numeral system is still used to indicate chapters and volumes of books, in the designation of centuries and, in chronological order, of popes and kings of the same name. Seven letters of the alphabet are used:

September 26, 2026 Read

ENEM 2022 – Solved mathematics: questions 136 to 150

Source: INEP – proof and template of the 2nd day, Notebook 5 – Yellow (1st application). The ENEM Mathematics and Technologies test 2022 was administered on the 2nd day and will from question 136 to 180. This post addresses questions 136 to 150. The numbering used here is that of the Notebook 5 – Yellow; us In other notebooks the same questions appear in a different order. Template Question Subject Template 136 Percentage and weighted average of salaries D 137 Ratio and time unit conversion B 138 Ratio between time units A 139 Distance × time graph reading A 140 Percentage increases and discounts A 141 Guidance and mesh paths C 142 Cylinder volume E 143 Ratio between quantities in graphs B 144 Big numbers: billion E 145 Shortest path: reflection and similarity D 146 Cost Comparison A 147 Sphere volume and proportion E 148 Cone volume and percentage C 149 Percentage: 75% drop C 150 Combinatorial analysis: combinations B Question 136: Salary increase and average salary (percentage) Statement At a university, professors work who are functionally for your highest degree: master or doctor. There are currently 60 masters and 40 doctors. The monthly salaries of master teachers and doctors are, respectively, R$ 8 000,00 and R$ 12 000,00.

September 26, 2026 Read

ENEM 2024 – Solved mathematics: questions 136 to 150

Source: INEP – proof and template of the 2nd day, Notebook 5 – Yellow. The ENEM Mathematics and Technologies test 2024 was administered on the 2nd day and will from question 136 to 180. This post addresses questions 136 to 150. The numbering used here is that of the Notebook 5 – Yellow; us In other notebooks the same questions appear in a different order. Template Question Subject Template 136 Related function: production cost C 137 Statistics: median E 138 Proportional division C 139 Combinatorial analysis: combinations E 140 Probability of independent events B 141 Reason: demographic density E 142 Fractions and mixed numbers D 143 Increasing plots (logical reasoning) A 144 Arc length: pulleys and belt D 145 equilateral triangle: circumcenter C 146 Graph Reading: Sine Amplitude A 147 Areas of regular polygons A 148 Speed and furniture finding D 149 Probability without replacement E 150 Logarithm: Richter scale C Question 136: Cost of producing backpacks (related function) Statement A company produces custom school backpacks. This company has a cost total production, composed of a fixed cost, which does not depend on the number of backpacks, plus a variable cost, which is proportional to the number of backpacks produced. The total cost grows linearly, and the table shows this cost for three quantities of backpacks produced.

September 26, 2026 Read

ENEM 2024 – Solved mathematics: questions 151 to 165

Source: INEP – proof and template of the 2nd day, Notebook 5 – Yellow. The ENEM Mathematics and Technologies test 2024 was administered on the 2nd day and will from question 136 to 180. This post addresses questions 151 to 165. The numbering used here is that of the Notebook 5 – Yellow; us In other notebooks the same questions appear in a different order. Template Question Subject Template 151 Cylinder volume D 152 Affine function: trend line B 153 Reading Charts: Tank Filling B 154 Statistics: arithmetic mean C 155 Circular sector area E 156 Related function: price modeling B 157 Flow and percentage D 158 Proportional quantities: units C 159 Cost optimization C 160 1st degree equation C 161 Bar chart and percentage A 162 Orthogonal projection C 163 Scales and magnification A 164 Arithmetic mean B 165 Fundamental counting principle B Question 151: Larger volume of cylindrical packaging Statement An industry forms a partnership with a juice distributor to launch market two types of packaging. To manufacture these packaging, the industry has rectangular aluminum sheets, with dimensions 10 cm per 20 cm. Each of these sheets is used to form the lateral surface of the packaging, in the shape of a straight circular cylinder, which subsequently receives circular bottom and lid. The figure illustrates, depending on which of the two extensions will be used as height, the two options to form the possible packaging.

September 26, 2026 Read

ENEM 2025 – Solved mathematics: questions 151 to 165

Source: INEP – proof and template of the 2nd day, Notebook 5 – Yellow. The ENEM Mathematics and Technologies test 2025 was administered on the 2nd day and will from question 136 to 180. This post addresses questions 151 to 165. The numbering used here is that of the Notebook 5 – Yellow; us In other notebooks the same questions appear in a different order. Template Question Subject Template 151 Polyhedra: number of vertices B 152 Proportionality: production D 153 Numeric code (rules reading) D 154 Probability: permutations D 155 Reading charts: bars and sectors B 156 Trigonometry: law of cosines and directions A 157 Statistics: mean and standard deviation D 158 Average (running pace) E 159 Tangent function: period and displacement E 160 Mesh localization (orientation) A 161 Percentage and inequality of the 1st degree B 162 Combinatorial analysis: multiplicative principle B 163 Scale E 164 Reason C 165 Percentage: cost and profit C Question 151: Vertices of the gyroelongated pentagonal dome Statement The gyroelongated pentagonal dome is a Johnson polyhedron, whose faces are regular polygons, but it is not a polyhedron of Plato, Archimedes, prism or antiprism.

September 26, 2026 Read

ENEM 2025 – Solved mathematics: questions 166 to 180

Source: INEP – proof and template of the 2nd day, Notebook 5 – Yellow. The ENEM Mathematics and Technologies test 2025 was administered on the 2nd day and will from question 136 to 180. This post addresses questions 166 to 180. The numbering used here is that of the Notebook 5 – Yellow; us In other notebooks the same questions appear in a different order. Template Question Subject Template 166 Volume Units D 167 Agricultural production and line graph A 168 Volume: cylinder minus prism B 169 Probability with data E 170 Units of measurement A 171 Sequences: expression of the general term D 172 Chart reading and fuel cost D 173 Cost Comparison E 174 Volume of the parallelepiped B 175 Spatial geometry: cylinder in prism E 176 Logarithm and quadratic function A 177 Area of the semicircle A 178 Equation: installments of financing Canceled 179 Combinatorial analysis: combinations C 180 Average speed C Question 166: Ice cream cups (volume units) Statement The owner of an ice cream shop stores ice cream in 20 000 cm³ containers. He serves the ice cream in cups, in portions of 250 mL.

September 26, 2026 Read

FUVEST 2022 – 1st phase: Mathematics solved

Source: FUVEST – General Knowledge test, version V and rectified official template of the 1st phase (applied in 12/12/2021). The 1st phase of FUVEST 2022 had 90 questions from all subjects. This post brings the 12 mathematics questions, which in test V are questions 30 to 41. In the others versions (K, Q, X and Z) the same questions appear with different numbering and another order of alternatives, so always check the version of your test.

September 26, 2026 Read

FUVEST 2022 – 2nd phase: Mathematics solved

Source: FUVEST – 2nd phase test, 2nd day (notebook with all the questions of the day), questions M01 to M06. FUVEST does not publish a template for the 2nd phase; the resolutions are ours. On the 2nd day of the 2nd phase of FUVEST 2022 each candidate received 12 discursive questions of the specific disciplines of your career. Mathematics had 6 questions (M01 to M06). Those who only did 4 math questions received M01 to M04; who only did 3, M01 to M03. Each resolution below was checked by a second path (another method or account on the computer).

September 26, 2026 Read

FUVEST 2023 – 1st phase: Mathematics solved

Source: FUVEST – proof (version V) and official answer key of the 1st phase (General Knowledge Test, applied in 04/12/2022). The 1st phase of FUVEST 2023 had 90 questions from all subjects. This post brings the mathematics questions: 21, 22, 24, 36, 58 and 84 to 90. The numbering is that of Proof V; in other versions (K, Q, X and Z) the same questions appear in another order. The question 58 is interdisciplinary (CO₂ emissions), but is resolved with graph reading and area estimation, that’s why it’s here.

September 26, 2026 Read

FUVEST 2023 – 2nd phase: Mathematics solved

Source: FUVEST – 2nd phase test, 2nd day (questions M01 to M06) and document expected approaches. The 2nd phase of FUVEST 2023 lasted two days. The 1st day was Portuguese and Writing; in 2nd day, the test had 6 questions from each subject, and Mathematics goes from M01 to M06. Each candidate answered only the subjects of their career (in some careers, only the first questions of each subject). FUVEST does not publish a template for the 2nd phase; the resolutions are ours. She disclosed an “expected approaches” document with the path expected of the candidates. Each resolution below was checked by a second path and compared to this document.

September 26, 2026 Read