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
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 $A$.