Entrance Exam

ITA 2026 – 2nd phase: Mathematics solved

Source: ITA – 2nd phase Mathematics exam. Files on the ITA past exams page. The 2nd phase Mathematics exam of ITA 2026 has 10 written questions. ITA does not publish official solutions for the written part (the 2nd phase answer key only covers the multiple-choice Portuguese questions); the solutions below are ours, and each answer was checked by a second route (substitution, another formula or a computer check). Answers Question Topic Answer 01 Lattice points in the octahedron 231 02 Powers of a triangular matrix $n = 15$ 03 Parallelogram and inscribed circle 30 04 Red area of the target $\pi(\alpha^2 + 32\alpha + 528)$ 05 Four spheres in a larger sphere $\frac{22 + 9\sqrt{6}}{16}$ 06 Triangular matrices (inconsistent statement) see comment 07 Degree-4 polynomial and remainder $-1$, $\frac{1}{2}$, $1 \pm 2i$ 08 Increasing triples with sum 100 784 09 Tangents to the hyperbola through a point $y = \pm 2\sqrt{6}\,x - 8$; $(\pm\sqrt{6}, 4)$ 10 Intersecting circles and common area radii $2\sqrt{3}$ and $6$; area $10\pi - 12\sqrt{3}$ Question 01: Lattice points in the octahedron Statement Consider a regular octahedron centred at the origin, with three vertices at $(5, 0, 0)$, $(0, 5, 0)$ and $(0, 0, 5)$. How many points in the interior and on the surface of this solid have all integer coordinates?

October 28, 2026 Read

ITA 2026 – 1st phase: Mathematics solved

Source: ITA – 1st phase exam and official answer key. Files on the ITA past exams page. The 1st phase of ITA 2026 had 48 multiple-choice questions: 12 in Mathematics, 12 in Physics, 12 in Chemistry and 12 in English. This post covers the mathematics questions, numbered 1 to 12 as in the exam. Exam conventions. $i^2 = -1$; $\bar{z}$ is the conjugate of $z$; $]a, b[$ is the open interval $a < x < b$.

October 28, 2026 Read

ITA 2025 – 2nd phase: Mathematics solved

Source: ITA – 2nd phase Mathematics exam. Files on the ITA past exams page. The 2nd phase Mathematics exam of ITA 2025 has 10 written questions. ITA does not publish official solutions for the written part (the 2nd phase answer key only covers the multiple-choice Portuguese questions); the solutions below are ours, and each answer was checked by a second route (substitution, another formula or a computer check). Answers Question Topic Answer 1 Remainder of the division by $x^2 - x + 1$ $a = -1$, $b = 3$ 2 Ellipse: maximum of $y/x$ $r = \sqrt{3}$ 3 Trigonometric system $\sin(\alpha + \beta) = \frac{5 + \sqrt{7}}{8}$ 4 Excircle $15\sqrt{3}$ 5 First digit of $3^{100}$ 5 6 Probability: fourth head on toss $n$ $n = 6$ and $n = 7$ 7 Polynomial with a double root and a right triangle $a = \pm\frac{3\sqrt[4]{2}}{2}$, $b = \mp\frac{1}{\sqrt[4]{2}}$ 8 Sum of determinants 1013 9 Integer matrices with $A^{-1} = A^T$ 3840 10 Projection of the cube perpendicular to the diagonal $4\sqrt{3}$ Question 1: The remainder of the division by x² − x + 1 Statement Find the real values $a$ and $b$ such that the polynomial $p(x) = x^{57} + ax^{14} + bx^7 + 1$, when divided by $x^2 - x + 1$, leaves remainder $2x + 1$.

October 28, 2026 Read

ITA 2025 – 1st phase: Mathematics solved

Source: ITA – 1st phase exam and official answer key. Files on the ITA past exams page. The 1st phase of ITA 2025 had 48 multiple-choice questions: 12 in Mathematics, 12 in Physics, 12 in Chemistry and 12 in English. This post covers the mathematics questions, numbered 1 to 12 as in the exam. Exam conventions. The horizontal and vertical axes are $O_x$ and $O_y$, and $O$ is the origin; $\mathbb{N} = \{1, 2, 3, \dots\}$; $i^2 = -1$; $\overline{AB}$ is the segment and $AB$ the line through $A$ and $B$.

October 28, 2026 Read

ITA 2024 – 2nd phase: Mathematics solved

Source: ITA – 2nd phase Mathematics exam. Files on the ITA past exams page. The 2nd phase Mathematics exam of ITA 2024 has 10 written questions. ITA does not publish official solutions for the 2nd phase; the solutions below are ours, and each answer was checked by a second route (substitution, another formula or a computer check). Answers Question Topic Answer 1 Arithmetic and geometric progressions $n = 4$ 2 Paths over the bridges (Euler path) 32 (16 in each direction) 3 Inequality with absolute values of logarithms $-4 < x < -\frac14$ or $\frac14 < x < 4$ 4 Positive definite matrix $-1 < b < 1$ 5 Polynomial symmetric about $x = 1$ $1 \pm \sqrt{3}$ and $1 \pm i\sqrt{6}$ 6 Longest segment inside a solid $\frac{4\sqrt{161}}{5}$ 7 Circle tangent to the $y$-axis $(x - 4 - \sqrt{6})^2 + (y - 3 + \sqrt{6})^2 = (4 + \sqrt{6})^2$ 8 Trapezoid with vertices on the parabola $\frac{483}{64}$ 9 Complex roots and area $4\sqrt{5}$ 10 Tangent of the half angle $\frac{1 + \sqrt{5}}{2}$, $\frac{\sqrt{5}}{5}$ and $\frac{5 - \sqrt{5}}{10}$ Question 1: How many terms of the geometric progression add up to −80? Statement Let $(a_n)$ be an arithmetic progression and $(b_n)$ a geometric progression. If the common difference of $(a_n)$ is $r$, $r \ne 0$, the ratio of $(b_n)$ is $q = 1/r$, $a_1 = b_1 = 4$ and

October 28, 2026 Read

ITA 2024 – 1st phase: Mathematics solved

Source: ITA – 1st phase exam and official answer key. Files on the ITA past exams page. The 1st phase of ITA 2024 had 60 multiple-choice questions, 12 per subject (Physics, Portuguese, English, Mathematics and Chemistry). This post covers the mathematics questions, numbered 37 to 48 as in the exam. Exam conventions. $\mathbb{N} = \{1, 2, 3, \dots\}$; $A - B$ is the difference of the sets $A$ and $B$; $i^2 = -1$; $M_n(\mathbb{R})$ is the set of $n \times n$ real matrices; $\overline{AB}$ is the segment and $AB$ the line through $A$ and $B$; $m(\overline{AB})$ is the length of $\overline{AB}$; $A\hat{B}C$ is the angle with vertex $B$; $\det A$ is the determinant and $A^T$ the transpose of $A$.

October 28, 2026 Read

ITA 2023 – 2nd phase: Mathematics solved

Source: ITA – 2nd phase Mathematics exam. Files on the ITA past exams page. The 2nd phase Mathematics exam of ITA 2023 has 10 written questions. ITA does not publish official solutions for the 2nd phase; the solutions below are ours, and each answer was checked by a second route (substitution, another formula or a computer check). Answers Question Topic Answer 1 Homogeneous linear system $b = 2a$; $\{(-15t, -16t, 17t) : t \in \mathbb{R}\}$ 2 Matrices: when $M$ is invertible $\alpha \notin \{0, 2, 3 - \sqrt{15}, 3 + \sqrt{15}\}$ 3 Logarithmic inequality with a variable base $]-2, 0[$ 4 Polynomial division and roots $q = x^6 + x^5 - x - 1$, $r = 0$; roots $\operatorname{cis}\frac{\pi}{5}, \operatorname{cis}\frac{3\pi}{5}, \operatorname{cis}\frac{7\pi}{5}, \operatorname{cis}\frac{9\pi}{5}$ 5 Trigonometry: sum to product $\sin(\alpha - \beta) = \frac{2AB}{A^2 + B^2}$ 6 Radius of the incircle $\frac{6\sqrt{7} - 3\sqrt{3}}{10}$ 7 Ellipse: a 60° angle between the focal radii $\left(\pm\frac{4\sqrt{2}}{3}, \pm\frac{1}{3}\right)$ 8 Equilateral cylinder with a sphere $400\pi$ cm² and $\frac{4000\pi}{3}$ cm³ 9 Law of sines: sides of the triangle $6 + \sqrt{11}$ and $6 - \sqrt{11}$ 10 Counting: pairs of diagonals of a decagon 175 Question 1: When the homogeneous system has a non-trivial solution Statement Let $a$ and $b$ be positive real numbers. Consider the linear system in the unknowns $x$, $y$ and $z$:

October 28, 2026 Read

ITA 2023 – 1st phase: Mathematics solved

Source: ITA – 1st phase exam and official answer key. Files on the ITA past exams page. The 1st phase of ITA 2023 had 60 multiple-choice questions, 12 per subject (Physics, Portuguese, English, Mathematics and Chemistry). This post covers the mathematics questions, numbered 37 to 48 as in the exam. Exam conventions. $\mathbb{N} = \{1, 2, 3, \dots\}$; $i^2 = -1$; $M_n(\mathbb{R})$ is the set of $n \times n$ real matrices; $\overline{AB}$ is the segment and $AB$ the line through $A$ and $B$; $m(\overline{AB})$ is the length of $\overline{AB}$; $A\hat{O}B$ is the angle with vertex $O$; $\det A$ is the determinant of $A$.

October 28, 2026 Read

ITA 2022 – 2nd phase: Mathematics solved

Source: ITA – 2nd phase Mathematics exam. Files on the ITA past exams page. The 2nd phase Mathematics exam of ITA 2022 has 10 written questions. ITA does not publish official solutions for the 2nd phase; the solutions below are ours, and each answer was checked by a second route (substitution, another formula or a numerical check). The notation conventions are the same as in the 1st phase: $i^2 = -1$, $m(\overline{AB})$ is the length of segment $\overline{AB}$ and $A\hat{O}B$ is the angle with vertex $O$.

October 28, 2026 Read

ITA 2022 – 1st phase: Mathematics solved

Source: ITA – 1st phase exam and official answer key. Files on the ITA past exams page. The 1st phase of ITA 2022 had 70 multiple-choice questions: 15 in Physics, 15 in Portuguese, 10 in English, 15 in Mathematics and 15 in Chemistry. This post covers the mathematics questions, numbered 41 to 55 as in the exam. Exam conventions. $\mathbb{N} = \{1, 2, 3, \dots\}$; $i$ is the imaginary unit ($i^2 = -1$); $M_n(\mathbb{R})$ is the set of $n \times n$ real matrices; $\overline{AB}$ is the segment with endpoints $A$ and $B$, and $m(\overline{AB})$ is its length; $A\hat{O}B$ is the angle with vertex $O$ formed by the rays $\overrightarrow{OA}$ and $\overrightarrow{OB}$.

October 28, 2026 Read

FUVEST 2026 – 2nd phase: Mathematics solved

Source: FUVEST – test on the 2nd day of the 2nd phase (specific tests; Mathematics has questions M01 to M06) and expected Mathematics answers. Files in the FUVEST collection 2026. In the 2nd phase of FUVEST 2026, the 2nd day consisted of specific essay tests for each area; The Mathematics test has six questions, M01 to M06, each with items a), b) and c). For this test, FUVEST published the expected answers; the resolutions below are ours, we checked each one in a second way and all match the official expected answers.

September 28, 2026 Read

FUVEST 2026 – 1st phase: Mathematics solved

Source: FUVEST – test V1 and template of the General Knowledge Test (1st phase), applied in 23/11/2025. Files in FUVEST collection 2026. The 1st phase of FUVEST 2026 had 90 multiple choice questions from all disciplines. This post brings mathematics questions. The numbering used here is that of Proof V1; in versions V2, V3 and V4 the same issues appear in another order (the official template includes the correspondence table). Template Question Subject Template 3 Logical reasoning: election with tiebreaker Canceled 4 Logarithms: Shannon entropy A 6 Spatial geometry: regular pyramid E 7 Trigonometry: double arc equation D 18 Exponential function and logarithms E 30 Spatial geometry: cylinder volume B 36 Trigonometry in the right triangle E 37 Analytical geometry: circle equation A 55 Perfect numbers, primes and progressions B 74 Function graph transformations D Question 3: How many votes guarantee the election? Question canceled by FUVEST. In the official answer sheet, question 3 from Test V1 appears with an asterisk (nullified). FUVEST did not disclose the reason for the cancellation.

September 28, 2026 Read