<p>In <img alt="$\Delta ABC$" height="13" src="https://latex.artofproblemsolving.com/3/b/6/3b6d5d23a7e91fb9f6c248fb4cd43111165540f9.png" width="57" />, <img alt="$\angle ABC = 90^\circ$" height="13" src="https://latex.artofproblemsolving.com/c/b/7/cb7357b61f4675a08fba468cf3435375a034c5d0.png" width="103" /> and <img alt="$BA = BC = \sqrt{2}$" height="21" src="https://latex.artofproblemsolving.com/3/9/b/39b207baa53bbc22cebe8d56c0312dd786d93bfe.png" width="132" />. Points <img alt="$P_1, P_2, \dots, P_{2024}$" height="18" src="https://latex.artofproblemsolving.com/f/3/f/f3fcaa089b53c8f304e51b4fe0291f8c65cc78e8.png" width="128" /> lie on hypotenuse <img alt="$\overline{AC}$" height="18" src="https://latex.artofproblemsolving.com/1/2/5/12517bbc31cb9026c71299b7a0d232d07e60f284.png" width="30" /> so that <img alt="$AP_1= P_1P_2 = P_2P_3 = \dots = P_{2023}P_{2024} = P_{2024}C$" height="17" src="https://latex.artofproblemsolving.com/b/e/7/be799cc2c4eb262effbf2a6b196232643b1d4068.png" width="385" />. What is the length of the vector sum<img alt="\[\overrightarrow{BP_1} + \overrightarrow{BP_2} + \overrightarrow{BP_3} + \dots + \overrightarrow{BP_{2024}}?\]" height="25" src="https://latex.artofproblemsolving.com/d/a/a/daafe683a70517576b26bb2a80fdb220002fc49c.png" width="273" /></p>

Question No: 344

In $\Delta ABC$ , $\angle ABC = 90^\circ$ 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 $\overrightarrow{BP_1} + \overrightarrow{BP_2} + \overrightarrow{BP_3} + \dots + \overrightarrow{BP_{2024}}?$

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Tags: Algebra Geometry

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