2024 AMC 12A Problems/Problem 7
Revision as of 17:36, 8 November 2024 by Technodoggo (talk | contribs) (Created page with "In <math>\Delta ABC</math>, <math>\angle ABC = 90^\circ</math> and <math>BA = BC = \sqrt{2}</math>. Points <math>P_1, P_2, \dots, P_{2024}</math> lie on hypotenuse <math>\over...")
In 
, 
and 
. Points 
lie on hypotenuse 
so that 
. What is the length of the vector sum
![\[\overrightarrow{BP_1} + \overrightarrow{BP_2} + \overrightarrow{BP_3} + \dots + \overrightarrow{BP_{2024}}?\]](http://latex.artofproblemsolving.com/d/a/a/daafe683a70517576b26bb2a80fdb220002fc49c.png)
Solution 1 (technical vector bash)
Let us find an expression for the 
- and 
-components of 
. Note that 
, so 
. All of the vectors 
and so on up to 
are equal; moreover, they equal 
.
We now note that 
(
copies of 
added together). Furthermore, note that 
We want 
's length, which can be determined from the - and -components. Note that the two values should actually be the same - in this problem, everything is symmetric with respect to the line 
, so the magnitudes of the - and -components should be identical. The -component is easier to calculate.
![\[\sum_{i=1}^{2024}\left(\overrightarrow{BP_i}\right)_x=\sum_{i=1}^{2024}\dfrac{i\sqrt2}{2025}=\dfrac{\sqrt2}{2025}\sum_{i=1}^{2024}i=\dfrac{\sqrt2}{2025}\cdot\dfrac{2024\cdot2025}2=1012\sqrt2.\]](http://latex.artofproblemsolving.com/7/b/7/7b7a0cb72970f4bf95c85216bd0c2340b37b4f8d.png)
One can similarly evaulate the -component and obtain an identical answer; thus, our desired length is 
.
~Technodoggo
