Competition problem — Geometry — Maths Olympiad Prep

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Geometry Difficulty 4.3 AIME Find the answer United States

In $\triangle ABC$ , $\angle ABC = 90^\circ$ and $BA = BC = \sqrt{2}$ . Points $P_1, P_2, \dots, P_{2024}$ lie on hypotenuse $\overline{AC}$ so that $AP_1 = P_1P_2 = P_2P_3 = \dots = P_{2023}P_{2024} = P_{2024}C$ . What is the length of the vector sum
$\overrightarrow{BP_1} + \overrightarrow{BP_2} + \overrightarrow{BP_3} + \dots + \overrightarrow{BP_{2024}}?$

Your answer

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Solution

Answer (D): For $1 \le i \le 2024$ , the vector sum $\overrightarrow{BP_i} + \overrightarrow{BP_{2025-i}}$ is the vector pointing from the apex of isosceles right triangle $\triangle ABC$ to the reflection of the apex across the hypotenuse, as seen in the figure below.

Its length is 2 times the height of the triangle, namely $2 \cdot 1 = 2$ . Each pair contributes a vector of length 2, all pointing in the same direction, to the total sum. With 1012 such pairs, the length of the resultant vector is $2 \cdot 1012 = 2024$ .

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Source: MathNet, licensed CC-BY-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.