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?
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 $O$.
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 $\overrightarrow{OB}$.