Subcontests
(25)Sideburns
In the figure below, 3 of the 6 disks are to be painted blue, 2 are to be painted red, and 1 is to be painted green. Two paintings that can be obtained from one another by a rotation or a reflection of the entire figure are considered the same. How many different paintings are possible?[asy]
size(100);
pair A, B, C, D, E, F;
A = (0,0);
B = (1,0);
C = (2,0);
D = rotate(60, A)*B;
E = B + D;
F = rotate(60, A)*C;
draw(Circle(A, 0.5));
draw(Circle(B, 0.5));
draw(Circle(C, 0.5));
draw(Circle(D, 0.5));
draw(Circle(E, 0.5));
draw(Circle(F, 0.5));
[/asy]<spanclass=′latex−bold′>(A)</span>6<spanclass=′latex−bold′>(B)</span>8<spanclass=′latex−bold′>(C)</span>9<spanclass=′latex−bold′>(D)</span>12<spanclass=′latex−bold′>(E)</span>15 Tamara's garden
Tamara has three rows of two 6-feet by 2-feet flower beds in her garden. The beds are separated and also surrounded by 1-foot-wide walkways, as shown on the diagram. What is the total area of the walkways, in square feet?
[asy]
unitsize(0.7cm);
path p1 = (0,0)--(15,0)--(15,10)--(0,10)--cycle;
fill(p1,lightgray);
draw(p1);
for (int i = 1; i <= 8; i += 7) {
for (int j = 1; j <= 7; j += 3 ) {
path p2 = (i,j)--(i+6,j)--(i+6,j+2)--(i,j+2)--cycle;
draw(p2);
fill(p2,white);
}
}
draw((0,8)--(1,8),Arrows);
label("1",(0.5,8),S);
draw((7,8)--(8,8),Arrows);
label("1",(7.5,8),S);
draw((14,8)--(15,8),Arrows);
label("1",(14.5,8),S);
draw((11,0)--(11,1),Arrows);
label("1",(11,0.5),W);
draw((11,3)--(11,4),Arrows);
label("1",(11,3.5),W);
draw((11,6)--(11,7),Arrows);
label("1",(11,6.5),W);
draw((11,9)--(11,10),Arrows);
label("1",(11,9.5),W);
label("6",(4,1),N);
label("2",(1,2),E);
[/asy]
<spanclass=′latex−bold′>(A)</span>72<spanclass=′latex−bold′>(B)</span>78<spanclass=′latex−bold′>(C)</span>90<spanclass=′latex−bold′>(D)</span>120<spanclass=′latex−bold′>(E)</span>150 Fibonacci Mod Three?
Define a sequence recursively by F0=0, F1=1, and Fn= the remainder when Fn−1+Fn−2 is divided by 3, for all n≥2. Thus the sequence starts 0,1,1,2,0,2…. What is F2017+F2018+F2019+F2020+F2021+F2022+F2023+F2024?<spanclass=′latex−bold′>(A)</span> 6<spanclass=′latex−bold′>(B)</span> 7<spanclass=′latex−bold′>(C)</span> 8<spanclass=′latex−bold′>(D)</span> 9<spanclass=′latex−bold′>(E)</span> 10 Extraneous Information
The number 21!=51,090,942,171,709,440,000 has over 60,000 positive integer divisors. One of them is chosen at random. What is the probability that it is odd?<spanclass=′latex−bold′>(A)</span>211<spanclass=′latex−bold′>(B)</span>191<spanclass=′latex−bold′>(C)</span>181<spanclass=′latex−bold′>(D)</span>21<spanclass=′latex−bold′>(E)</span>2111 Easy Incircles
In △ABC, AB=6,AC=8,BC=10, and D is the midpoint of BC. What is the sum of the radii of the circles inscribed in △ADB and △ADC?<spanclass=′latex−bold′>(A)</span>5<spanclass=′latex−bold′>(B)</span>411<spanclass=′latex−bold′>(C)</span>22<spanclass=′latex−bold′>(D)</span>617<spanclass=′latex−bold′>(E)</span>3 Reflecting on....life
Points A(11,9) and B(2,−3) are vertices of △ABC with AB=AC. The altitude from A meets the opposite side at D(−1,3). What are the coordinates of point C?<spanclass=′latex−bold′>(A)</span>(−8,9)<spanclass=′latex−bold′>(B)</span>(−4,8)<spanclass=′latex−bold′>(C)</span>(−4,9)<spanclass=′latex−bold′>(D)</span>(−2,3)<spanclass=′latex−bold′>(E)</span>(−1,0)