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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

Graphs and maps: the drawing in which only connections matter

Königsberg in an engraving of 1652, with the river Pregel, the two islands and the seven bridges. To solve the problem of the bridges, Euler threw away almost everything this picture shows.Merian-Erben · Wikimedia Commons · public domain The city of Königsberg, in Prussia, was cut through by the river Pregel, which formed two islands, and seven bridges joined the four parts of the city. The inhabitants wondered whether it was possible to take a walk crossing each bridge exactly once. In 1736 Euler proved that it was not. The History book tells the proof in the chapter on the bridges of Königsberg. What matters here is what Euler did with the drawing.

October 19, 2026 Read

The scale of a chart: when the drawing lies

A bar chart seems to need no explanation: taller bar, bigger number. In the language of chapter 2, it is an icon: the height resembles the quantity. But behind every chart there is a convention the eye does not read: the scale. And when the convention changes, the icon goes on speaking with the same confidence, now saying something else. The cut axis A company’s sales were $96$, $98$, $101$ and $104$ in four consecutive years. Modest growth: $8.3\%$ in three years. Now draw the bars with the vertical axis starting at $95$ instead of $0$. The first bar has height $1$; the last, height $9$. The last one looks nine times larger than the first.

October 19, 2026 Read

Charts that saved lives: Playfair, Snow, Nightingale and Minard

Until the end of the eighteenth century, numbers were published in tables. A table is exact, but it forces the reader to compare row by row, holding values in their head. The data chart replaces that reading with a glance: differences, trends and peaks appear before a single number is read. Four charts show how this invention changed what could be seen. Playfair invents the line chart England’s exports and imports with Africa, 1700 to 1780, in William Playfair’s Commercial and Political Atlas (1786). The area between the two lines shows the balance of trade.William Playfair · Wikimedia Commons · public domain In 1786 the Scotsman William Playfair, an engineer and pamphleteer with an eventful life, published The Commercial and Political Atlas, with 44 charts on English trade. There, for the first time in a systematic way, appear the line chart with time on the horizontal axis and the bar chart. In 1801 he added the pie chart. Playfair justified the drawings by saying that the eye grasps the shape of a series of numbers far better than memory holds the numbers.

October 19, 2026 Read

From Oresme to the graph: drawing a quality that varies

A graph of speed against time looks like the most natural thing in the world. But think about what it does: it turns time into horizontal length and speed into vertical length, two quantities that are not lengths. It is a bold translation, and someone had to invent it. Oresme’s latitudes Around 1350, the Frenchman Nicole Oresme, a master in Paris and later bishop of Lisieux, wrote a treatise on the configurations of qualities. A quality, such as the heat of a bar or the speed of a body, can vary from one point to another or from one instant to another. Oresme proposed drawing it:

October 19, 2026 Read