Subcontests
(25)When is the product 0?
Two different numbers are randomly selected from the set −2,−1,0,3,4,5 and multiplied together. What is the probability that the product is 0?<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 Guess-the-Length-of-the-Name Game
The following bar graph represents the length (in letters) of the names of 19 people. What is the median length of these names? <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(0.9cm); draw((-0.5,0)--(10,0), linewidth(1.5)); draw((-0.5,1)--(10,1)); draw((-0.5,2)--(10,2)); draw((-0.5,3)--(10,3)); draw((-0.5,4)--(10,4)); draw((-0.5,5)--(10,5)); draw((-0.5,6)--(10,6)); draw((-0.5,7)--(10,7)); label("frequency",(-0.5,8)); label("0", (-1, 0)); label("1", (-1, 1)); label("2", (-1, 2)); label("3", (-1, 3)); label("4", (-1, 4)); label("5", (-1, 5)); label("6", (-1, 6)); label("7", (-1, 7)); filldraw((0,0)--(0,7)--(1,7)--(1,0)--cycle, black); filldraw((2,0)--(2,3)--(3,3)--(3,0)--cycle, black); filldraw((4,0)--(4,1)--(5,1)--(5,0)--cycle, black); filldraw((6,0)--(6,4)--(7,4)--(7,0)--cycle, black); filldraw((8,0)--(8,4)--(9,4)--(9,0)--cycle, black); label("3", (0.5, -0.5)); label("4", (2.5, -0.5)); label("5", (4.5, -0.5)); label("6", (6.5, -0.5)); label("7", (8.5, -0.5)); label("name length", (4.5,-1.5)); [/asy] 911 is not allowed!
An ATM password at Fred's Bank is composed of four digits from 0 to 9, with repeated digits allowable. If no password may begin with the sequence 9,1,1, then how many passwords are possible?(A)\mbox{ }30\qquad(B)\mbox{ }7290\qquad(C)\mbox{ }9000\qquad(D)\mbox{ }9990\qquad(E)\mbox{ }9999 Batman Returns! (kind of)
Rectangle DEFA below is a 3×4 rectangle with DC=CB=BA. The area of the "bat wings" is
[asy]
size(180);
defaultpen(fontsize(11pt));
draw((0,0)--(3,0)--(3,4)--(0,4)--(0,0)--(2,4)--(3,0));
draw((3,0)--(1,4)--(0,0));
fill((0,0)--(1,4)--(1.5,3)--cycle, black);
fill((3,0)--(2,4)--(1.5,3)--cycle, black);
label("D",(0,4),NW);
label("C",(1,4),N);
label("B",(2,4),N);
label("A",(3,4),NE);
label("E",(0,0),SW);
label("F",(3,0),SE);[/asy]
<spanclass=′latex−bold′>(A)</span>2<spanclass=′latex−bold′>(B)</span>221<spanclass=′latex−bold′>(C)</span>3<spanclass=′latex−bold′>(D)</span>321<spanclass=′latex−bold′>(E)</span>5 Semicircle Inscribed in an Isosceles Triangle
A semicircle is inscribed in an isosceles triangle with base 16 and height 15 so that the diameter of the semicircle is contained in the base of the triangle as shown. What is the radius of the semicircle?[asy]
unitsize(0.25cm);
pair A, B, C, O;
A = (-8, 0);
B = (8, 0);
C = (0, 15);
O = (0, 0);draw(arc(O, 120/17, 0, 180));
draw(A--B--C--cycle);
[/asy]<spanclass=′latex−bold′>(A)</span>43<spanclass=′latex−bold′>(B)</span>17120<spanclass=′latex−bold′>(C)</span>10<spanclass=′latex−bold′>(D)</span>2172<spanclass=′latex−bold′>(E)</span>2173