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Difference between revisions of "2015 AMC 10B Problems"

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==Problem 1==
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njovihboxicuhoizuhogiuzshfoiuhseiuahpuhqwiuthpouhppzxpiguzhoigzhsodifuhzxoiuhvzoxiubhiocnhioubhiocvbhmiocvhbnoixhcvobiuzhoiuhzxoiuvhzi oudhfoiausasguhaeohgaowpiehgaoweighpaoweighaopwegihapoweighporihgoxjfohisjdfoihjsdoijaposihpqowehproiqhweptoihqweoptihqwopithqopwih topqiwehgoihpoibhpzoixhbpoxbupozixycboipyzoaigoahweopithapowiehtaopispboiyzxoibhpzxogaosinfpoisdnpfoiandspfoaishgpoiasbposydpoiahofins epogiaywepgoiahwepgoihawpeoihpaowiethpaoiwehgpoiahwpoibhpaoiwehpoiawegpoaweotpihaweawepgoihpaiohapaobiwyepogiaywepgoiahwepgoi hawpeoihpaowiethpaoiwehgpoiahwpoibhpaoiwehpoiawegpoaweotpihaweawepgoihpaiohapaobiwyepogiaywepgoiahwepgoihawpeoihpaowiethpaoiw ehgpoiahwpoibhpaoiwehpoiawegpoaweotpihaweawepgoihpaiohapaobiwyepogiaywepgoiahwepgoihawpeoihpaowiethpaoiwehgpoiahwpoibhpaoiweh poiawegpoaweotpihaweawepgoihpaiohapaobiwyepogiaywepgoiahwepgoihawpeoihpaowiethpaoiwehgpoiahwpoibhpaoiwehpoiawegpoaweotpihawe awepgoihpaiohapaobiwyepogiaywepgoiahwepgoihawpeoihpaowiethpaoiwehgpoiahwpoibhpaoiwehpoiawegpoaweotpihaweawepgoihpaiohapaobiwy epogiaywepgoiahwepgoihawpeoihpaowiethpaoiwehgpoiahwpoibhpaoiwehpoiawegpoaweotpihaweawepgoihpaiohapaobiwyepogiaywepgoiahwepgoi hawpeoihpaowiethpaoiwehgpoiahwpoibhpaoiwehpoiawegpoaweotpihaweawepgoihpaiohapaobiwyepogiaywepgoiahwepgoihawpeoihpaowiethpaoiw ehgpoiahwpoibhpaoiwehpoiawegpoaweotpihaweawegahwepogiahwepogihapwoibhpofyisdfyutasdvtzxciuvytarwtpnpnbpbpoiwawegvsaubiausdfifuck
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What is the value of <math>2-(-2)^{-2}</math>?
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<math> \textbf{(A) } -2 \qquad\textbf{(B) } \dfrac{1}{16} \qquad\textbf{(C) } \dfrac{7}{4} \qquad\textbf{(D) } \dfrac{9}{4} \qquad\textbf{(E) } 6 </math>
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[[2015 AMC 10B Problems/Problem 1|Solution]]
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==Problem 2==
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Marie does three equally time-consuming tasks in a row without taking breaks. She begins the first task at 1:00 PM and finishes the second task at 2:40 PM. When does she finish the third task?
 +
 
 +
<math> \textbf{(A) }\text{3:10 PM}\qquad\textbf{(B) }\text{3:30 PM}\qquad\textbf{(C) }\text{4:00 PM}\qquad\textbf{(D) }\text{4:10 PM}\qquad\textbf{(E) }\text{4:30 PM} </math>
 +
 
 +
[[2015 AMC 10B Problems/Problem 2|Solution]]
 +
 +
==Problem 3==
 +
 
 +
Isaac has written down one integer two times and another integer three times. The sum of the five numbers is <math>100</math>, and one of the numbers is <math>28</math>. What is the other number?
 +
 
 +
<math>\textbf{(A) } 8 \qquad\textbf{(B) } 11 \qquad\textbf{(C) } 14 \qquad\textbf{(D) } 15 \qquad\textbf{(E) } 18 </math>
 +
 
 +
[[2015 AMC 10B Problems/Problem 3|Solution]]
 +
 
 +
==Problem 4==
 +
 
 +
Four siblings ordered an extra large pizza. Alex ate <math>\frac15</math>, Beth <math>\frac13</math>, and Cyril <math>\frac14</math> of the pizza. Dan got the leftovers. What is the sequence of the siblings in decreasing order of the part of pizza they consumed?
 +
 
 +
<math>\textbf{(A) } \text{Alex, Beth, Cyril, Dan}</math>
 +
<math>\textbf{(B) } \text{Beth, Cyril, Alex, Dan}</math>
 +
<math>\textbf{(C) } \text{Beth, Cyril, Dan, Alex}</math>
 +
<math>\textbf{(D) } \text{Beth, Dan, Cyril, Alex}</math>
 +
<math>\textbf{(E) } \text{Dan, Beth, Cyril, Alex}</math>
 +
 
 +
[[2015 AMC 10B Problems/Problem 4|Solution]]
 +
 
 +
==Problem 5==
 +
 
 +
David, Hikmet, Jack, Marta, Rand, and Todd were in a <math>12</math>-person race with <math>6</math> other people. Rand finished <math>6</math> places ahead of Hikmet. Marta finished <math>1</math> place behind Jack. David finished <math>2</math> places behind Hikmet. Jack finished <math>2</math> places behind Todd. Todd finished <math>1</math> place behind Rand. Marta finished in <math>6</math>th place. Who finished in <math>8</math>th place?
 +
 
 +
<math>\textbf{(A) } \text{David} \qquad\textbf{(B) } \text{Hikmet} \qquad\textbf{(C) } \text{Jack} \qquad\textbf{(D) } \text{Rand} \qquad\textbf{(E) } \text{Todd} </math>
 +
 
 +
[[2015 AMC 10B Problems/Problem 5|Solution]]
 +
 
 +
==Problem 6==
 +
 
 +
Marley practices exactly one sport each day of the week. She runs three days a week but never on two consecutive days. On Monday she plays basketball and two days later golf. She swims and plays tennis, but she never plays tennis the day after running or swimming. Which day of the week does Marley swim?
 +
 
 +
<math>\textbf{(A) } \text{Sunday} \qquad\textbf{(B) } \text{Tuesday} \qquad\textbf{(C) } \text{Thursday} \qquad\textbf{(D) } \text{Friday} \qquad\textbf{(E) } \text{Saturday} </math>
 +
 
 +
[[2015 AMC 10B Problems/Problem 6|Solution]]
 +
 
 +
==Problem 7==
 +
 
 +
Consider the operation "minus the reciprocal of," defined by <math>a\diamond b=a-\frac{1}{b}</math>. What is <math>((1\diamond2)\diamond3)-(1\diamond(2\diamond3))</math>?
 +
 
 +
<math>\textbf{(A) } -\dfrac{7}{30} \qquad\textbf{(B) } -\dfrac{1}{6} \qquad\textbf{(C) } 0 \qquad\textbf{(D) } \dfrac{1}{6} \qquad\textbf{(E) } \dfrac{7}{30} </math>
 +
 
 +
[[2015 AMC 10B Problems/Problem 7|Solution]]
 +
 
 +
==Problem 8==
 +
 
 +
The letter F shown below is rotated <math>90^\circ</math> clockwise around the origin, then reflected in the <math>y</math>-axis, and then rotated a half turn around the origin. What is the final image?
 +
<asy>import cse5;pathpen=black;pointpen=black; size(1inch); D((0,-2)--MP("y",(0,7),N)); D((-3,0)--MP("x",(5,0),E)); D((1,0)--(1,2)--(2,2)--(2,3)--(1,3)--(1,4)--(3,4)--(3,5)--(0,5)); import cse5;pathpen=black;pointpen=black; unitsize(0.25cm); D((0,-2)--MP("y",(0,7),N)); D(MP("\textbf{(A) }",(-3,0),W)--MP("x",(5,0),E)); D((1,0)--(1,2)--(2,2)--(2,3)--(1,3)--(1,4)--(3,4)--(3,5)--(0,5)); // D((18,-2)--MP("y",(18,7),N)); D(MP("\textbf{(B) }",(13,0),W)--MP("x",(21,0),E)); D((17,0)--(17,2)--(16,2)--(16,3)--(17,3)--(17,4)--(15,4)--(15,5)--(18,5)); // D((36,-2)--MP("y",(36,7),N)); D(MP("\textbf{(C) }",(29,0),W)--MP("x",(38,0),E)); D((31,0)--(31,1)--(33,1)--(33,2)--(34,2)--(34,1)--(35,1)--(35,3)--(36,3)); // D((0,-17)--MP("y",(0,-8),N)); D(MP("\textbf{(D) }",(-3,-15),W)--MP("x",(5,-15),E)); D((3,-15)--(3,-14)--(1,-14)--(1,-13)--(2,-13)--(2,-12)--(1,-12)--(1,-10)--(0,-10)); // D((15,-17)--MP("y",(15,-8),N)); D(MP("\textbf{(E) }",(13,-15),W)--MP("x",(22,-15),E)); D((15,-14)--(17,-14)--(17,-13)--(18,-13)--(18,-14)--(19,-14)--(19,-12)--(20,-12)--(20,-15)); </asy>
 +
 
 +
[[2015 AMC 10B Problems/Problem 8|Solution]]
 +
 
 +
==Problem 9==
 +
 
 +
The shaded region below is called a shark's fin falcata, a figure studied by Leonardo da Vinci. It is bounded by the portion of the circle of radius <math>3</math> and center <math>(0,0)</math> that lies in the first quadrant, the portion of the circle with radius <math>\tfrac{3}{2}</math> and center <math>(0,\tfrac{3}{2})</math> that lies in the first quadrant, and the line segment from <math>(0,0)</math> to <math>(3,0)</math>. What is the area of the shark's fin falcata?
 +
 
 +
<asy>import cse5;pathpen=black;pointpen=black; size(1.5inch); D(MP("x",(3.5,0),S)--(0,0)--MP("\frac{3}{2}",(0,3/2),W)--MP("y",(0,3.5),W)); path P=(0,0)--MP("3",(3,0),S)..(3*dir(45))..MP("3",(0,3),W)--(0,3)..(3/2,3/2)..cycle; draw(P,linewidth(2)); fill(P,gray); </asy>
 +
 
 +
<math>\textbf{(A) } \dfrac{4\pi}{5} \qquad\textbf{(B) } \dfrac{9\pi}{8} \qquad\textbf{(C) } \dfrac{4\pi}{3} \qquad\textbf{(D) } \dfrac{7\pi}{5} \qquad\textbf{(E) } \dfrac{3\pi}{2} </math>
 +
 
 +
[[2015 AMC 10B Problems/Problem 9|Solution]]
 +
 
 +
==Problem 10==
 +
 
 +
What are the sign and units digit of the product of all the odd negative integers strictly greater than <math>-2015</math>?
 +
 
 +
<math>\textbf{(A) } \text{It is a negative number ending with a 1.}</math>
 +
 
 +
<math>\textbf{(B) } \text{It is a positive number ending with a 1.}</math>
 +
 
 +
<math>\textbf{(C) } \text{It is a negative number ending with a 5.}</math>
 +
 
 +
<math>\textbf{(D) } \text{It is a positive number ending with a 5.}</math>
 +
 
 +
<math>\textbf{(E) } \text{It is a negative number ending with a 0.}</math>
 +
 
 +
[[2015 AMC 10B Problems/Problem 10|Solution]]
 +
 
 +
==Problem 11==
 +
 
 +
Among the positive integers less than <math>100</math>, each of whose digits is a prime number, one is selected at random. What is the probablility that the selected number is prime?
 +
 
 +
<math>\textbf{(A) } \dfrac{8}{99} \qquad\textbf{(B) } \dfrac{2}{5} \qquad\textbf{(C) } \dfrac{9}{20} \qquad\textbf{(D) } \dfrac{1}{2} \qquad\textbf{(E) } \dfrac{9}{16} </math>
 +
 
 +
[[2015 AMC 10B Problems/Problem 11|Solution]]
 +
 
 +
==Problem 12==
 +
 
 +
For how many integers <math>x</math> is the point <math>(x,-x)</math> inside or on the circle of radius <math>10</math> centered at <math>(5,5)</math>?
 +
 
 +
<math>\textbf{(A) } 11 \qquad\textbf{(B) } 12 \qquad\textbf{(C) } 13 \qquad\textbf{(D) } 14 \qquad\textbf{(E) } 15 </math>
 +
 
 +
[[2015 AMC 10B Problems/Problem 12|Solution]]
 +
 
 +
==Problem 13==
 +
 
 +
The line <math>12x+5y=60</math> forms a triangle with the coordinate axes. What is the sum of the lengths of the altitudes of this triangle?
 +
 
 +
<math>\textbf{(A) } 20 \qquad\textbf{(B) } \dfrac{360}{17} \qquad\textbf{(C) } \dfrac{107}{5} \qquad\textbf{(D) } \dfrac{43}{2} \qquad\textbf{(E) } \dfrac{281}{13} </math>
 +
 
 +
[[2015 AMC 10B Problems/Problem 13|Solution]]
 +
 
 +
==Problem 14==
 +
 
 +
Let <math>a</math>, <math>b</math>, and <math>c</math> be three distinct one-digit numbers. What is the maximum value of the sum of the roots of the equation <math>(x-a)(x-b)+(x-b)(x-c)=0</math>?
 +
 
 +
<math>\textbf{(A) } 15 \qquad\textbf{(B) } 15.5 \qquad\textbf{(C) } 16 \qquad\textbf{(D) } 16.5 \qquad\textbf{(E) } 17 </math>
 +
 
 +
[[2015 AMC 10B Problems/Problem 14|Solution]]
 +
 
 +
==Problem 15==
 +
 
 +
The town of Hamlet has <math>3</math> people for each horse, <math>4</math> sheep for each cow, and <math>3</math> ducks for each person. Which of the following could not possibly be the total number of people, horses, sheep, cows, and ducks in Hamlet?
 +
 
 +
<math>\textbf{(A) } 41 \qquad\textbf{(B) } 47 \qquad\textbf{(C) } 59 \qquad\textbf{(D) } 61 \qquad\textbf{(E) } 66 </math>
 +
 
 +
[[2015 AMC 10B Problems/Problem 15|Solution]]
 +
 
 +
==Problem 16==
 +
 
 +
Al, Bill, and Cal will each randomly be assigned a whole number from <math>1</math> to <math>10</math>, inclusive, with no two of them getting the same number. What is the probability that Al's number will be a whole number multiple of Bill's and Bill's number will be a whole number multiple of Cal's?
 +
 
 +
<math>\textbf{(A) } \dfrac{9}{1000} \qquad\textbf{(B) } \dfrac{1}{90} \qquad\textbf{(C) } \dfrac{1}{80} \qquad\textbf{(D) } \dfrac{1}{72} \qquad\textbf{(E) } \dfrac{2}{121} </math>
 +
 
 +
[[2015 AMC 10B Problems/Problem 16|Solution]]
 +
 
 +
==Problem 17==
 +
 
 +
The centers of the faces of the right rectangular prism shown below are joined to create an octahedron, What is the volume of the octahedron?
 +
 
 +
<asy>import three; size(2inch); currentprojection=orthographic(4,2,2); draw((0,0,0)--(0,0,3),dashed); draw((0,0,0)--(0,4,0),dashed); draw((0,0,0)--(5,0,0),dashed); draw((5,4,3)--(5,0,3)--(5,0,0)--(5,4,0)--(0,4,0)--(0,4,3)--(0,0,3)--(5,0,3)); draw((0,4,3)--(5,4,3)--(5,4,0)); label("3",(5,0,3)--(5,0,0),W); label("4",(5,0,0)--(5,4,0),S); label("5",(5,4,0)--(0,4,0),SE); </asy>
 +
 
 +
<math>\textbf{(A) } \dfrac{75}{12} \qquad\textbf{(B) } 10 \qquad\textbf{(C) } 12 \qquad\textbf{(D) } 10\sqrt2 \qquad\textbf{(E) } 15 </math>
 +
 
 +
[[2015 AMC 10B Problems/Problem 17|Solution]]
 +
 
 +
==Problem 18==
 +
 
 +
Johann has <math>64</math> fair coins. He flips all the coins. Any coin that lands on tails is tossed again. Coins that land on tails on the second toss are tossed a third time. What is the expected number of coins that are now heads?
 +
 
 +
<math>\textbf{(A) } 32 \qquad\textbf{(B) } 40 \qquad\textbf{(C) } 48 \qquad\textbf{(D) } 56 \qquad\textbf{(E) } 64 </math>
 +
 
 +
[[2015 AMC 10B Problems/Problem 18|Solution]]
 +
 
 +
==Problem 19==
 +
 
 +
In <math>\triangle{ABC}</math>, <math>\angle{C} = 90^{\circ}</math> and <math>AB = 12</math>. Squares <math>ABXY</math> and <math>ACWZ</math> are constructed outside of the triangle. The points <math>X, Y, Z</math>, and <math>W</math> lie on a circle. What is the perimeter of the triangle?
 +
 
 +
<math> \textbf{(A)}\ 12+9\sqrt{3}\qquad\textbf{(B)}\ 18+6\sqrt{3}\qquad\textbf{(C)}\ 12+12\sqrt{2}\qquad\textbf{(D)}\ 30\qquad\textbf{(E)}\ 32 </math>
 +
 
 +
[[2015 AMC 10B Problems/Problem 19|Solution]]
 +
 
 +
==Problem 20==
 +
 
 +
Erin the ant starts at a given corner of a cube and crawls along exactly <math>7</math> edges in such a way that she visits every corner exactly once and then finds that she is unable to return along an edge to her starting point. How many paths are there meeting these conditions?
 +
 
 +
<math> \textbf{(A) }\text{6}\qquad\textbf{(B) }\text{9}\qquad\textbf{(C) }\text{12}\qquad\textbf{(D) }\text{18}\qquad\textbf{(E) }\text{24} </math>
 +
 
 +
[[2015 AMC 10B Problems/Problem 20|Solution]]
 +
 
 +
==Problem 21==
 +
 
 +
Cozy the Cat and Dash the Dog are going up a staircase with a certain number of steps. However, instead of walking up the steps one at a time, both Cozy and Dash jump. Cozy goes two steps up with each jump (though if necessary, he will just jump the last step). Dash goes five steps up with each jump (though if necessary, he will just jump the last steps if there are fewer than 5 steps left). Suppose the Dash takes 19 fewer jumps than Cozy to reach the top of the staircase. Let <math>s</math> denote the sum of all possible numbers of steps this staircase can have. What is the sum of the digits of <math>s</math>?
 +
 
 +
<math>\textbf{(A) } 9 \qquad\textbf{(B) } 11 \qquad\textbf{(C) } 12 \qquad\textbf{(D) } 13 \qquad\textbf{(E) } 15 </math>
 +
 
 +
[[2015 AMC 10B Problems/Problem 21|Solution]]
 +
 
 +
==Problem 22==
 +
 
 +
In the figure shown below, <math>ABCDE</math> is a regular pentagon and <math>AG=1</math>. What is <math>FG+JH+CD</math>?
 +
<asy>import cse5;pathpen=black;pointpen=black; size(2inch); pair A=dir(90), B=dir(18), C=dir(306), D=dir(234), E=dir(162); D(MP("A",A,A)--MP("B",B,B)--MP("C",C,C)--MP("D",D,D)--MP("E",E,E)--cycle,linewidth(1.5)); D(A--C--E--B--D--cycle); pair F=IP(A--D,B--E), G=IP(B--E,C--A), H=IP(C--A,B--D), I=IP(D--B,E--C), J=IP(C--E,D--A); D(MP("F",F,dir(126))--MP("I",I,dir(270))--MP("G",G,dir(54))--MP("J",J,dir(198))--MP("H",H,dir(342))--cycle); </asy>
 +
 
 +
<math>\textbf{(A) } 3 \qquad\textbf{(B) } 12-4\sqrt5 \qquad\textbf{(C) } \dfrac{5+2\sqrt5}{3} \qquad\textbf{(D) } 1+\sqrt5 \qquad\textbf{(E) } \dfrac{11+11\sqrt5}{10} </math>
 +
 
 +
[[2015 AMC 10B Problems/Problem 22|Solution]]
 +
 
 +
==Problem 23==
 +
 
 +
Let <math>n</math> be a positive integer greater than 4 such that the decimal representation of <math>n!</math> ends in <math>k</math> zeros and the decimal representation of <math>(2n)!</math> ends in <math>3k</math> zeros. Let <math>s</math> denote the sum of the four least possible values of <math>n</math>. What is the sum of the digits of <math>s</math>?
 +
 
 +
<math> \textbf{(A) }7\qquad\textbf{(B) }8\qquad\textbf{(C) }9\qquad\textbf{(D) }10\qquad\textbf{(E) }11 </math>
 +
 
 +
[[2015 AMC 10B Problems/Problem 23|Solution]]
 +
 
 +
==Problem 24==
 +
 
 +
Aaron the ant walks on the coordinate plane according to the following rules. He starts at the origin <math>p_0=(0,0)</math> facing to the east and walks one unit, arriving at <math>p_1=(1,0)</math>. For <math>n=1,2,3,\dots</math>, right after arriving at the point <math>p_n</math>, if Aaron can turn <math>90^\circ</math> left and walk one unit to an unvisited point <math>p_{n+1}</math>, he does that. Otherwise, he walks one unit straight ahead to reach <math>p_{n+1}</math>. Thus the sequence of points continues <math>p_2=(1,1), p_3=(0,1), p_4=(-1,1), p_5=(-1,0)</math>, and so on in a counterclockwise spiral pattern. What is <math>p_{2015}</math>?
 +
 
 +
<math> \textbf{(A) } (-22,-13)\qquad\textbf{(B) } (-13,-22)\qquad\textbf{(C) } (-13,22)\qquad\textbf{(D) } (13,-22)\qquad\textbf{(E) } (22,-13) </math>
 +
 
 +
[[2015 AMC 10B Problems/Problem 24|Solution]]
 +
 
 +
==Problem 25==
 +
 
 +
A rectangular box measures <math>a \times b \times c</math>, where <math>a,</math> <math>b,</math> and <math>c</math> are integers and <math>1 \leq a \leq b \leq c</math>. The volume and surface area of the box are numerically equal. How many ordered triples <math>(a,b,c)</math> are possible?
 +
 
 +
<math> \textbf{(A) }4\qquad\textbf{(B) }10\qquad\textbf{(C) }12\qquad\textbf{(D) }21\qquad\textbf{(E) }26 </math>
 +
 
 +
[[2015 AMC 10B Problems/Problem 25|Solution]]
 +
 
 +
== See also ==
 +
* [[AMC 10]]
 +
* [[AMC 10 Problems and Solutions]]
 +
* [[Mathematics competition resources]]
 +
{{MAA Notice}}

Revision as of 21:24, 17 March 2015

Problem 1

What is the value of $2-(-2)^{-2}$?

$\textbf{(A) } -2 \qquad\textbf{(B) } \dfrac{1}{16} \qquad\textbf{(C) } \dfrac{7}{4} \qquad\textbf{(D) } \dfrac{9}{4} \qquad\textbf{(E) } 6$

Solution

Problem 2

Marie does three equally time-consuming tasks in a row without taking breaks. She begins the first task at 1:00 PM and finishes the second task at 2:40 PM. When does she finish the third task?

$\textbf{(A) }\text{3:10 PM}\qquad\textbf{(B) }\text{3:30 PM}\qquad\textbf{(C) }\text{4:00 PM}\qquad\textbf{(D) }\text{4:10 PM}\qquad\textbf{(E) }\text{4:30 PM}$

Solution

Problem 3

Isaac has written down one integer two times and another integer three times. The sum of the five numbers is $100$, and one of the numbers is $28$. What is the other number?

$\textbf{(A) } 8 \qquad\textbf{(B) } 11 \qquad\textbf{(C) } 14 \qquad\textbf{(D) } 15 \qquad\textbf{(E) } 18$

Solution

Problem 4

Four siblings ordered an extra large pizza. Alex ate $\frac15$, Beth $\frac13$, and Cyril $\frac14$ of the pizza. Dan got the leftovers. What is the sequence of the siblings in decreasing order of the part of pizza they consumed?

$\textbf{(A) } \text{Alex, Beth, Cyril, Dan}$ $\textbf{(B) } \text{Beth, Cyril, Alex, Dan}$ $\textbf{(C) } \text{Beth, Cyril, Dan, Alex}$ $\textbf{(D) } \text{Beth, Dan, Cyril, Alex}$ $\textbf{(E) } \text{Dan, Beth, Cyril, Alex}$

Solution

Problem 5

David, Hikmet, Jack, Marta, Rand, and Todd were in a $12$-person race with $6$ other people. Rand finished $6$ places ahead of Hikmet. Marta finished $1$ place behind Jack. David finished $2$ places behind Hikmet. Jack finished $2$ places behind Todd. Todd finished $1$ place behind Rand. Marta finished in $6$th place. Who finished in $8$th place?

$\textbf{(A) } \text{David} \qquad\textbf{(B) } \text{Hikmet} \qquad\textbf{(C) } \text{Jack} \qquad\textbf{(D) } \text{Rand} \qquad\textbf{(E) } \text{Todd}$

Solution

Problem 6

Marley practices exactly one sport each day of the week. She runs three days a week but never on two consecutive days. On Monday she plays basketball and two days later golf. She swims and plays tennis, but she never plays tennis the day after running or swimming. Which day of the week does Marley swim?

$\textbf{(A) } \text{Sunday} \qquad\textbf{(B) } \text{Tuesday} \qquad\textbf{(C) } \text{Thursday} \qquad\textbf{(D) } \text{Friday} \qquad\textbf{(E) } \text{Saturday}$

Solution

Problem 7

Consider the operation "minus the reciprocal of," defined by $a\diamond b=a-\frac{1}{b}$. What is $((1\diamond2)\diamond3)-(1\diamond(2\diamond3))$?

$\textbf{(A) } -\dfrac{7}{30} \qquad\textbf{(B) } -\dfrac{1}{6} \qquad\textbf{(C) } 0 \qquad\textbf{(D) } \dfrac{1}{6} \qquad\textbf{(E) } \dfrac{7}{30}$

Solution

Problem 8

The letter F shown below is rotated $90^\circ$ clockwise around the origin, then reflected in the $y$-axis, and then rotated a half turn around the origin. What is the final image? [asy]import cse5;pathpen=black;pointpen=black; size(1inch); D((0,-2)--MP("y",(0,7),N)); D((-3,0)--MP("x",(5,0),E)); D((1,0)--(1,2)--(2,2)--(2,3)--(1,3)--(1,4)--(3,4)--(3,5)--(0,5)); import cse5;pathpen=black;pointpen=black; unitsize(0.25cm); D((0,-2)--MP("y",(0,7),N)); D(MP("\textbf{(A) }",(-3,0),W)--MP("x",(5,0),E)); D((1,0)--(1,2)--(2,2)--(2,3)--(1,3)--(1,4)--(3,4)--(3,5)--(0,5)); // D((18,-2)--MP("y",(18,7),N)); D(MP("\textbf{(B) }",(13,0),W)--MP("x",(21,0),E)); D((17,0)--(17,2)--(16,2)--(16,3)--(17,3)--(17,4)--(15,4)--(15,5)--(18,5)); // D((36,-2)--MP("y",(36,7),N)); D(MP("\textbf{(C) }",(29,0),W)--MP("x",(38,0),E)); D((31,0)--(31,1)--(33,1)--(33,2)--(34,2)--(34,1)--(35,1)--(35,3)--(36,3)); // D((0,-17)--MP("y",(0,-8),N)); D(MP("\textbf{(D) }",(-3,-15),W)--MP("x",(5,-15),E)); D((3,-15)--(3,-14)--(1,-14)--(1,-13)--(2,-13)--(2,-12)--(1,-12)--(1,-10)--(0,-10)); // D((15,-17)--MP("y",(15,-8),N)); D(MP("\textbf{(E) }",(13,-15),W)--MP("x",(22,-15),E)); D((15,-14)--(17,-14)--(17,-13)--(18,-13)--(18,-14)--(19,-14)--(19,-12)--(20,-12)--(20,-15)); [/asy]

Solution

Problem 9

The shaded region below is called a shark's fin falcata, a figure studied by Leonardo da Vinci. It is bounded by the portion of the circle of radius $3$ and center $(0,0)$ that lies in the first quadrant, the portion of the circle with radius $\tfrac{3}{2}$ and center $(0,\tfrac{3}{2})$ that lies in the first quadrant, and the line segment from $(0,0)$ to $(3,0)$. What is the area of the shark's fin falcata?

[asy]import cse5;pathpen=black;pointpen=black; size(1.5inch); D(MP("x",(3.5,0),S)--(0,0)--MP("\frac{3}{2}",(0,3/2),W)--MP("y",(0,3.5),W)); path P=(0,0)--MP("3",(3,0),S)..(3*dir(45))..MP("3",(0,3),W)--(0,3)..(3/2,3/2)..cycle; draw(P,linewidth(2)); fill(P,gray); [/asy]

$\textbf{(A) } \dfrac{4\pi}{5} \qquad\textbf{(B) } \dfrac{9\pi}{8} \qquad\textbf{(C) } \dfrac{4\pi}{3} \qquad\textbf{(D) } \dfrac{7\pi}{5} \qquad\textbf{(E) } \dfrac{3\pi}{2}$

Solution

Problem 10

What are the sign and units digit of the product of all the odd negative integers strictly greater than $-2015$?

$\textbf{(A) } \text{It is a negative number ending with a 1.}$

$\textbf{(B) } \text{It is a positive number ending with a 1.}$

$\textbf{(C) } \text{It is a negative number ending with a 5.}$

$\textbf{(D) } \text{It is a positive number ending with a 5.}$

$\textbf{(E) } \text{It is a negative number ending with a 0.}$

Solution

Problem 11

Among the positive integers less than $100$, each of whose digits is a prime number, one is selected at random. What is the probablility that the selected number is prime?

$\textbf{(A) } \dfrac{8}{99} \qquad\textbf{(B) } \dfrac{2}{5} \qquad\textbf{(C) } \dfrac{9}{20} \qquad\textbf{(D) } \dfrac{1}{2} \qquad\textbf{(E) } \dfrac{9}{16}$

Solution

Problem 12

For how many integers $x$ is the point $(x,-x)$ inside or on the circle of radius $10$ centered at $(5,5)$?

$\textbf{(A) } 11 \qquad\textbf{(B) } 12 \qquad\textbf{(C) } 13 \qquad\textbf{(D) } 14 \qquad\textbf{(E) } 15$

Solution

Problem 13

The line $12x+5y=60$ forms a triangle with the coordinate axes. What is the sum of the lengths of the altitudes of this triangle?

$\textbf{(A) } 20 \qquad\textbf{(B) } \dfrac{360}{17} \qquad\textbf{(C) } \dfrac{107}{5} \qquad\textbf{(D) } \dfrac{43}{2} \qquad\textbf{(E) } \dfrac{281}{13}$

Solution

Problem 14

Let $a$, $b$, and $c$ be three distinct one-digit numbers. What is the maximum value of the sum of the roots of the equation $(x-a)(x-b)+(x-b)(x-c)=0$?

$\textbf{(A) } 15 \qquad\textbf{(B) } 15.5 \qquad\textbf{(C) } 16 \qquad\textbf{(D) } 16.5 \qquad\textbf{(E) } 17$

Solution

Problem 15

The town of Hamlet has $3$ people for each horse, $4$ sheep for each cow, and $3$ ducks for each person. Which of the following could not possibly be the total number of people, horses, sheep, cows, and ducks in Hamlet?

$\textbf{(A) } 41 \qquad\textbf{(B) } 47 \qquad\textbf{(C) } 59 \qquad\textbf{(D) } 61 \qquad\textbf{(E) } 66$

Solution

Problem 16

Al, Bill, and Cal will each randomly be assigned a whole number from $1$ to $10$, inclusive, with no two of them getting the same number. What is the probability that Al's number will be a whole number multiple of Bill's and Bill's number will be a whole number multiple of Cal's?

$\textbf{(A) } \dfrac{9}{1000} \qquad\textbf{(B) } \dfrac{1}{90} \qquad\textbf{(C) } \dfrac{1}{80} \qquad\textbf{(D) } \dfrac{1}{72} \qquad\textbf{(E) } \dfrac{2}{121}$

Solution

Problem 17

The centers of the faces of the right rectangular prism shown below are joined to create an octahedron, What is the volume of the octahedron?

[asy]import three; size(2inch); currentprojection=orthographic(4,2,2); draw((0,0,0)--(0,0,3),dashed); draw((0,0,0)--(0,4,0),dashed); draw((0,0,0)--(5,0,0),dashed); draw((5,4,3)--(5,0,3)--(5,0,0)--(5,4,0)--(0,4,0)--(0,4,3)--(0,0,3)--(5,0,3)); draw((0,4,3)--(5,4,3)--(5,4,0)); label("3",(5,0,3)--(5,0,0),W); label("4",(5,0,0)--(5,4,0),S); label("5",(5,4,0)--(0,4,0),SE); [/asy]

$\textbf{(A) } \dfrac{75}{12} \qquad\textbf{(B) } 10 \qquad\textbf{(C) } 12 \qquad\textbf{(D) } 10\sqrt2 \qquad\textbf{(E) } 15$

Solution

Problem 18

Johann has $64$ fair coins. He flips all the coins. Any coin that lands on tails is tossed again. Coins that land on tails on the second toss are tossed a third time. What is the expected number of coins that are now heads?

$\textbf{(A) } 32 \qquad\textbf{(B) } 40 \qquad\textbf{(C) } 48 \qquad\textbf{(D) } 56 \qquad\textbf{(E) } 64$

Solution

Problem 19

In $\triangle{ABC}$, $\angle{C} = 90^{\circ}$ and $AB = 12$. Squares $ABXY$ and $ACWZ$ are constructed outside of the triangle. The points $X, Y, Z$, and $W$ lie on a circle. What is the perimeter of the triangle?

$\textbf{(A)}\ 12+9\sqrt{3}\qquad\textbf{(B)}\ 18+6\sqrt{3}\qquad\textbf{(C)}\ 12+12\sqrt{2}\qquad\textbf{(D)}\ 30\qquad\textbf{(E)}\ 32$

Solution

Problem 20

Erin the ant starts at a given corner of a cube and crawls along exactly $7$ edges in such a way that she visits every corner exactly once and then finds that she is unable to return along an edge to her starting point. How many paths are there meeting these conditions?

$\textbf{(A) }\text{6}\qquad\textbf{(B) }\text{9}\qquad\textbf{(C) }\text{12}\qquad\textbf{(D) }\text{18}\qquad\textbf{(E) }\text{24}$

Solution

Problem 21

Cozy the Cat and Dash the Dog are going up a staircase with a certain number of steps. However, instead of walking up the steps one at a time, both Cozy and Dash jump. Cozy goes two steps up with each jump (though if necessary, he will just jump the last step). Dash goes five steps up with each jump (though if necessary, he will just jump the last steps if there are fewer than 5 steps left). Suppose the Dash takes 19 fewer jumps than Cozy to reach the top of the staircase. Let $s$ denote the sum of all possible numbers of steps this staircase can have. What is the sum of the digits of $s$?

$\textbf{(A) } 9 \qquad\textbf{(B) } 11 \qquad\textbf{(C) } 12 \qquad\textbf{(D) } 13 \qquad\textbf{(E) } 15$

Solution

Problem 22

In the figure shown below, $ABCDE$ is a regular pentagon and $AG=1$. What is $FG+JH+CD$? [asy]import cse5;pathpen=black;pointpen=black; size(2inch); pair A=dir(90), B=dir(18), C=dir(306), D=dir(234), E=dir(162); D(MP("A",A,A)--MP("B",B,B)--MP("C",C,C)--MP("D",D,D)--MP("E",E,E)--cycle,linewidth(1.5)); D(A--C--E--B--D--cycle); pair F=IP(A--D,B--E), G=IP(B--E,C--A), H=IP(C--A,B--D), I=IP(D--B,E--C), J=IP(C--E,D--A); D(MP("F",F,dir(126))--MP("I",I,dir(270))--MP("G",G,dir(54))--MP("J",J,dir(198))--MP("H",H,dir(342))--cycle); [/asy]

$\textbf{(A) } 3 \qquad\textbf{(B) } 12-4\sqrt5 \qquad\textbf{(C) } \dfrac{5+2\sqrt5}{3} \qquad\textbf{(D) } 1+\sqrt5 \qquad\textbf{(E) } \dfrac{11+11\sqrt5}{10}$

Solution

Problem 23

Let $n$ be a positive integer greater than 4 such that the decimal representation of $n!$ ends in $k$ zeros and the decimal representation of $(2n)!$ ends in $3k$ zeros. Let $s$ denote the sum of the four least possible values of $n$. What is the sum of the digits of $s$?

$\textbf{(A) }7\qquad\textbf{(B) }8\qquad\textbf{(C) }9\qquad\textbf{(D) }10\qquad\textbf{(E) }11$

Solution

Problem 24

Aaron the ant walks on the coordinate plane according to the following rules. He starts at the origin $p_0=(0,0)$ facing to the east and walks one unit, arriving at $p_1=(1,0)$. For $n=1,2,3,\dots$, right after arriving at the point $p_n$, if Aaron can turn $90^\circ$ left and walk one unit to an unvisited point $p_{n+1}$, he does that. Otherwise, he walks one unit straight ahead to reach $p_{n+1}$. Thus the sequence of points continues $p_2=(1,1), p_3=(0,1), p_4=(-1,1), p_5=(-1,0)$, and so on in a counterclockwise spiral pattern. What is $p_{2015}$?

$\textbf{(A) } (-22,-13)\qquad\textbf{(B) } (-13,-22)\qquad\textbf{(C) } (-13,22)\qquad\textbf{(D) } (13,-22)\qquad\textbf{(E) } (22,-13)$

Solution

Problem 25

A rectangular box measures $a \times b \times c$, where $a,$ $b,$ and $c$ are integers and $1 \leq a \leq b \leq c$. The volume and surface area of the box are numerically equal. How many ordered triples $(a,b,c)$ are possible?

$\textbf{(A) }4\qquad\textbf{(B) }10\qquad\textbf{(C) }12\qquad\textbf{(D) }21\qquad\textbf{(E) }26$

Solution

See also

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