19
Part of 2022 AMC 10
Problems(2)
When u take an L
Source: AMC 10A Problem 19
11/11/2022
Define as the least common multiple of all the integers from to inclusive. There is a unique integer such that . What is the remainder when is divided by
AMCAMC 10AMC 10 A2022 AMC 10a2022 AMCleast common multiple
MAA is cruel
Source: 2022 AMC 12B #18/2022 AMC 10B #19
11/17/2022
Each square in a grid is either filled or empty, and has up to eight adjacent neighboring squares, where neighboring squares share either a side or a corner. The grid is transformed by the following rules:[*] Any filled square with two or three filled neighbors remains filled.
[*] Any empty square with exactly three filled neighbors becomes a filled square.
[*] All other squares remain empty or become empty.A sample transformation is shown in the figure below.[asy]
import geometry;
unitsize(0.6cm); void ds(pair x) {
filldraw(x -- (1,0) + x -- (1,1) + x -- (0,1)+x -- cycle,gray+opacity(0.5),invisible);
} ds((1,1));
ds((2,1));
ds((3,1));
ds((1,3)); for (int i = 0; i <= 5; ++i) {
draw((0,i)--(5,i));
draw((i,0)--(i,5));
} label("Initial", (2.5,-1));
draw((6,2.5)--(8,2.5),Arrow); ds((10,2));
ds((11,1));
ds((11,0)); for (int i = 0; i <= 5; ++i) {
draw((9,i)--(14,i));
draw((i+9,0)--(i+9,5));
} label("Transformed", (11.5,-1));
[/asy]Suppose the grid has a border of empty squares surrounding a subgrid. How many initial configurations will lead to a transformed grid consisting of a single filled square in the center after a single transformation? (Rotations and reflections of the same configuration are considered different.)
[asy]
import geometry;
unitsize(0.6cm); void ds(pair x) {
filldraw(x -- (1,0) + x -- (1,1) + x -- (0,1)+x -- cycle,gray+opacity(0.5),invisible);
} for (int i = 1; i < 4; ++ i) {
for (int j = 1; j < 4; ++j) {
label("?",(i + 0.5, j + 0.5));
}
} for (int i = 0; i <= 5; ++i) {
draw((0,i)--(5,i));
draw((i,0)--(i,5));
} label("Initial", (2.5,-1));
draw((6,2.5)--(8,2.5),Arrow); ds((11,2)); for (int i = 0; i <= 5; ++i) {
draw((9,i)--(14,i));
draw((i+9,0)--(i+9,5));
} label("Transformed", (11.5,-1));
[/asy]
AMCAMC 12AMC 102022 AMC2022 AMC 10B2022 AMC 12Bcounting