2024 AMC 12A Problems/Problem 7

Problem

In , and . Points lie on hypotenuse so that . What is the length of the vector sum

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.

One can similarly evaulate the -component and obtain an identical answer; thus, our desired length is .

~Technodoggo

Solution 2

Notice that the average vector sum is 1. Multiplying the 2024 by 1, our answer is

~MC

Solution 3 (Pair Sum)

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Let point reflect over

We can see that for all , As a result, ~lptoggled image and edited by ~luckuso

Solution 4

Using Pythagoream theorom, we see the length of the hypotenuse is . There are 2024 points on , that are equal spaced, so there are 2025 line segments along that hypotenuse. is the length of each line segment. We get Someone please clean this up lol ~helpmebro

Solution 5 (Complex Number)

~luckuso

See also

These problems are copyrighted © by the Mathematical Association of America, as part of the American Mathematics Competitions. Error creating thumbnail: File missing