AMC 8 Daily Practice Round 6

Complete problem set with solutions and individual problem pages

Problem 19 Easy

As shown in the diagram, a regular hexagon ABCDEF (with all sides equal) is positioned on a number line. Points E and F correspond to numbers-3 and -1, respectively. When the hexagon is rotated clockwise around a vertex, after 1 rotation, point A aligns with the number 1. Continuing this rotation pattern, identify which vertex of the hexagon corresponds to the number 4041 on the number line.

  • A.

    E

  • B.

    D

  • C.

    C

  • D.

    B

  • E.

    A

Answer:A

From the first rotation, point A moves to 1, indicating the side length of the hexagon is 2 (distance between -1 and 1).

Observing the rotation pattern:

- After 1 rotation: Point A → 1

- After 2 rotations: Point B → 3

- After 3 rotations: Point C → 5

- After 4 rotations: Point D → 7

- After 5 rotations: Point E → 9

- After 6 rotations: Point F → 11

- After 7 rotations: Point A → 13

This establishes a periodic pattern with period 6.

The general formula for the position after n rotations is:

\text{Position} = -1 + 2n

Set -1 + 2n = 4041\Rightarrow \quad2n = 4042 \quad \Rightarrow \quad n = 2021

Determine the position within the 6-rotation cycle:   2021 \div 6 = 336 \text{ remainder } 5

A remainder of 5 corresponds to the 5th vertex in the cycle (E).

Final result: \boxed{E}

Related Problems
Problem 17 AMC 8 Daily Practice Round 6
Problem 18 AMC 8 Daily Practice Round 6
Problem 20 AMC 8 Daily Practice Round 6
Problem 21 AMC 8 Daily Practice Round 6
Think Academy Powered by ChatGPT