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 $B$.