18
Part of 2021 AMC 10 Fall
Problems(2)
Des Moines
Source: AMC 10A #18 2022
11/11/2021
A farmer's rectangular field is partitioned into by grid of rectangular sections as shown in the figure. In each section the farmer will plant one crop: corn, wheat, soybeans, or potatoes. The farmer does not want to grow corn and wheat in any two sections that share a border, and the farmer does not want to grow soybeans and potatoes in any two sections that share a border. Given these restrictions, in how many ways can the farmer choose crops to plant in each of the four sections of the field?
[asy]
draw((0,0)--(100,0)--(100,50)--(0,50)--cycle);
draw((50,0)--(50,50));
draw((0,25)--(100,25));
[/asy]
AMCAMC 10AMC 10 A
Rotating Paper Problem
Source: 2021 AMC 12B #15, AMC 10B #18
11/17/2021
Three identical square sheets of paper each with side length are stacked on top of each other. The middle sheet is rotated clockwise about its center and the top sheet is rotated clockwise about its center, resulting in the -sided polygon shown in the figure below. The area of this polygon can be expressed in the form , where , , and are positive integers, and is not divisible by the square of any prime. What is [asy]
size(160);
defaultpen(linewidth(1.1));
path square = (1,1)--(1,-1)--(-1,-1)--(-1,1)--cycle;
filldraw(square,white);
filldraw(rotate(30)*square,white);
filldraw(rotate(60)*square,white);
dot((0,0),linewidth(7));
[/asy]
AMCAMC 12AMC 12 BAMC 10AMC 10 B