10
Part of 2019 MIG
Problems(2)
2019 Individual #10
Source:
5/1/2022
John defines the function . What is the value of ?
2019 Individual
2019 Team #10
Source:
5/1/2022
people, numbered through counterclockwise, sit around a circular table. They begin playing a game. Each person is initially considered "alive". Starting with person , the first person eliminates the closest "alive" person to their right (so Person eliminates Person ). Then the next "alive" person, moving counterclockwise, eliminates the closest "alive" person to their right (so since Person is eliminated, Person eliminates Person ). This process continues until there is only "alive" person remaining. What is the number of the last "alive" person?
[asy]
usepackage("cancel", "makeroom, thicklines");
usepackage("bm");
size(15cm);
picture p;
draw(p, circle((0,0), 5));
for(int i = 0; i < 4; ++i) {
label(p, "", 5 * dir(-20 * i - 100), 2 * dir(-20 * i - 100));
label(p, "", 5 * dir(20 * i - 80), 2 * dir(20 * i - 80));
}
int n = 20;
for(int i = 0; i <= n; ++i) {
label(p, scale(2)*"", 6 *dir(180 / n * i));
}
draw(p, arc((0,0), 8 * dir(-80), 8 * dir(0)), EndArrow);
add(shift(-20, 0) * p);
draw((-11, 0)--(-8,0), EndArrow);
picture q;
draw(q, circle((0,0), 5));
for(int i = 0; i < 4; ++i) {
label(q, "", 5 * dir(-20 * i - 100), 2 * dir(-20 * i - 100));
if(i != 1) label(q, "", 5 * dir(20 * i - 80), 2 * dir(20 * i - 80));
}
int n = 20;
for(int i = 0; i <= n; ++i) {
label(q, scale(2)*"", 6 *dir(180 / n * i));
}
draw(q, arc((0,0), 8 * dir(-80), 8 * dir(0)), EndArrow);
for(int i = 0; i < 1; i+=2) {
//label(q, "\bm\xcancel{~}", 5 * dir(-20 * i - 100), 2 * dir(-20 * i - 100));
label(q, "\xcancel{2}", 5 * dir(20 * (i + 1) - 80), 2 * dir(20 * (i + 1) - 80));
}
add(q);
draw((9,0)--(12,0), EndArrow);
picture r;
draw(r, circle((0,0), 5));
for(int i = 0; i < 4; ++i) {
if(i % 2 == 1) label(r, "", 5 * dir(-20 * i - 100), 2 * dir(-20 * i - 100));
if(i % 2 != 1) label(r, "", 5 * dir(20 * i - 80), 2 * dir(20 * i - 80));
}
int n = 20;
for(int i = 0; i <= n; ++i) {
label(r, scale(2)*"", 6 *dir(180 / n * i));
}
draw(r, arc((0,0), 8 * dir(-80), 8 * dir(0)), EndArrow);
for(int i = 0; i < 4; i+=2) {
label(r, "\xcancel{" + string(40 - i) +"}", 5 * dir(-20 * i - 100), 2 * dir(-20 * i - 100));
label(r, "\xcancel{" + string(i + 1) + "}", 5 * dir(20 * (i + 1) - 80), 2 * dir(20 * (i + 1) - 80));
}
add(shift(20, 0) * r);
[/asy]
In the last step here, Person eliminates Person . Next turn, Person eliminates the closest person to his right, Person .
2019 Team