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 $2x + 1$.
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 $B$.
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 $z$:
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 $A$.