MathDB

2012 AMC 10

Part of AMC 10

Subcontests

(25)
18
2

Area of a Closed Curve

The closed curve in the figure is made up of 99 congruent circular arcs each of length 2π3\frac{2\pi}{3}, where each of the centers of the corresponding circles is among the vertices of a regular hexagon of side 22. What is the area enclosed by the curve?
[asy] size(170); defaultpen(fontsize(6pt)); dotfactor=4; label("∘\circ",(0,1)); label("∘\circ",(0.865,0.5)); label("∘\circ",(-0.865,0.5)); label("∘\circ",(0.865,-0.5)); label("∘\circ",(-0.865,-0.5)); label("∘\circ",(0,-1)); dot((0,1.5)); dot((-0.4325,0.75)); dot((0.4325,0.75)); dot((-0.4325,-0.75)); dot((0.4325,-0.75)); dot((-0.865,0)); dot((0.865,0)); dot((-1.2975,-0.75)); dot((1.2975,-0.75)); draw(Arc((0,1),0.5,210,-30)); draw(Arc((0.865,0.5),0.5,150,270)); draw(Arc((0.865,-0.5),0.5,90,-150)); draw(Arc((0.865,-0.5),0.5,90,-150)); draw(Arc((0,-1),0.5,30,150)); draw(Arc((-0.865,-0.5),0.5,330,90)); draw(Arc((-0.865,0.5),0.5,-90,30)); [/asy]
<spanclass=′latex−bold′>(A)</span> 2π+6<spanclass=′latex−bold′>(B)</span> 2π+43<spanclass=′latex−bold′>(C)</span> 3π+4<spanclass=′latex−bold′>(D)</span> 2π+33+2<spanclass=′latex−bold′>(E)</span> π+63 <span class='latex-bold'>(A)</span>\ 2\pi+6\qquad<span class='latex-bold'>(B)</span>\ 2\pi+4\sqrt3 \qquad<span class='latex-bold'>(C)</span>\ 3\pi+4 \qquad<span class='latex-bold'>(D)</span>\ 2\pi+3\sqrt3+2 \qquad<span class='latex-bold'>(E)</span>\ \pi+6\sqrt3

Disease and Blood Test

Suppose that one of every 500500 people in a certain population has a particular disease, which displays no symptoms. A blood test is available for screening for this disease. For a person who has this disease, the test always turns out positive. For a person who does not have the disease, however, there is a 2%2\% false positive rate; in other words, for such people, 98%98\% of the time the test will turn out negative, but 2%2\% of the time the test will turn out positive and will incorrectly indicate that the person has the disease. Let pp be the probability that a person who is chosen at random from the population and gets a positive test result actually has the disease. Which of the following is closest to pp?
<spanclass=′latex−bold′>(A)</span> 198<spanclass=′latex−bold′>(B)</span> 19<spanclass=′latex−bold′>(C)</span> 111<spanclass=′latex−bold′>(D)</span> 4999<spanclass=′latex−bold′>(E)</span> 9899 <span class='latex-bold'>(A)</span>\ \frac{1}{98}\qquad<span class='latex-bold'>(B)</span>\ \frac{1}{9}\qquad<span class='latex-bold'>(C)</span>\ \frac{1}{11}\qquad<span class='latex-bold'>(D)</span>\ \frac{49}{99}\qquad<span class='latex-bold'>(E)</span>\ \frac{98}{99}
15
2

Three Unit Squares

Three unit squares and two line segments connecting two pairs of vertices are shown. What is the area of △ABC\triangle ABC?
[asy] size(200); defaultpen(linewidth(.6pt)+fontsize(12pt)); dotfactor=4; draw((0,0)--(0,2)); draw((0,0)--(1,0)); draw((1,0)--(1,2)); draw((0,1)--(2,1)); draw((0,0)--(1,2)); draw((0,2)--(2,1)); draw((0,2)--(2,2)); draw((2,1)--(2,2)); label("AA",(0,2),NW); label("BB",(1,2),N); label("CC",(4/5,1.55),W); dot((0,2)); dot((1,2)); dot((4/5,1.6)); dot((2,1)); dot((0,0)); [/asy]
<spanclass=′latex−bold′>(A)</span> 16<spanclass=′latex−bold′>(B)</span> 15<spanclass=′latex−bold′>(C)</span> 29<spanclass=′latex−bold′>(D)</span> 13<spanclass=′latex−bold′>(E)</span> 24 <span class='latex-bold'>(A)</span>\ \frac{1}{6}\qquad<span class='latex-bold'>(B)</span>\ \frac{1}{5}\qquad<span class='latex-bold'>(C)</span>\ \frac{2}{9}\qquad<span class='latex-bold'>(D)</span>\ \frac{1}{3}\qquad<span class='latex-bold'>(E)</span>\ \frac{\sqrt2}{4}

Round-Robin Tournament

In a round-robin tournament with 66 teams, each team plays one game against each other team, and each game results in one team winning and one team losing. At the end of the tournament, the teams are ranked by the number of games won. What is the maximum number of teams that could be tied for the most wins at the end of the tournament?
<spanclass=′latex−bold′>(A)</span> 2<spanclass=′latex−bold′>(B)</span> 3<spanclass=′latex−bold′>(C)</span> 4<spanclass=′latex−bold′>(D)</span> 5<spanclass=′latex−bold′>(E)</span> 6 <span class='latex-bold'>(A)</span>\ 2\qquad<span class='latex-bold'>(B)</span>\ 3\qquad<span class='latex-bold'>(C)</span>\ 4\qquad<span class='latex-bold'>(D)</span>\ 5\qquad<span class='latex-bold'>(E)</span>\ 6
25
2

Three Real Numbers Chosen From [0,n]

Real numbers x,yx,y, and zz are chosen independently and at random from the interval [0,n][0,n] for some positive integer nn. The probability that no two of x,yx,y, and zz are within 11 unit of each other is greater than 12\tfrac{1}{2}. What is the smallest possible value of nn?
<spanclass=′latex−bold′>(A)</span> 7<spanclass=′latex−bold′>(B)</span> 8<spanclass=′latex−bold′>(C)</span> 9<spanclass=′latex−bold′>(D)</span> 10<spanclass=′latex−bold′>(E)</span> 11 <span class='latex-bold'>(A)</span>\ 7 \qquad<span class='latex-bold'>(B)</span>\ 8 \qquad<span class='latex-bold'>(C)</span>\ 9 \qquad<span class='latex-bold'>(D)</span>\ 10 \qquad<span class='latex-bold'>(E)</span>\ 11

A bug on a hexagonal lattice

A bug travels from AA to BB along the segments in the hexagonal lattice pictured below. The segments marked with an arrow can be traveled only in the direction of the arrow, and the bug never travels the same segment more than once. How many different paths are there?
[asy] size(10cm); draw((0.0,0.0)--(1.0,1.7320508075688772)--(3.0,1.7320508075688772)--(4.0,3.4641016151377544)--(6.0,3.4641016151377544)--(7.0,5.196152422706632)--(9.0,5.196152422706632)--(10.0,6.928203230275509)--(12.0,6.928203230275509)); draw((0.0,0.0)--(1.0,1.7320508075688772)--(3.0,1.7320508075688772)--(4.0,3.4641016151377544)--(6.0,3.4641016151377544)--(7.0,5.196152422706632)--(9.0,5.196152422706632)--(10.0,6.928203230275509)--(12.0,6.928203230275509)); draw((3.0,-1.7320508075688772)--(4.0,0.0)--(6.0,0.0)--(7.0,1.7320508075688772)--(9.0,1.7320508075688772)--(10.0,3.4641016151377544)--(12.0,3.464101615137755)--(13.0,5.196152422706632)--(15.0,5.196152422706632)); draw((6.0,-3.4641016151377544)--(7.0,-1.7320508075688772)--(9.0,-1.7320508075688772)--(10.0,0.0)--(12.0,0.0)--(13.0,1.7320508075688772)--(15.0,1.7320508075688776)--(16.0,3.464101615137755)--(18.0,3.4641016151377544)); draw((9.0,-5.196152422706632)--(10.0,-3.464101615137755)--(12.0,-3.464101615137755)--(13.0,-1.7320508075688776)--(15.0,-1.7320508075688776)--(16.0,0)--(18.0,0.0)--(19.0,1.7320508075688772)--(21.0,1.7320508075688767)); draw((12.0,-6.928203230275509)--(13.0,-5.196152422706632)--(15.0,-5.196152422706632)--(16.0,-3.464101615137755)--(18.0,-3.4641016151377544)--(19.0,-1.7320508075688772)--(21.0,-1.7320508075688767)--(22.0,0)); draw((0.0,-0.0)--(1.0,-1.7320508075688772)--(3.0,-1.7320508075688772)--(4.0,-3.4641016151377544)--(6.0,-3.4641016151377544)--(7.0,-5.196152422706632)--(9.0,-5.196152422706632)--(10.0,-6.928203230275509)--(12.0,-6.928203230275509)); draw((3.0,1.7320508075688772)--(4.0,-0.0)--(6.0,-0.0)--(7.0,-1.7320508075688772)--(9.0,-1.7320508075688772)--(10.0,-3.4641016151377544)--(12.0,-3.464101615137755)--(13.0,-5.196152422706632)--(15.0,-5.196152422706632)); draw((6.0,3.4641016151377544)--(7.0,1.7320508075688772)--(9.0,1.7320508075688772)--(10.0,-0.0)--(12.0,-0.0)--(13.0,-1.7320508075688772)--(15.0,-1.7320508075688776)--(16.0,-3.464101615137755)--(18.0,-3.4641016151377544)); draw((9.0,5.1961524)--(10.0,3.464101)--(12.0,3.46410)--(13.0,1.73205)--(15.0,1.732050)--(16.0,0)--(18.0,-0.0)--(19.0,-1.7320)--(21.0,-1.73205080)); draw((12.0,6.928203)--(13.0,5.1961524)--(15.0,5.1961524)--(16.0,3.464101615)--(18.0,3.4641016)--(19.0,1.7320508)--(21.0,1.732050)--(22.0,0)); dot((0,0)); dot((22,0)); label("AA",(0,0),WNW); label("BB",(22,0),E); filldraw((2.0,1.7320508075688772)--(1.6,1.2320508075688772)--(1.75,1.7320508075688772)--(1.6,2.232050807568877)--cycle,black); filldraw((5.0,3.4641016151377544)--(4.6,2.9641016151377544)--(4.75,3.4641016151377544)--(4.6,3.9641016151377544)--cycle,black); filldraw((8.0,5.196152422706632)--(7.6,4.696152422706632)--(7.75,5.196152422706632)--(7.6,5.696152422706632)--cycle,black); filldraw((11.0,6.928203230275509)--(10.6,6.428203230275509)--(10.75,6.928203230275509)--(10.6,7.428203230275509)--cycle,black); filldraw((4.6,0.0)--(5.0,-0.5)--(4.85,0.0)--(5.0,0.5)--cycle,white); filldraw((8.0,1.732050)--(7.6,1.2320)--(7.75,1.73205)--(7.6,2.2320)--cycle,black); filldraw((11.0,3.4641016)--(10.6,2.9641016)--(10.75,3.46410161)--(10.6,3.964101)--cycle,black); filldraw((14.0,5.196152422706632)--(13.6,4.696152422706632)--(13.75,5.196152422706632)--(13.6,5.696152422706632)--cycle,black); filldraw((8.0,-1.732050)--(7.6,-2.232050)--(7.75,-1.7320508)--(7.6,-1.2320)--cycle,black); filldraw((10.6,0.0)--(11,-0.5)--(10.85,0.0)--(11,0.5)--cycle,white); filldraw((14.0,1.7320508075688772)--(13.6,1.2320508075688772)--(13.75,1.7320508075688772)--(13.6,2.232050807568877)--cycle,black); filldraw((17.0,3.464101615137755)--(16.6,2.964101615137755)--(16.75,3.464101615137755)--(16.6,3.964101615137755)--cycle,black); filldraw((11.0,-3.464101615137755)--(10.6,-3.964101615137755)--(10.75,-3.464101615137755)--(10.6,-2.964101615137755)--cycle,black); filldraw((14.0,-1.7320508075688776)--(13.6,-2.2320508075688776)--(13.75,-1.7320508075688776)--(13.6,-1.2320508075688776)--cycle,black); filldraw((16.6,0)--(17,-0.5)--(16.85,0)--(17,0.5)--cycle,white); filldraw((20.0,1.7320508075688772)--(19.6,1.2320508075688772)--(19.75,1.7320508075688772)--(19.6,2.232050807568877)--cycle,black); filldraw((14.0,-5.196152422706632)--(13.6,-5.696152422706632)--(13.75,-5.196152422706632)--(13.6,-4.696152422706632)--cycle,black); filldraw((17.0,-3.464101615137755)--(16.6,-3.964101615137755)--(16.75,-3.464101615137755)--(16.6,-2.964101615137755)--cycle,black); filldraw((20.0,-1.7320508075688772)--(19.6,-2.232050807568877)--(19.75,-1.7320508075688772)--(19.6,-1.2320508075688772)--cycle,black); filldraw((2.0,-1.7320508075688772)--(1.6,-1.2320508075688772)--(1.75,-1.7320508075688772)--(1.6,-2.232050807568877)--cycle,black); filldraw((5.0,-3.4641016)--(4.6,-2.964101)--(4.75,-3.4641)--(4.6,-3.9641016)--cycle,black); filldraw((8.0,-5.1961524)--(7.6,-4.6961524)--(7.75,-5.19615242)--(7.6,-5.696152422)--cycle,black); filldraw((11.0,-6.9282032)--(10.6,-6.4282032)--(10.75,-6.928203)--(10.6,-7.428203)--cycle,black);[/asy]
<spanclass=′latex−bold′>(A)</span> 2112<spanclass=′latex−bold′>(B)</span> 2304<spanclass=′latex−bold′>(C)</span> 2368<spanclass=′latex−bold′>(D)</span> 2384<spanclass=′latex−bold′>(E)</span> 2400 <span class='latex-bold'>(A)</span>\ 2112\qquad<span class='latex-bold'>(B)</span>\ 2304\qquad<span class='latex-bold'>(C)</span>\ 2368\qquad<span class='latex-bold'>(D)</span>\ 2384\qquad<span class='latex-bold'>(E)</span>\ 2400
24
2

Three Positive Integers, Two Equations

Let a,b,a,b, and cc be positive integers with a≥b≥ca\ge b\ge c such that \begin{align*} a^2-b^2-c^2+ab&=2011\text{ and}\\ a^2+3b^2+3c^2-3ab-2ac-2bc&=-1997\end{align*} What is aa?
<spanclass=′latex−bold′>(A)</span> 249<spanclass=′latex−bold′>(B)</span> 250<spanclass=′latex−bold′>(C)</span> 251<spanclass=′latex−bold′>(D)</span> 252<spanclass=′latex−bold′>(E)</span> 253 <span class='latex-bold'>(A)</span>\ 249 \qquad<span class='latex-bold'>(B)</span>\ 250 \qquad<span class='latex-bold'>(C)</span>\ 251 \qquad<span class='latex-bold'>(D)</span>\ 252 \qquad<span class='latex-bold'>(E)</span>\ 253

Listening to Songs

Amy, Beth, and Jo listen to four different songs and discuss which ones they like. No song is liked by all three. Furthermore, for each of the three pairs of the girls, there is at least one song liked by those two girls but disliked by the third. In how many different ways is this possible?
<spanclass=′latex−bold′>(A)</span> 108<spanclass=′latex−bold′>(B)</span> 132<spanclass=′latex−bold′>(C)</span> 671<spanclass=′latex−bold′>(D)</span> 846<spanclass=′latex−bold′>(E)</span> 1105 <span class='latex-bold'>(A)</span>\ 108\qquad<span class='latex-bold'>(B)</span>\ 132\qquad<span class='latex-bold'>(C)</span>\ 671\qquad<span class='latex-bold'>(D)</span>\ 846\qquad<span class='latex-bold'>(E)</span>\ 1105
23
2

Internet Friends

Adam, Benin, Chiang, Deshawn, Esther, and Fiona have internet accounts. Some, but not all, of them are internet friends with each other, and none of them has an internet friend outside this group. Each of them has the same number of internet friends. In how many different ways can this happen?
<spanclass=′latex−bold′>(A)</span> 60<spanclass=′latex−bold′>(B)</span> 170<spanclass=′latex−bold′>(C)</span> 290<spanclass=′latex−bold′>(D)</span> 320<spanclass=′latex−bold′>(E)</span> 660 <span class='latex-bold'>(A)</span>\ 60 \qquad<span class='latex-bold'>(B)</span>\ 170 \qquad<span class='latex-bold'>(C)</span>\ 290 \qquad<span class='latex-bold'>(D)</span>\ 320 \qquad<span class='latex-bold'>(E)</span>\ 660

Slicing a Cube

A solid tetrahedron is sliced off a solid wooden unit cube by a plane passing through two nonadjacent vertices on one face and one vertex on the opposite face not adjacent to either of the first two vertices. The tetrahedron is discarded and the remaining portion of the cube is placed on a table with the cut surface face down. What is the height of this object?
<spanclass=′latex−bold′>(A)</span> 33<spanclass=′latex−bold′>(B)</span> 223<spanclass=′latex−bold′>(C)</span> 1<spanclass=′latex−bold′>(D)</span> 233<spanclass=′latex−bold′>(E)</span> 2 <span class='latex-bold'>(A)</span>\ \dfrac{\sqrt{3}}{3}\qquad<span class='latex-bold'>(B)</span>\ \dfrac{2\sqrt{2}}{3}\qquad<span class='latex-bold'>(C)</span>\ 1\qquad<span class='latex-bold'>(D)</span>\ \dfrac{2\sqrt{3}}{3}\qquad<span class='latex-bold'>(E)</span>\ \sqrt{2}
22
2

Sum of Odd and Even Positive Integers

The sum of the first mm positive odd integers is 212212 more than the sum of the first nn positive even integers. What is the sum of all possible values of nn?
<spanclass=′latex−bold′>(A)</span> 255<spanclass=′latex−bold′>(B)</span> 256<spanclass=′latex−bold′>(C)</span> 257<spanclass=′latex−bold′>(D)</span> 258<spanclass=′latex−bold′>(E)</span> 259 <span class='latex-bold'>(A)</span>\ 255 \qquad<span class='latex-bold'>(B)</span>\ 256 \qquad<span class='latex-bold'>(C)</span>\ 257 \qquad<span class='latex-bold'>(D)</span>\ 258 \qquad<span class='latex-bold'>(E)</span>\ 259

Ways to List Integers

Let (a1,a2,…,a10)(a_1,a_2, \dots ,a_{10}) be a list of the first 1010 positive integers such that for each 2≤i≤102 \le i \le 10 either ai+1a_i+1 or ai−1a_i-1 or both appear somewhere before aia_i in the list. How many such lists are there?
<spanclass=′latex−bold′>(A)</span> 120<spanclass=′latex−bold′>(B)</span> 512<spanclass=′latex−bold′>(C)</span> 1024<spanclass=′latex−bold′>(D)</span> 181,440<spanclass=′latex−bold′>(E)</span> 362,880 <span class='latex-bold'>(A)</span>\ 120\qquad<span class='latex-bold'>(B)</span>\ 512\qquad<span class='latex-bold'>(C)</span>\ 1024\qquad<span class='latex-bold'>(D)</span>\ 181,440\qquad<span class='latex-bold'>(E)</span>\ 362,880
21
2

3D Geometry Area

Let points A=(0,0,0)A=(0,0,0), B=(1,0,0)B=(1,0,0), C=(0,2,0)C=(0,2,0), and D=(0,0,3)D=(0,0,3). Points E,F,GE,F,G, and HH are midpoints of line segments BD‾,AB‾,AC‾\overline{BD},\overline{AB},\overline{AC}, and DC‾\overline{DC} respectively. What is the area of EFGHEFGH?
<spanclass=′latex−bold′>(A)</span> 2<spanclass=′latex−bold′>(B)</span> 253<spanclass=′latex−bold′>(C)</span> 354<spanclass=′latex−bold′>(D)</span> 3<spanclass=′latex−bold′>(E)</span> 273 <span class='latex-bold'>(A)</span>\ \sqrt2 \qquad<span class='latex-bold'>(B)</span>\ \frac{2\sqrt5}{3} \qquad<span class='latex-bold'>(C)</span>\ \frac{3\sqrt5}{4} \qquad<span class='latex-bold'>(D)</span>\ \sqrt3 \qquad<span class='latex-bold'>(E)</span>\ \frac{2\sqrt7}{3}

Four Points in a Plane

Four distinct points are arranged in a plane so that the segments connecting them has lengths a,a,a,a,2a,a,a,a,a,2a, and bb. What is the ratio of bb to aa?
<spanclass=′latex−bold′>(A)</span> 3<spanclass=′latex−bold′>(B)</span> 2<spanclass=′latex−bold′>(C)</span> 5<spanclass=′latex−bold′>(D)</span> 3<spanclass=′latex−bold′>(E)</span> π <span class='latex-bold'>(A)</span>\ \sqrt{3}\qquad<span class='latex-bold'>(B)</span>\ 2\qquad<span class='latex-bold'>(C)</span>\ \sqrt{5}\qquad<span class='latex-bold'>(D)</span>\ 3\qquad<span class='latex-bold'>(E)</span>\ \pi
20
2

Rotating a Partitioned Square

A 3×33\times3 square is partitioned into 99 unit squares. Each unit square is painted either white or black with each color being equally likely, chosen independently and at random. The square is the rotated 90∘90^\circ clockwise about its center, and every white square in a position formerly occupied by a black square is painted black. The colors of all other squares are left unchanged. What is the probability that the grid is now entirely black?
<spanclass=′latex−bold′>(A)</span> 49512<spanclass=′latex−bold′>(B)</span> 764<spanclass=′latex−bold′>(C)</span> 1211024<spanclass=′latex−bold′>(D)</span> 81512<spanclass=′latex−bold′>(E)</span> 932 <span class='latex-bold'>(A)</span>\ \dfrac{49}{512} \qquad<span class='latex-bold'>(B)</span>\ \dfrac{7}{64} \qquad<span class='latex-bold'>(C)</span>\ \dfrac{121}{1024} \qquad<span class='latex-bold'>(D)</span>\ \dfrac{81}{512} \qquad<span class='latex-bold'>(E)</span>\ \dfrac{9}{32}

Bernado and Silvia

Bernado and Silvia play the following game. An integer between 0 and 999, inclusive, is selected and given to Bernado. Whenever Bernado receives a number, he doubles it and passes the result to Silvia. Whenever Silvia receives a number, she adds 50 to it and passes the result to Bernado. The winner is the last person who produces a number less than 1000. Let NN be the smallest initial number that results in a win for Bernado. What is the sum of the digits of NN?
<spanclass=′latex−bold′>(A)</span> 7<spanclass=′latex−bold′>(B)</span> 8<spanclass=′latex−bold′>(C)</span> 9<spanclass=′latex−bold′>(D)</span> 10<spanclass=′latex−bold′>(E)</span> 11<span class='latex-bold'>(A)</span>\ 7 \qquad<span class='latex-bold'>(B)</span>\ 8 \qquad<span class='latex-bold'>(C)</span>\ 9 \qquad<span class='latex-bold'>(D)</span>\ 10 \qquad<span class='latex-bold'>(E)</span>\ 11
19
2

How Long is the Lunch Break?

Paula the painter and her two helpers each paint at constant, but different, rates. They always start at 8:00 AM\text{8:00 AM}, and all three always take the same amount of time to eat lunch. On Monday the three of them painted 50%50\% of a house, quitting at 4:00 PM\text{4:00 PM}. On Tuesday, when Paula wasn't there, the two helpers painted only 24%24\% of the house and quit at 2:12 PM\text{2:12 PM}. On Wednesday Paula worked by herself and finished the house by working until 7:12 PM\text{7:12 PM}. How long, in minutes, was each day's lunch break?
<spanclass=′latex−bold′>(A)</span> 30<spanclass=′latex−bold′>(B)</span> 36<spanclass=′latex−bold′>(C)</span> 42<spanclass=′latex−bold′>(D)</span> 48<spanclass=′latex−bold′>(E)</span> 60 <span class='latex-bold'>(A)</span>\ 30 \qquad<span class='latex-bold'>(B)</span>\ 36 \qquad<span class='latex-bold'>(C)</span>\ 42 \qquad<span class='latex-bold'>(D)</span>\ 48 \qquad<span class='latex-bold'>(E)</span>\ 60

Area of Quadrilateral in Rectangle

In rectangle ABCDABCD, AB=6AB=6, AD=30AD=30, and GG is the midpoint of AD‾\overline{AD}. Segment ABAB is extended 22 units beyond BB to point EE, and FF is the intersection of ED‾\overline{ED} and BC‾\overline{BC}. What is the area of BFDGBFDG?
<spanclass=′latex−bold′>(A)</span> 1332<spanclass=′latex−bold′>(B)</span> 67<spanclass=′latex−bold′>(C)</span> 1352<spanclass=′latex−bold′>(D)</span> 68<spanclass=′latex−bold′>(E)</span> 1372 <span class='latex-bold'>(A)</span>\ \frac{133}{2}\qquad<span class='latex-bold'>(B)</span>\ 67\qquad<span class='latex-bold'>(C)</span>\ \frac{135}{2}\qquad<span class='latex-bold'>(D)</span>\ 68\qquad<span class='latex-bold'>(E)</span>\ \frac{137}{2}
17
2

Two Relatively Prime Integers

Let aa and bb be relatively prime integers with a>b>0a>b>0 and a3−b3(a−b)3=733\tfrac{a^3-b^3}{(a-b)^3}=\tfrac{73}{3}. What is a−ba-b?
<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 <span class='latex-bold'>(A)</span>\ 1 \qquad<span class='latex-bold'>(B)</span>\ 2 \qquad<span class='latex-bold'>(C)</span>\ 3 \qquad<span class='latex-bold'>(D)</span>\ 4 \qquad<span class='latex-bold'>(E)</span>\ 5

Paper Disk =&gt; Cones

Jesse cuts a circular paper disk of radius 1212 along two radii to form two sectors, the smaller having a central angle of 120120 degrees. He makes two circular cones, using each sector to form the lateral surface of a cone. What is the ratio of the volume of the smaller cone to that of the larger?
<spanclass=′latex−bold′>(A)</span> 18<spanclass=′latex−bold′>(B)</span> 14<spanclass=′latex−bold′>(C)</span> 1010<spanclass=′latex−bold′>(D)</span> 56<spanclass=′latex−bold′>(E)</span> 105 <span class='latex-bold'>(A)</span>\ \frac{1}{8} \qquad<span class='latex-bold'>(B)</span>\ \frac{1}{4} \qquad<span class='latex-bold'>(C)</span>\ \frac{\sqrt{10}}{10} \qquad<span class='latex-bold'>(D)</span>\ \frac{\sqrt{5}}{6} \qquad<span class='latex-bold'>(E)</span>\ \frac{\sqrt{10}}{5}
16
2

Three Runners Stopping

Three runners start running simultaneously from the same point on a 500500-meter circular track. They each run clockwise around the course maintaining constant speeds of 4.44.4, 4.84.8, and 5.05.0 meters per second. The runners stop once they are all together again somewhere on the circular course. How many seconds do the runners run?
<spanclass=′latex−bold′>(A)</span> 1,000<spanclass=′latex−bold′>(B)</span> 1,250<spanclass=′latex−bold′>(C)</span> 2,500<spanclass=′latex−bold′>(D)</span> 5,000<spanclass=′latex−bold′>(E)</span> 10,000 <span class='latex-bold'>(A)</span>\ 1,000 \qquad<span class='latex-bold'>(B)</span>\ 1,250 \qquad<span class='latex-bold'>(C)</span>\ 2,500 \qquad<span class='latex-bold'>(D)</span>\ 5,000 \qquad<span class='latex-bold'>(E)</span>\ 10,000

Region Bounded by Circles Area

Three circles with radius 22 are mutually tangent. What is the total area of the circles and the region bounded by them, as shown in the figure?
[asy] filldraw((0,0)--(2,0)--(1,sqrt(3))--cycle,gray,gray); filldraw(circle((1,sqrt(3)),1),gray); filldraw(circle((0,0),1),gray); filldraw(circle((2,0),1),grey); [/asy]
<spanclass=′latex−bold′>(A)</span> 10π+43<spanclass=′latex−bold′>(B)</span> 13π−3<spanclass=′latex−bold′>(C)</span> 12π+3<spanclass=′latex−bold′>(D)</span> 10π+9<spanclass=′latex−bold′>(E)</span> 13π <span class='latex-bold'>(A)</span>\ 10\pi+4\sqrt3\qquad<span class='latex-bold'>(B)</span>\ 13\pi-\sqrt3\qquad<span class='latex-bold'>(C)</span>\ 12\pi+\sqrt3\qquad<span class='latex-bold'>(D)</span>\ 10\pi+9\qquad<span class='latex-bold'>(E)</span>\ 13\pi
14
2

Nonstandard Checkerboards

Chubby makes nonstandard checkerboards that have 3131 squares on each side. The checkerboards have a black square in every corner and alternate red and black squares along every row and column. How many black squares are there on such a checkerboard?
<spanclass=′latex−bold′>(A)</span> 480<spanclass=′latex−bold′>(B)</span> 481<spanclass=′latex−bold′>(C)</span> 482<spanclass=′latex−bold′>(D)</span> 483<spanclass=′latex−bold′>(E)</span> 484 <span class='latex-bold'>(A)</span>\ 480 \qquad<span class='latex-bold'>(B)</span>\ 481 \qquad<span class='latex-bold'>(C)</span>\ 482 \qquad<span class='latex-bold'>(D)</span>\ 483 \qquad<span class='latex-bold'>(E)</span>\ 484

Equilateral Triangles in Square

Two equilateral triangles are contained in a square whose side length is 232\sqrt3. The bases of these triangles are the opposite sides of the square, and their intersection is a rhombus. What is the area of the rhombus?
<spanclass=′latex−bold′>(A)</span> 32<spanclass=′latex−bold′>(B)</span> 3<spanclass=′latex−bold′>(C)</span> 22−1<spanclass=′latex−bold′>(D)</span> 83−12<spanclass=′latex−bold′>(E)</span> 433 <span class='latex-bold'>(A)</span>\ \frac{3}{2}\qquad<span class='latex-bold'>(B)</span>\ \sqrt3\qquad<span class='latex-bold'>(C)</span>\ 2\sqrt2-1\qquad<span class='latex-bold'>(D)</span>\ 8\sqrt3-12\qquad<span class='latex-bold'>(E)</span>\ \frac{4\sqrt3}{3}
12
2

What Day of the Week?

A year is a leap year if and only if the year number is divisible by 400400 (such as 20002000) or is divisible by 44 but not by 100100 (such as 20122012). The 200th200\text{th} anniversary of the birth of novelist Charles Dickens was celebrated on February 77, 20122012, a Tuesday. On what day of the week was Dickens born?
<spanclass=′latex−bold′>(A)</span> Friday<spanclass=′latex−bold′>(B)</span> Saturday<spanclass=′latex−bold′>(C)</span> Sunday<spanclass=′latex−bold′>(D)</span> Monday<spanclass=′latex−bold′>(E)</span> Tuesday <span class='latex-bold'>(A)</span>\ \text{Friday} \qquad<span class='latex-bold'>(B)</span>\ \text{Saturday} \qquad<span class='latex-bold'>(C)</span>\ \text{Sunday} \qquad<span class='latex-bold'>(D)</span>\ \text{Monday} \qquad<span class='latex-bold'>(E)</span>\ \text{Tuesday}

Finding Distance Between Points

Point BB is due east of point AA. Point CC is due north of point BB. The distance between points AA and CC is 10210\sqrt{2} meters, and ∠BAC=45∘\angle BAC=45^{\circ}. Point DD is 2020 meters due north of point CC. The distance ADAD is between which two integers?
<spanclass=′latex−bold′>(A)</span> 30 and 31<spanclass=′latex−bold′>(B)</span> 31 and 32<spanclass=′latex−bold′>(C)</span> 32 and 33<spanclass=′latex−bold′>(D)</span> 33 and 34<spanclass=′latex−bold′>(E)</span> 34 and 35 <span class='latex-bold'>(A)</span>\ 30\text{ and }31\qquad<span class='latex-bold'>(B)</span>\ 31\text{ and }32\qquad<span class='latex-bold'>(C)</span>\ 32\text{ and }33\qquad<span class='latex-bold'>(D)</span>\ 33\text{ and }34\qquad<span class='latex-bold'>(E)</span>\ 34\text{ and }35
13
2

Iterative Averages of the First Five Natural Numbers

An iterative average of the numbers 11, 22, 33, 44, and 55 is computed in the following way. Arrange the five numbers in some order. Find the mean of the first two numbers, then find the mean of that with the third number, then the mean of that with the fourth number, and finally the mean of that with the fifth number. What is the difference between the largest and smallest possible values that can be obtained using this procedure?
<spanclass=′latex−bold′>(A)</span> 3116<spanclass=′latex−bold′>(B)</span> 2<spanclass=′latex−bold′>(C)</span> 178<spanclass=′latex−bold′>(D)</span> 3<spanclass=′latex−bold′>(E)</span> 6516 <span class='latex-bold'>(A)</span>\ \frac{31}{16}\qquad<span class='latex-bold'>(B)</span>\ 2\qquad<span class='latex-bold'>(C)</span>\ \frac{17}{8}\qquad<span class='latex-bold'>(D)</span>\ 3\qquad<span class='latex-bold'>(E)</span>\ \frac{65}{16}

Clea Walking Down Escalator

It takes Clea 6060 seconds to walk down an escalator when it is not operating and only 2424 seconds to walk down the escalator when it is operating. How many seconds does it take Clea to ride down the operating escalator when she just stands on it?
<spanclass=′latex−bold′>(A)</span> 36<spanclass=′latex−bold′>(B)</span> 40<spanclass=′latex−bold′>(C)</span> 42<spanclass=′latex−bold′>(D)</span> 48<spanclass=′latex−bold′>(E)</span> 52 <span class='latex-bold'>(A)</span>\ 36\qquad<span class='latex-bold'>(B)</span>\ 40\qquad<span class='latex-bold'>(C)</span>\ 42\qquad<span class='latex-bold'>(D)</span>\ 48\qquad<span class='latex-bold'>(E)</span>\ 52
11
2

Externally Tangent Circles

Externally tangent circles with centers at points AA and BB have radii of lengths 55 and 33, respectively. A line externally tangent to both circles intersects ray ABAB at point CC. What is BCBC?
<spanclass=′latex−bold′>(A)</span> 4<spanclass=′latex−bold′>(B)</span> 4.8<spanclass=′latex−bold′>(C)</span> 10.2<spanclass=′latex−bold′>(D)</span> 12<spanclass=′latex−bold′>(E)</span> 14.4 <span class='latex-bold'>(A)</span>\ 4 \qquad<span class='latex-bold'>(B)</span>\ 4.8 \qquad<span class='latex-bold'>(C)</span>\ 10.2 \qquad<span class='latex-bold'>(D)</span>\ 12 \qquad<span class='latex-bold'>(E)</span>\ 14.4

Dessert Chef and Yummy Desserts

A dessert chef prepares the dessert for every day of a week starting with Sunday. The dessert each day is either cake, pie, ice cream, or pudding. The same dessert may not be served two days in a row. There must be cake on Friday because of a birthday. How many different dessert menus for the week are possible?
<spanclass=′latex−bold′>(A)</span> 729<spanclass=′latex−bold′>(B)</span> 972<spanclass=′latex−bold′>(C)</span> 1024<spanclass=′latex−bold′>(D)</span> 2187<spanclass=′latex−bold′>(E)</span> 2304 <span class='latex-bold'>(A)</span>\ 729\qquad<span class='latex-bold'>(B)</span>\ 972\qquad<span class='latex-bold'>(C)</span>\ 1024\qquad<span class='latex-bold'>(D)</span>\ 2187\qquad<span class='latex-bold'>(E)</span>\ 2304
10
2

An Arithmetic Sequence of Sectors

Mary divides a circle into 1212 sectors. The central angles of these sectors, measured in degrees, are all integers and they form an arithmetic sequence. What is the degree measure of the smallest possible sector angle?
<spanclass=′latex−bold′>(A)</span> 5<spanclass=′latex−bold′>(B)</span> 6<spanclass=′latex−bold′>(C)</span> 8<spanclass=′latex−bold′>(D)</span> 10<spanclass=′latex−bold′>(E)</span> 12 <span class='latex-bold'>(A)</span>\ 5\qquad<span class='latex-bold'>(B)</span>\ 6\qquad<span class='latex-bold'>(C)</span>\ 8\qquad<span class='latex-bold'>(D)</span>\ 10\qquad<span class='latex-bold'>(E)</span>\ 12

Counting Ordered Pairs

How many ordered pairs of positive integers (M,N)(M,N) staisfy the equation M6=6N\frac{M}{6}=\frac{6}{N}?
<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 <span class='latex-bold'>(A)</span>\ 6\qquad<span class='latex-bold'>(B)</span>\ 7\qquad<span class='latex-bold'>(C)</span>\ 8\qquad<span class='latex-bold'>(D)</span>\ 9\qquad<span class='latex-bold'>(E)</span>\ 10
9
2

Odd Dice

A pair of six-sided fair dice are labeled so that one die has only even numbers (two each of 22, 44, and 66), and the other die has only odd numbers (two each of 11, 33, and 55). The pair of dice is rolled. What is the probability that the sum of the numbers on top of the two dice is 77?
<spanclass=′latex−bold′>(A)</span> 16<spanclass=′latex−bold′>(B)</span> 15<spanclass=′latex−bold′>(C)</span> 14<spanclass=′latex−bold′>(D)</span> 13<spanclass=′latex−bold′>(E)</span> 12 <span class='latex-bold'>(A)</span>\ \dfrac{1}{6} \qquad<span class='latex-bold'>(B)</span>\ \dfrac{1}{5} \qquad<span class='latex-bold'>(C)</span>\ \dfrac{1}{4} \qquad<span class='latex-bold'>(D)</span>\ \dfrac{1}{3} \qquad<span class='latex-bold'>(E)</span>\ \dfrac{1}{2}

Minimum Even Integers

Two integers have a sum of 2626. When two more integers are added to the first two integers the sum is 4141. Finally when two more integers are added to the sum of the previous four integers the sum is 5757. What is the minimum number of even integers among the 66 integers?
<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{{ <span class='latex-bold'>(A)</span>\ 1\qquad<span class='latex-bold'>(B)</span>\ 2\qquad<span class='latex-bold'>(C)</span>\ 3\qquad<span class='latex-bold'>(D)</span>\ 4}\qquad<span class='latex-bold'>(E)</span>\ 5}
8
2

Sums by Pairs

The sums of three whole numbers taken in pairs are 1212, 1717, and 1919. What is the middle number?
<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 <span class='latex-bold'>(A)</span>\ 4\qquad<span class='latex-bold'>(B)</span>\ 5\qquad<span class='latex-bold'>(C)</span>\ 6\qquad<span class='latex-bold'>(D)</span>\ 7\qquad<span class='latex-bold'>(E)</span>\ 8

Sum of Integer Solutions

What is the sum of all integer solutions to 1<(x−2)2<251<(x-2)^2<25?
<spanclass=′latex−bold′>(A)</span> 10<spanclass=′latex−bold′>(B)</span> 12<spanclass=′latex−bold′>(C)</span> 15<spanclass=′latex−bold′>(D)</span> 19<spanclass=′latex−bold′>(E)</span> 25 <span class='latex-bold'>(A)</span>\ 10\qquad<span class='latex-bold'>(B)</span>\ 12\qquad<span class='latex-bold'>(C)</span>\ 15\qquad<span class='latex-bold'>(D)</span>\ 19\qquad<span class='latex-bold'>(E)</span>\ 25
7
2

A Bag of Marbles

In a bag of marbles, 35\frac{3}{5} of the marbles are blue and the rest are red. If the number of red marbles is doubled and the number of blue marbles stays the same, what fraction of the marbles will be red?
<spanclass=′latex−bold′>(A)</span> 25<spanclass=′latex−bold′>(B)</span> 37<spanclass=′latex−bold′>(C)</span> 47<spanclass=′latex−bold′>(D)</span> 35<spanclass=′latex−bold′>(E)</span> 45 <span class='latex-bold'>(A)</span>\ \dfrac{2}{5} \qquad<span class='latex-bold'>(B)</span>\ \dfrac{3}{7} \qquad<span class='latex-bold'>(C)</span>\ \dfrac{4}{7} \qquad<span class='latex-bold'>(D)</span>\ \dfrac{3}{5} \qquad<span class='latex-bold'>(E)</span>\ \dfrac{4}{5}

Chipmunk, Squirrel, Acorn

For a science project, Sammy observed a chipmunk and a squirrel stashing acorns in holes. The chipmunk hid 33 acorns in each of the holes it dug. The squirrel hid 44 acorns in each of the holes it dug. They each hid the same number of acorns, although the squirrel needed 44 fewer holes. How many acorns did the chipmunk hide?
<spanclass=′latex−bold′>(A)</span> 30<spanclass=′latex−bold′>(B)</span> 36<spanclass=′latex−bold′>(C)</span> 42<spanclass=′latex−bold′>(D)</span> 48<spanclass=′latex−bold′>(E)</span> 54{{ <span class='latex-bold'>(A)</span>\ 30\qquad<span class='latex-bold'>(B)</span>\ 36\qquad<span class='latex-bold'>(C)</span>\ 42\qquad<span class='latex-bold'>(D)</span>\ 48}\qquad<span class='latex-bold'>(E)</span>\ 54}
6
2

Find the Sum of Two Numbers

The product of two positive numbers is 99. The reciprocal of one of these numbers is 44 times the reciprocal of the other number. What is the sum of the two numbers?
<spanclass=′latex−bold′>(A)</span> 103<spanclass=′latex−bold′>(B)</span> 203<spanclass=′latex−bold′>(C)</span> 7<spanclass=′latex−bold′>(D)</span> 152<spanclass=′latex−bold′>(E)</span> 8 <span class='latex-bold'>(A)</span>\ \dfrac{10}{3} \qquad<span class='latex-bold'>(B)</span>\ \dfrac{20}{3} \qquad<span class='latex-bold'>(C)</span>\ 7 \qquad<span class='latex-bold'>(D)</span>\ \dfrac{15}{2} \qquad<span class='latex-bold'>(E)</span>\ 8

Estimating x-y

In order to estimate the value of x−yx-y where xx and yy are real numbers with x>y>0x>y>0, Xiaoli rounded xx up by a small amount, rounded yy down by the same amount, and then subtracted her values. Which of the following statements is necessarily correct?
<spanclass=′latex−bold′>(A)</span> Her estimate is larger than x−y<spanclass=′latex−bold′>(B)</span> Her estimate is smaller than x−y<spanclass=′latex−bold′>(C)</span> Her estimate equals x−y<spanclass=′latex−bold′>(D)</span> Her estimate equals y−x<spanclass=′latex−bold′>(E)</span> Her estimate is 0 <span class='latex-bold'>(A)</span>\ \text{Her estimate is larger than }x-y\\ <span class='latex-bold'>(B)</span>\ \text{Her estimate is smaller than }x-y\\ <span class='latex-bold'>(C)</span>\ \text{Her estimate equals }x-y\\ <span class='latex-bold'>(D)</span>\ \text{Her estimate equals }y-x\\ <span class='latex-bold'>(E)</span>\ \text{Her estimate is }0
5
2

How Many Cats and Kittens?

Last year 100100 adult cats, half of whom were female, were brought into the Smallville Animal Shelter. Half of the adult female cats were accompanied by a litter of kittens. The average number of kittens per litter was 44. What was the total number of cats and kittens received by the shelter last year?
<spanclass=′latex−bold′>(A)</span> 150<spanclass=′latex−bold′>(B)</span> 200<spanclass=′latex−bold′>(C)</span> 250<spanclass=′latex−bold′>(D)</span> 300<spanclass=′latex−bold′>(E)</span> 400 <span class='latex-bold'>(A)</span>\ 150 \qquad<span class='latex-bold'>(B)</span>\ 200 \qquad<span class='latex-bold'>(C)</span>\ 250 \qquad<span class='latex-bold'>(D)</span>\ 300 \qquad<span class='latex-bold'>(E)</span>\ 400

Anna's Dinner

Anna enjoys dinner at a restaurant in Washington, D.C., where the sales tax on meals is 10%10\%. She leaves a 15%15\% tip on the prices of her meal before the sales tax is added, and the tax is calculated on the pre-tip amount. She spends a total of 27.5027.50 for dinner. What is the cost of here dinner without tax or tip?
<spanclass=′latex−bold′>(A)</span> $18<spanclass=′latex−bold′>(B)</span> $20<spanclass=′latex−bold′>(C)</span> $21<spanclass=′latex−bold′>(D)</span> $22<spanclass=′latex−bold′>(E)</span> $24 <span class='latex-bold'>(A)</span>\ \$18\qquad<span class='latex-bold'>(B)</span>\ \$20\qquad<span class='latex-bold'>(C)</span>\ \$21\qquad<span class='latex-bold'>(D)</span>\ \$22\qquad<span class='latex-bold'>(E)</span>\ \$24
4
2

Smallest Possible Measure of an Angle

Let ∠ABC=24∘\angle ABC=24^\circ and ∠ABD=20∘\angle ABD =20^\circ. What is the smallest possible degree measure for ∠CBD\angle CBD?
<spanclass=′latex−bold′>(A)</span> 0<spanclass=′latex−bold′>(B)</span> 2<spanclass=′latex−bold′>(C)</span> 4<spanclass=′latex−bold′>(D)</span> 6<spanclass=′latex−bold′>(E)</span> 12 <span class='latex-bold'>(A)</span>\ 0 \qquad<span class='latex-bold'>(B)</span>\ 2 \qquad<span class='latex-bold'>(C)</span>\ 4 \qquad<span class='latex-bold'>(D)</span>\ 6 \qquad<span class='latex-bold'>(E)</span>\ 12

Placing Marbles Into Bags

When Ringo places his marbles into bags with 66 marbles per bag, he has 44 marbles left over. When Paul does the same with his marbles, he has 33 marbles left over. Ringo and Paul pool their marbles and place them into as many bags as possible, with 66 marbles per bag. How many marbles will be left over?
<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 <span class='latex-bold'>(A)</span>\ 1\qquad<span class='latex-bold'>(B)</span>\ 2\qquad<span class='latex-bold'>(C)</span>\ 3\qquad<span class='latex-bold'>(D)</span>\ 4\qquad<span class='latex-bold'>(E)</span>\ 5
3
2

How Far Did the Bug Crawl?

A bug crawls along a number line, starting at −2-2. It crawls to −6-6, then turns around and crawls to 55. How many units does the bug crawl altogether?
<spanclass=′latex−bold′>(A)</span> 9<spanclass=′latex−bold′>(B)</span> 11<spanclass=′latex−bold′>(C)</span> 13<spanclass=′latex−bold′>(D)</span> 14<spanclass=′latex−bold′>(E)</span> 15 <span class='latex-bold'>(A)</span>\ 9 \qquad<span class='latex-bold'>(B)</span>\ 11 \qquad<span class='latex-bold'>(C)</span>\ 13 \qquad<span class='latex-bold'>(D)</span>\ 14 \qquad<span class='latex-bold'>(E)</span>\ 15

Reflecting Point over Line

The point in the xyxy-plane with coordinates (1000,2012)(1000,2012) is reflected across line y=2000y=2000. What are the coordinates of the reflected point?
<spanclass=′latex−bold′>(A)</span> (998,2012)<spanclass=′latex−bold′>(B)</span> (1000,1988)<spanclass=′latex−bold′>(C)</span> (1000,2024)<spanclass=′latex−bold′>(D)</span> (1000,4012)<spanclass=′latex−bold′>(E)</span> (1012,2012) <span class='latex-bold'>(A)</span>\ (998,2012) \qquad<span class='latex-bold'>(B)</span>\ (1000,1988)\qquad<span class='latex-bold'>(C)</span>\ (1000,2024)\qquad<span class='latex-bold'>(D)</span>\ (1000,4012)\qquad<span class='latex-bold'>(E)</span>\ (1012,2012)
2
2

A Square Cut in Half

A square with side length 88 is cut in half, creating two congruent rectangles. What are the dimensions of one of these rectangles?
<spanclass=′latex−bold′>(A)</span> 2 by 4<spanclass=′latex−bold′>(B)</span> 2 by 6<spanclass=′latex−bold′>(C)</span> 2 by 8<spanclass=′latex−bold′>(D)</span> 4 by 4<spanclass=′latex−bold′>(E)</span> 4 by 8 <span class='latex-bold'>(A)</span>\ 2\text{ by }4 \qquad<span class='latex-bold'>(B)</span>\ 2\text{ by }6 \qquad<span class='latex-bold'>(C)</span>\ 2\text{ by }8 \qquad<span class='latex-bold'>(D)</span>\ 4\text{ by }4 \qquad<span class='latex-bold'>(E)</span>\ 4\text{ by }8

Circle Inscribed in Rectangle

A circle of radius 55 is inscribed in a rectangle as shown. The ratio of the the length of the rectangle to its width is 2 : 12\ :\ 1. What is the area of the rectangle?
[asy] draw((0,0)--(0,10)--(20,10)--(20,0)--cycle); draw(circle((10,5),5)); [/asy]
<spanclass=′latex−bold′>(A)</span> 50<spanclass=′latex−bold′>(B)</span> 100<spanclass=′latex−bold′>(C)</span> 125<spanclass=′latex−bold′>(D)</span> 150<spanclass=′latex−bold′>(E)</span> 200 <span class='latex-bold'>(A)</span>\ 50\qquad<span class='latex-bold'>(B)</span>\ 100\qquad<span class='latex-bold'>(C)</span>\ 125\qquad<span class='latex-bold'>(D)</span>\ 150\qquad<span class='latex-bold'>(E)</span>\ 200
1
2

Frosting Cupcakes

Cagney can frost a cupcake every 2020 seconds and Lacey can frost a cupcake every 3030 seconds. Working together, how many cupcakes can they frost in 55 minutes?
<spanclass=′latex−bold′>(A)</span> 10<spanclass=′latex−bold′>(B)</span> 15<spanclass=′latex−bold′>(C)</span> 20<spanclass=′latex−bold′>(D)</span> 25<spanclass=′latex−bold′>(E)</span> 30 <span class='latex-bold'>(A)</span>\ 10 \qquad<span class='latex-bold'>(B)</span>\ 15 \qquad<span class='latex-bold'>(C)</span>\ 20 \qquad<span class='latex-bold'>(D)</span>\ 25 \qquad<span class='latex-bold'>(E)</span>\ 30

Rabbits and Students

Each third-grade classroom at Pearl Creek Elementary has 1818 students and 22 pet rabbits. How many more students than rabbits are there in all 44 of the third-grade classrooms?
<spanclass=′latex−bold′>(A)</span> 48<spanclass=′latex−bold′>(B)</span> 56<spanclass=′latex−bold′>(C)</span> 64<spanclass=′latex−bold′>(D)</span> 72<spanclass=′latex−bold′>(E)</span> 80{{ <span class='latex-bold'>(A)</span>\ 48\qquad<span class='latex-bold'>(B)</span>\ 56\qquad<span class='latex-bold'>(C)</span>\ 64\qquad<span class='latex-bold'>(D)</span>\ 72}\qquad<span class='latex-bold'>(E)</span>\ 80}