Subcontests
(25)Parallel Edges in a Cube
How many pairs of parallel edges, such as AB and GH or EH and FG, does a cube have?<spanclass=′latex−bold′>(A)</span>6<spanclass=′latex−bold′>(B)</span>12<spanclass=′latex−bold′>(C)</span>18<spanclass=′latex−bold′>(D)</span>24<spanclass=′latex−bold′>(E)</span>36[asy] import three;
currentprojection=orthographic(1/2,-1,1/2); /* three - currentprojection, orthographic */
draw((0,0,0)--(1,0,0)--(1,1,0)--(0,1,0)--cycle);
draw((0,0,0)--(0,0,1)); draw((0,1,0)--(0,1,1));
draw((1,1,0)--(1,1,1)); draw((1,0,0)--(1,0,1));
draw((0,0,1)--(1,0,1)--(1,1,1)--(0,1,1)--cycle);
label("D",(0,0,0),S);
label("A",(0,0,1),N);
label("H",(0,1,0),S);
label("E",(0,1,1),N);
label("C",(1,0,0),S);
label("B",(1,0,1),N);
label("G",(1,1,0),S);
label("F",(1,1,1),N);
[/asy] Equiangular Hexagons
In the given figure hexagon ABCDEF is equiangular, ABJI and FEHG are squares with areas 18 and 32 respectively, △JBK is equilateral and FE=BC. What is the area of △KBC?<spanclass=′latex−bold′>(A)</span>62<spanclass=′latex−bold′>(B)</span>9<spanclass=′latex−bold′>(C)</span>12<spanclass=′latex−bold′>(D)</span>92<spanclass=′latex−bold′>(E)</span>32[asy]
draw((-4,6*sqrt(2))--(4,6*sqrt(2)));
draw((-4,-6*sqrt(2))--(4,-6*sqrt(2)));
draw((-8,0)--(-4,6*sqrt(2)));
draw((-8,0)--(-4,-6*sqrt(2)));
draw((4,6*sqrt(2))--(8,0));
draw((8,0)--(4,-6*sqrt(2)));
draw((-4,6*sqrt(2))--(4,6*sqrt(2))--(4,8+6*sqrt(2))--(-4,8+6*sqrt(2))--cycle);
draw((-8,0)--(-4,-6*sqrt(2))--(-4-6*sqrt(2),-4-6*sqrt(2))--(-8-6*sqrt(2),-4)--cycle);
label("I",(-4,8+6*sqrt(2)),dir(100)); label("J",(4,8+6*sqrt(2)),dir(80));
label("A",(-4,6*sqrt(2)),dir(280)); label("B",(4,6*sqrt(2)),dir(250));
label("C",(8,0),W); label("D",(4,-6*sqrt(2)),NW); label("E",(-4,-6*sqrt(2)),NE); label("F",(-8,0),E);
draw((4,8+6*sqrt(2))--(4,6*sqrt(2))--(4+4*sqrt(3),4+6*sqrt(2))--cycle);
label("K",(4+4*sqrt(3),4+6*sqrt(2)),E);
draw((4+4*sqrt(3),4+6*sqrt(2))--(8,0),dashed);
label("H",(-4-6*sqrt(2),-4-6*sqrt(2)),S);
label("G",(-8-6*sqrt(2),-4),W);
label("32",(-10,-8),N);
label("18",(0,6*sqrt(2)+2),N);
[/asy] Shaded Octagon
Point O is the center of the regular octagon ABCDEFGH, and X is the midpoint of the side AB. What fraction of the area of the octagon is shaded?<spanclass=′latex−bold′>(A)</span>3211<spanclass=′latex−bold′>(B)</span>83<spanclass=′latex−bold′>(C)</span>3213<spanclass=′latex−bold′>(D)</span>167<spanclass=′latex−bold′>(E)</span>3215[asy]
pair A,B,C,D,E,F,G,H,O,X;
A=dir(45);
B=dir(90);
C=dir(135);
D=dir(180);
E=dir(-135);
F=dir(-90);
G=dir(-45);
H=dir(0);
O=(0,0);
X=midpoint(A--B);fill(X--B--C--D--E--O--cycle,rgb(0.75,0.75,0.75));
draw(A--B--C--D--E--F--G--H--cycle);dot("A",A,dir(45));
dot("B",B,dir(90));
dot("C",C,dir(135));
dot("D",D,dir(180));
dot("E",E,dir(-135));
dot("F",F,dir(-90));
dot("G",G,dir(-45));
dot("H",H,dir(0));
dot("X",X,dir(135/2));
dot("O",O,dir(0));
draw(E--O--X);
[/asy] X Marks the Spot
An arithmetic sequence is a sequence in which each term after the first is obtained by adding a constant to the previous term. For example, 2,5,8,11,14 is an arithmetic sequence with five terms, in which the first term is 2 and the constant added is 3. Each row and each column in this 5×5 array is an arithmetic sequence with five terms. What is the value of X?<spanclass=′latex−bold′>(A)</span>21<spanclass=′latex−bold′>(B)</span>31<spanclass=′latex−bold′>(C)</span>36<spanclass=′latex−bold′>(D)</span>40<spanclass=′latex−bold′>(E)</span>42[asy]
size(3.85cm);
label("X",(2.5,2.1),N);
for (int i=0; i<=5; ++i)
draw((i,0)--(i,5), linewidth(.5));for (int j=0; j<=5; ++j)
draw((0,j)--(5,j), linewidth(.5));
void draw_num(pair ll_corner, int num)
{
label(string(num), ll_corner + (0.5, 0.5), p = fontsize(19pt));
}draw_num((0,0), 17);
draw_num((4, 0), 81);draw_num((0, 4), 1);draw_num((4,4), 25);
void foo(int x, int y, string n)
{
label(n, (x+0.5,y+0.5), p = fontsize(19pt));
}foo(2, 4, " ");
foo(3, 4, " ");
foo(0, 3, " ");
foo(2, 3, " ");
foo(1, 2, " ");
foo(3, 2, " ");
foo(1, 1, " ");
foo(2, 1, " ");
foo(3, 1, " ");
foo(4, 1, " ");
foo(2, 0, " ");
foo(3, 0, " ");
foo(0, 1, " ");
foo(0, 2, " ");
foo(1, 0, " ");
foo(1, 3, " ");
foo(1, 4, " ");
foo(3, 3, " ");
foo(4, 2, " ");
foo(4, 3, " ");
[/asy] Inscribed square
One-inch squares are cut from the corners of this 5 inch square. What is the area in square inches of the largest square that can be fitted into the remaining space?<spanclass=′latex−bold′>(A)</span>9<spanclass=′latex−bold′>(B)</span>1221<spanclass=′latex−bold′>(C)</span>15<spanclass=′latex−bold′>(D)</span>1521<spanclass=′latex−bold′>(E)</span>17[asy]
draw((0,0)--(0,5)--(5,5)--(5,0)--cycle);
filldraw((0,4)--(1,4)--(1,5)--(0,5)--cycle, gray);
filldraw((0,0)--(1,0)--(1,1)--(0,1)--cycle, gray);
filldraw((4,0)--(4,1)--(5,1)--(5,0)--cycle, gray);
filldraw((4,4)--(4,5)--(5,5)--(5,4)--cycle, gray);
[/asy] License Plates in Mathland
In the small country of Mathland, all automobile license plates have four symbols. The first must be a vowel (A, E, I, O, or U), the second and third must be two different letters among the 21 non-vowels, and the fourth must be a digit (0 through 9). If the symbols are chosen at random subject to these conditions, what is the probability that the plate will read "AMC8"?<spanclass=′latex−bold′>(A)</span>22,0501<spanclass=′latex−bold′>(B)</span>21,0001<spanclass=′latex−bold′>(C)</span>10,5001<spanclass=′latex−bold′>(D)</span>2,1001<spanclass=′latex−bold′>(E)</span>1,0501 Basketball Scores
Billy's basketball team scored the following points over the course of the first 11 games of the season:
42,47,53,53,58,58,58,61,64,65,73
If his team scores 40 in the 12th game, which of the following statistics will show an increase?<spanclass=′latex−bold′>(A)</span>range<spanclass=′latex−bold′>(B)</span>median<spanclass=′latex−bold′>(C)</span>mean<spanclass=′latex−bold′>(D)</span>mode<spanclass=′latex−bold′>(E)</span>mid-range Sorting Slips of Paper
Tom has twelve slips of paper which he wants to put into five cups labeled A, B, C, D, E. He wants the sum of the numbers on the slips in each cup to be an integer. Furthermore, he wants the five integers to be consecutive and increasing from A to E. The numbers on the papers are 2, 2, 2, 2.5, 2.5, 3, 3, 3, 3, 3.5, 4, and 4.5. If a slip with 2 goes into cup E and a slip with 3 goes into cup B, then the slip with 3.5 must go into what cup?<spanclass=′latex−bold′>(A)</span>A<spanclass=′latex−bold′>(B)</span>B<spanclass=′latex−bold′>(C)</span>C<spanclass=′latex−bold′>(D)</span>D<spanclass=′latex−bold′>(E)</span>E Subsets and Means
How many subsets of two elements can be removed from the set {1,2,3,4,5,6,7,8,9,10,11} so that the mean (average) of the remaining numbers is 6?<spanclass=′latex−bold′>(A)</span> 1<spanclass=′latex−bold′>(B)</span> 2<spanclass=′latex−bold′>(C)</span> 3<spanclass=′latex−bold′>(D)</span> 5<spanclass=′latex−bold′>(E)</span> 6 2015 AMC 8 #19
A triangle with vertices as A=(1,3), B=(5,1), and C=(4,4) is plotted on a 6×5 grid. What fraction of the grid is covered by the triangle?<spanclass=′latex−bold′>(A)</span>61<spanclass=′latex−bold′>(B)</span>51<spanclass=′latex−bold′>(C)</span>41<spanclass=′latex−bold′>(D)</span>31<spanclass=′latex−bold′>(E)</span>21[asy]draw((1,0)--(1,5),linewidth(.5));
draw((2,0)--(2,5),linewidth(.5));
draw((3,0)--(3,5),linewidth(.5));
draw((4,0)--(4,5),linewidth(.5));
draw((5,0)--(5,5),linewidth(.5));
draw((6,0)--(6,5),linewidth(.5));
draw((0,1)--(6,1),linewidth(.5));
draw((0,2)--(6,2),linewidth(.5));
draw((0,3)--(6,3),linewidth(.5));
draw((0,4)--(6,4),linewidth(.5));
draw((0,5)--(6,5),linewidth(.5));
draw((0,0)--(0,6),EndArrow);
draw((0,0)--(7,0),EndArrow);
draw((1,3)--(4,4)--(5,1)--cycle);
label("y",(0,6),W); label("x",(7,0),S);
label("A",(1,3),dir(230)); label("B",(5,1),SE); label("C",(4,4),dir(50));
[/asy]