Subcontests
(25)Magic Square
In the magic square shown, the sums of the numbers in each row, column, and diagonal are the same. Five of these numbers are represented by v, w, x, y, and z. Find y \plus{} z.
<spanclass=′latex−bold′>(A)</span> 43<spanclass=′latex−bold′>(B)</span> 44<spanclass=′latex−bold′>(C)</span> 45<spanclass=′latex−bold′>(D)</span> 46<spanclass=′latex−bold′>(E)</span> 47
[asy]unitsize(10mm);
defaultpen(linewidth(1pt));
for(int i=0; i<=3; ++i)
{
draw((0,i)--(3,i));
draw((i,0)--(i,3));
}label("25",(0.5,0.5));
label("z",(1.5,0.5));
label("21",(2.5,0.5));
label("18",(0.5,1.5));
label("x",(1.5,1.5));
label("y",(2.5,1.5));
label("v",(0.5,2.5));
label("24",(1.5,2.5));
label("w",(2.5,2.5));[/asy] Colored Grid
Consider the dark square in an array of unit squares, part of which is shown. The first ring of squares around this center square contains 8 unit squares. The second ring contains 16 unit squares. If we continue this process, the number of unit squares in the 100th ring is
<spanclass=′latex−bold′>(A)</span> 396<spanclass=′latex−bold′>(B)</span> 404<spanclass=′latex−bold′>(C)</span> 800<spanclass=′latex−bold′>(D)</span> 10,000<spanclass=′latex−bold′>(E)</span> 10,404
[asy]unitsize(3mm);defaultpen(linewidth(1pt));fill((2,2)--(2,7)--(7,7)--(7,2)--cycle, mediumgray);
fill((3,3)--(6,3)--(6,6)--(3,6)--cycle, gray);
fill((4,4)--(5,4)--(5,5)--(4,5)--cycle, black);for(real i=0; i<=9; ++i)
{
draw((i,0)--(i,9));
draw((0,i)--(9,i));
}[/asy] Lines of Symmetry
How many of the twelve pentominoes pictured below have at least one line of symmetry?<spanclass=′latex−bold′>(A)</span> 3<spanclass=′latex−bold′>(B)</span> 4<spanclass=′latex−bold′>(C)</span> 5<spanclass=′latex−bold′>(D)</span> 6<spanclass=′latex−bold′>(E)</span> 7
[asy]unitsize(5mm);
defaultpen(linewidth(1pt));draw(shift(2,0)*unitsquare);
draw(shift(2,1)*unitsquare);
draw(shift(2,2)*unitsquare);
draw(shift(1,2)*unitsquare);
draw(shift(0,2)*unitsquare);draw(shift(2,4)*unitsquare);
draw(shift(2,5)*unitsquare);
draw(shift(2,6)*unitsquare);
draw(shift(1,5)*unitsquare);
draw(shift(0,5)*unitsquare);draw(shift(4,8)*unitsquare);
draw(shift(3,8)*unitsquare);
draw(shift(2,8)*unitsquare);
draw(shift(1,8)*unitsquare);
draw(shift(0,8)*unitsquare);draw(shift(6,8)*unitsquare);
draw(shift(7,8)*unitsquare);
draw(shift(8,8)*unitsquare);
draw(shift(9,8)*unitsquare);
draw(shift(9,9)*unitsquare);
draw(shift(6,5)*unitsquare);
draw(shift(7,5)*unitsquare);
draw(shift(8,5)*unitsquare);
draw(shift(7,6)*unitsquare);
draw(shift(7,4)*unitsquare);draw(shift(6,1)*unitsquare);
draw(shift(7,1)*unitsquare);
draw(shift(8,1)*unitsquare);
draw(shift(6,0)*unitsquare);
draw(shift(7,2)*unitsquare);draw(shift(11,8)*unitsquare);
draw(shift(12,8)*unitsquare);
draw(shift(13,8)*unitsquare);
draw(shift(14,8)*unitsquare);
draw(shift(13,9)*unitsquare);draw(shift(11,5)*unitsquare);
draw(shift(12,5)*unitsquare);
draw(shift(13,5)*unitsquare);
draw(shift(11,6)*unitsquare);
draw(shift(13,4)*unitsquare);draw(shift(11,1)*unitsquare);
draw(shift(12,1)*unitsquare);
draw(shift(13,1)*unitsquare);
draw(shift(13,2)*unitsquare);
draw(shift(14,2)*unitsquare);draw(shift(16,8)*unitsquare);
draw(shift(17,8)*unitsquare);
draw(shift(18,8)*unitsquare);
draw(shift(17,9)*unitsquare);
draw(shift(18,9)*unitsquare);draw(shift(16,5)*unitsquare);
draw(shift(17,6)*unitsquare);
draw(shift(18,5)*unitsquare);
draw(shift(16,6)*unitsquare);
draw(shift(18,6)*unitsquare);
draw(shift(16,0)*unitsquare);
draw(shift(17,0)*unitsquare);
draw(shift(17,1)*unitsquare);
draw(shift(18,1)*unitsquare);
draw(shift(18,2)*unitsquare);[/asy] Congruent squares and pentagons
The plane is tiled by congruent squares and congruent pentagons as indicated. The percent of the plane that is enclosed by the pentagons is closest to
<spanclass=′latex−bold′>(A)</span> 50<spanclass=′latex−bold′>(B)</span> 52<spanclass=′latex−bold′>(C)</span> 54<spanclass=′latex−bold′>(D)</span> 56<spanclass=′latex−bold′>(E)</span> 58
[asy]unitsize(3mm);
defaultpen(linewidth(0.8pt));path p1=(0,0)--(3,0)--(3,3)--(0,3)--(0,0);
path p2=(0,1)--(1,1)--(1,0);
path p3=(2,0)--(2,1)--(3,1);
path p4=(3,2)--(2,2)--(2,3);
path p5=(1,3)--(1,2)--(0,2);
path p6=(1,1)--(2,2);
path p7=(2,1)--(1,2);
path[] p=p1^^p2^^p3^^p4^^p5^^p6^^p7;
for(int i=0; i<3; ++i)
{
for(int j=0; j<3; ++j)
{
draw(shift(3*i,3*j)*p);
}
}[/asy] Sector and cone from it
Which of the cones listed below can be formed from a 252∘ sector of a circle of radius 10 by aligning the two straight sides?
[asy]import graph;unitsize(1.5cm);defaultpen(fontsize(8pt));draw(Arc((0,0),1,-72,180),linewidth(.8pt));draw(dir(288)--(0,0)--(-1,0),linewidth(.8pt));label("10",(-0.5,0),S);draw(Arc((0,0),0.1,-72,180));label("252∘",(0.05,0.05),NE);[/asy]
[asy]
import three;
picture mainframe;
defaultpen(fontsize(11pt));
picture conePic(picture pic, real r, real h, real sh)
{
size(pic, 3cm);
triple eye = (11, 0, 5);
currentprojection = perspective(eye);
real R = 1, y = 2;
triple center = (0, 0, 0);
triple radPt = (0, R, 0);
triple negRadPt = (0, -R, 0);
triple heightPt = (0, 0, y);
draw(pic, arc(center, radPt, negRadPt, heightPt, CW));
draw(pic, arc(center, radPt, negRadPt, heightPt, CCW), linetype("8 8"));
draw(pic, center--radPt, linetype("8 8"));
draw(pic, center--heightPt, linetype("8 8"));
draw(pic, negRadPt--heightPt--radPt); label(pic, (string) r, center--radPt, dir(270));
if (h != 0)
{
label(pic, (string) h, heightPt--center, dir(0));
}
if (sh != 0)
{
label(pic, (string) sh, heightPt--radPt, dir(0));
}
return pic;
}
picture pic1;
pic1 = conePic(pic1, 6, 0, 10);
picture pic2;
pic2 = conePic(pic2, 6, 10, 0);
picture pic3;
pic3 = conePic(pic3, 7, 0, 10);
picture pic4;
pic4 = conePic(pic4, 7, 10, 0);
picture pic5;
pic5 = conePic(pic5, 8, 0, 10);picture aux1; picture aux2; picture aux3;
add(aux1, pic1.fit(), (0,0), W);
label(aux1, "<spanclass=′latex−bold′>(A)</span>", (0,0), 22W, linewidth(4));
label(aux1, "<spanclass=′latex−bold′>(B)</span>", (0,0), 3E);
add(aux1, pic2.fit(), (0,0), 35E);add(aux2, aux1.fit(), (0,0), W);
label(aux2, "<spanclass=′latex−bold′>(C)</span>", (0,0), 3E);
add(aux2, pic3.fit(), (0,0), 35E);add(aux3, aux2.fit(), (0,0), W);
label(aux3, "<spanclass=′latex−bold′>(D)</span>", (0,0), 3E);
add(aux3, pic4.fit(), (0,0), 35E);add(mainframe, aux3.fit(), (0,0), W);
label(mainframe, "<spanclass=′latex−bold′>(E)</span>", (0,0), 3E);
add(mainframe, pic5.fit(), (0,0), 35E);
add(mainframe.fit(), (0,0), N);
[/asy] Telephone number!
A telephone number has the form ABC \minus{} DEF \minus{} GHIJ, where each letter represents a different digit. The digits in each part of the numbers are in decreasing order; that is, A>B>C, D>E>F, and G>H>I>J. Furthermore, D, E, and F are consecutive even digits; G, H, I, and J are consecutive odd digits; and A \plus{} B \plus{} C \equal{} 9. Find A.
<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