Subcontests
(18)Misplaced Complex Numbers
How many integers n≥2 are there such that whenever z1,z2,...,zn are complex numbers such that
∣z1∣=∣z2∣=...=∣zn∣=1 and z1+z2+...+zn=0,
then the numbers z1,z2,...,zn are equally spaced on the unit circle in the complex plane?<spanclass=′latex−bold′>(A)</span> 1<spanclass=′latex−bold′>(B)</span> 2<spanclass=′latex−bold′>(C)</span> 3<spanclass=′latex−bold′>(D)</span> 4<spanclass=′latex−bold′>(E)</span> 5 AMC Acronym
The acronym AMC is shown in the rectangular grid below with grid lines spaced 1 unit apart. In units, what is the sum of the lengths of the line segments that form the acronym AMC?[asy]
import olympiad;
unitsize(25);
for (int i = 0; i < 3; ++i) {
for (int j = 0; j < 9; ++j) {
pair A = (j,i);}
}
for (int i = 0; i < 3; ++i) {
for (int j = 0; j < 9; ++j) {
if (j != 8) {
draw((j,i)--(j+1,i), gray(0.6)+dashed);
}
if (i != 2) {
draw((j,i)--(j,i+1), gray(0.6)+dashed);
}
}
}
draw((0,0)--(2,2),linewidth(2));
draw((2,0)--(2,2),linewidth(2));
draw((1,1)--(2,1),linewidth(2));
draw((3,0)--(3,2),linewidth(2));
draw((5,0)--(5,2),linewidth(2));
draw((4,1)--(3,2),linewidth(2));
draw((4,1)--(5,2),linewidth(2));
draw((6,0)--(8,0),linewidth(2));
draw((6,2)--(8,2),linewidth(2));
draw((6,0)--(6,2),linewidth(2));
[/asy]<spanclass=′latex−bold′>(A)</span>17<spanclass=′latex−bold′>(B)</span>15+22<spanclass=′latex−bold′>(C)</span>13+42<spanclass=′latex−bold′>(D)</span>11+62<spanclass=′latex−bold′>(E)</span>21 Symmetric Grid
In the plane figure shown below, 3 of the unit squares have been shaded. What is the least number of additional unit squares that must be shaded so that the resulting figure has two lines of symmetry?[asy]
import olympiad;
unitsize(25);
filldraw((1,3)--(1,4)--(2,4)--(2,3)--cycle, gray(0.7));
filldraw((2,1)--(2,2)--(3,2)--(3,1)--cycle, gray(0.7));
filldraw((4,0)--(5,0)--(5,1)--(4,1)--cycle, gray(0.7));
for (int i = 0; i < 5; ++i) {
for (int j = 0; j < 6; ++j) {
pair A = (j,i);}
}
for (int i = 0; i < 5; ++i) {
for (int j = 0; j < 6; ++j) {
if (j != 5) {
draw((j,i)--(j+1,i));
}
if (i != 4) {
draw((j,i)--(j,i+1));
}
}
}
[/asy]<spanclass=′latex−bold′>(A)</span>4<spanclass=′latex−bold′>(B)</span>5<spanclass=′latex−bold′>(C)</span>6<spanclass=′latex−bold′>(D)</span>7<spanclass=′latex−bold′>(E)</span>8