Statistic

Image credits

Images from Wikimedia Commons, all in the public domain: Gauss at 1840: Christian Albrecht Jensen. Gauss at 1803: Johann Christian August Schwartz. Gauss's birthplace, photo by 1914. Gauss's diary: Göttinger Digitalisierungszentrum. Disquisitiones Arithmeticae, 1801. Giuseppe Piazzi: F. Bordiga. Sophie Germain: Berthe Chégaray. Gaussian heliotrope. Triangle network: Reinhold Gerstetter for the Deutsche Bundesbank. Theoria motus corporum coelestium, 1809. Wilhelm Weber: Rudolph Hoffmann. The images of the 10 mark note, the squaring of the circle, and the determinant and matrix notations are from Euler’s Eyes collection. The interactive figures were made for this website.

October 10, 2026 Read

ENEM 2021 – Solved mathematics: questions 166 to 180

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 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 Percentage increase C 167 Histogram and accumulated frequency C 168 Caesar cipher and letter frequency C 169 Statistics: median C 170 Percentage: reservoir volume D 171 Weighted average D 172 Weighted average and linear system C 173 Coefficient of variation E 174 Comparison of means E 175 Weighted average: monthly revenue B 176 Algebraic expression with percentage A 177 Probability of the complementary event B 178 Combinatorial analysis (cancelled) Canceled 179 Powers of the golden ratio E 180 Orthogonal projection of the cube E Question 166: Increase in sales of antidepressants Statement Depression is characterized by an imbalance in brain chemistry. The Neurons in a depressed person do not respond well to stimuli from neurotransmitters. Medicines that combat depression aim to restore brain chemistry. With the gradual increase in cases of depression, the sale of these medicines is increasingly evolving, as illustrated by the graphic.

September 26, 2026 Read

ENEM 2022 – Solved mathematics: questions 151 to 165

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 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 Graph reading and 2nd degree equation E 152 Odds: World Series C 153 Median from bar chart D 154 Proportion: game pack E 155 Linear system and maximization E 156 Proportionality and percentage C 157 Multiplicative principle B 158 Cube planning D 159 Checksum: remainder of division C 160 Scale A 161 Counting days in a tournament B 162 Odds: bingo E 163 Folding and symmetry C 164 Arithmetic mean B 165 Orthogonal projection onto the sphere E Question 151: How many times does the piston speed cancel out Statement A machine in operation has its temperature T monitored using a graphic record over time t. This machine has a piston whose speed V varies with the temperature T of the machine, according to the expression $V = T^2 - 4$. After the machine works during the time interval of 10 hours, its operator analyzes the graphic record, shown in the figure, to assess the need for possible adjustments, knowing that the machine malfunctions when the piston speed drops.

September 26, 2026 Read

ENEM 2023 – Solved mathematics: questions 136 to 150

Source: INEP – proof and template of the 2nd day, Notebook 5 – Yellow. The ENEM Mathematics and Technologies test 2023 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 Proportion and rounding up D 137 Trigonometry in the right triangle C 138 Scale E 139 Tables: caloric expenditure C 140 Discount proportion and percentage A 141 Conversion between calendars C 142 Pythagoras and the law of cosines: areas of circles C 143 Units of measurement B 144 Arithmetic progressions: sales comparison D 145 Statistics: mean, median and mode E 146 Cylinder flow and volume C 147 Capable arc (inscribed angle) B 148 Magic triangle and arithmetic progression E 149 Probability with weights E 150 Cylinder volume and water consumption A Question 136: Packets of bread for the walk (proportion) Statement Some studies prove that carbohydrates provide the body with energy, preserve the structural proteins of muscles during activity physics and also give strength to the brain to coordinate movements, which in fact has a positive impact on the practitioner’s development. The ideal is to consume 1 grams of carbohydrates for every minute of walking.

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 136 to 150

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 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 Circumference length C 137 Percentage: solution concentration E 138 Reasoning problem: discarded volume C 139 Statistics: median C 140 Multiplication and comparison A 141 Spatial geometry: faces of a polyhedron D 142 Graph reading D 143 Operations with decimals B 144 Spatial geometry: orthogonal projections E 145 Probability B 146 Reason and consumption C 147 Analytical geometry: perpendicular bisector A 148 Bar and line graph reading C 149 Percentage and graphs E 150 Scale C Question 136: Police on the bike path (length of circumference) Statement Around a circular lagoon, whose radius measures 1 km, there is a cycle path. Due to frequent bicycle thefts, the city hall plans to allocate police officers in strategic positions to patrol this cycle path, in order to make it fully protected. A point on the cycle path is considered protected if there is at least one police officer at most 200 m away from that point, positioned over the cycle path. The figure illustrates a point $P$ on the cycle path, that will be protected if there is at least one police officer positioned over the dark gray region.

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

FUVEST 2024 – 1st phase: Mathematics solved

Source: FUVEST – test of the 1st phase, version V and official template (rectified in 24/11/2023), available in the FUVEST collection 2024. The 1st phase of FUVEST 2024 (General Knowledge Test, 19/11/2023) had 90 questions from all disciplines. This post only contains 12 math questions. The numbering used here is that of Proof V; In versions K, Q, X and Z the same questions appear with other numbers. Template Question Subject Template 13 Matrices: product and social network B 31 Volume of cylinder and circular segment B 33 Percentage: cost variation A 34 Statistics: mode, median, mean and range A 35 Count: digital clock reflected B 48 Cobblestone Packaging D or E (both accepted) 49 Parallelogram and trigonometry C 59 Exponential and logarithmic function (inverse) D 60 Linear system and line graph E 63 Pythagorean Tree: geometric progression and logarithm C 84 Graph Reading: Homicide Rates E 85 Triangular and oblong numbers D Question 13: Matrices and connections in a social network Statement Matrices can be used to obtain information about a social network. To understand how this can be done, let’s consider as an example a small social network made up of 4 people: $P_1, P_2, P_3, P_4$. The matrix associated with this social network is the matrix $4 \times 4$:

September 26, 2026 Read