Analytic Geometry

The curve of Agnesi

Agnesi did not discover the curve that bears her name. But it was through her book that the curve became known, because she used it as an example of how to pass from a geometric construction to an equation, which was exactly what Descartes’s new method promised. The construction Start with a circle of diameter $OM = a$, with $O$ at the bottom and $M$ at the top, and draw the line tangent to the circle at $M$. Now:

October 28, 2026 Read

Fermat and analytic geometry: a curve in an equation

For almost two thousand years, geometry and algebra lived apart. Greek geometry worked with ruler, compass and figures. Algebra, which reached Europe through the Arabs, worked with equations. At the start of the 17th century, two people had, at the same time and without consulting each other, the idea of joining them: René Descartes and Pierre de Fermat. A sentence that created a field Around 1636 Fermat wrote a short Latin text, Introduction to plane and solid loci. “Locus” was the Greek name for a set of points satisfying a condition, like the circle, the locus of points at a fixed distance from the centre. Right at the start, Fermat wrote:

October 7, 2026 Read

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 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 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