2023 AMC 10A Problems/Problem 24
Contents
- 1 Problem
- 2 Solution 1
- 3 Solution 2 (Not rigorous)
- 4 Solution 2
- 5 Solution 3 (almost no words)
- 6 Solution 4 (educated guessing)
- 7 Video Solution ⚡️ 3 min solve ⚡️
- 8 Video Solution by MegaMath
- 9 Video Solution by CosineMethod [🔥Fast and Easy🔥]
- 10 Video Solution by OmegaLearn
- 11 Video Solution by epicbird08
- 12 Video Solution
- 13 Video Solution by TheBeautyofMath
- 14 See Also
Problem
Six regular hexagonal blocks of side length 1 unit are arranged inside a regular hexagonal frame. Each block lies along an inside edge of the frame and is aligned with two other blocks, as shown in the figure below. The distance from any corner of the frame to the nearest vertex of a block is unit. What is the area of the region inside the frame not occupied by the blocks?
Solution 1
Examining the red isosceles trapezoid with and as two bases, we know that the side lengths are from triangle.
We can conclude that the big hexagon has side length 3.
Thus the target area is: area of the big hexagon - 6 * area of the small hexagon.
~Technodoggo
Solution 1.1
We can extend the line of the parallelogram to the end until it touches the next hexagon and it will make a small equilateral triangle and a longer parallelogram. We can prove that one side of the tiny equilateral triangle is 4/7 by playing around with angles and the parallelogram because it is parallel, we can then use the whole side of the hexagon which is one and subtract 3/7 which is one side of the equilateral triangle which is 4/7. That means the whole side of the big hexagon length is 3 and we can continue with solution 1.
-BrandonChen
Solution 2 (Not rigorous)
Note that one can "slide' the small hexagons along their respective edges, and either by sliding them to the center or to the corners, and thus getting that the side length of the larger hexagon is 3. The rest proceeds the same as solution 1.
Solution 2.1 (Clarification)
Notice that when sliding the smaller hexagon along the edge, we see that the contact edge withe the smaller hexagon "in front" of it is , thus meaning the hexagon "in front" is pushed at a speed times the actual speed of the hexagon. We can preform a similar analysis on the hexagon that is being pushed and get that the speed at which that hexagon is moving is times the speed it pushed by. As we can see, the two factors cancel out and by the same argument, every small hexagon can move at the same speed while mantaining an edge of contact with the two adjacent hexagons.
Note
The number is irrelevant to solve the problem. In fact, if the smaller hexagons have side length , the side length of the large hexagon will always be .
Solution 2
We put the diagram to a complex plane, with the center of the outside hexagon at the origin. We denote by the length of each side of the outside hexagon.
The complex number of the upper left vertex of the upper right small hexagon is
The complex number of the upper right vertex of the top small hexagon is
The above two vertices are on the same vertical line. So their real part values are the same. By solving this equation, we get .
Therefore, the area of the region not occupied by the blocks is
~Steven Chen (Professor Chen Education Palace, www.professorchenedu.com)
Solution 3 (almost no words)
~~By afly ~(very minor change by Marshall_Huang)
Solution 4 (educated guessing)
Since the area of each hexagon is , the answer is just the area of the large hexagon minus . The side length of the large hexagon should be an integer or fraction since we are given integers and fractions, and solving this problem shouldn't require any square roots. Therefore, the correct answer plus should be the product of and a square number. Only choice satisfies this, since . Alternatively, we can also deduce is correct since it is the only fraction whose denominator is . Therefore, the answer is (probably)
-faliure167
Video Solution ⚡️ 3 min solve ⚡️
~Education, the Study of Everything
Video Solution by MegaMath
https://www.youtube.com/watch?v=Qe3eVv3aWEw&t=2s
Video Solution by CosineMethod [🔥Fast and Easy🔥]
https://www.youtube.com/watch?v=asRnZdiTYcI
Video Solution by OmegaLearn
Video Solution by epicbird08
~EpicBird08
Video Solution
~Steven Chen (Professor Chen Education Palace, www.professorchenedu.com)
Video Solution by TheBeautyofMath
~IceMatrix
See Also
2023 AMC 10A (Problems • Answer Key • Resources) | ||
Preceded by Problem 23 |
Followed by Problem 25 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | ||
All AMC 10 Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.