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Difference between revisions of "2014 AMC 12A Problems"

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{{AMC12 Problems|year=2014|ab=A}}
 +
 +
 
==Problem 1==
 
==Problem 1==
 
What is <math>10 \cdot \left(\tfrac{1}{2} + \tfrac{1}{5} + \tfrac{1}{10}\right)^{-1}?</math>
 
What is <math>10 \cdot \left(\tfrac{1}{2} + \tfrac{1}{5} + \tfrac{1}{10}\right)^{-1}?</math>
  
−
<math> \textbf{(A)}\ 3\qquad\textbf{(B)}\ 8\qquad\textbf{(C)}\ \frac{25}{2}\qquad\textbf{(D)}}\ \frac{170}{3}\qquad\textbf{(E)}\ 170</math>
+
<math> \textbf{(A)}\ 3\qquad\textbf{(B)}\ 8\qquad\textbf{(C)}\ \frac{25}{2}\qquad\textbf{(D)}\ \frac{170}{3}\qquad\textbf{(E)}\ 170</math>
  
−
[[2014 AMC 12A  Problems/Problem 1|Solution]]
+
[[2014 AMC 10A Problems/Problem 1|Solution]]
  
 
==Problem 2==
 
==Problem 2==
−
At the theater children get in for half price.  The price for <math>5</math> adult tickets and <math>4</math> child tickets is <math>24.50</math>.  How much would <math>8</math> adult tickets and <math>6</math> child tickets cost?
+
At the theater children get in for half price.  The price for <math>5</math> adult tickets and <math>4</math> child tickets is <math>\$24.50</math>.  How much would <math>8</math> adult tickets and <math>6</math> child tickets cost?
  
−
<math>\textbf{(A) }35\qquad
+
<math>\textbf{(A) }\$35\qquad
−
\textbf{(B) }38.50\qquad
+
\textbf{(B) }\$38.50\qquad
−
\textbf{(C) }40\qquad
+
\textbf{(C) }\$40\qquad
−
\textbf{(D) }42\qquad
+
\textbf{(D) }\$42\qquad
−
\textbf{(E) }42.50</math>
+
\textbf{(E) }\$42.50</math>
  
 
[[2014 AMC 12A  Problems/Problem 2|Solution]]
 
[[2014 AMC 12A  Problems/Problem 2|Solution]]
Line 20: Line 23:
 
Walking down Jane Street, Ralph passed four houses in a row, each painted a different color. He passed the orange house before the red house, and he passed the blue house before the yellow house. The blue house was not next to the yellow house. How many orderings of the colored houses are possible?
 
Walking down Jane Street, Ralph passed four houses in a row, each painted a different color. He passed the orange house before the red house, and he passed the blue house before the yellow house. The blue house was not next to the yellow house. How many orderings of the colored houses are possible?
  
−
<math> \textbf{(A)}\ 2\qquad\textbf{(B)}\ 3\qquad\textbf{(C)}\ 4\qquad\textbf{(D)}}\ 5\qquad\textbf{(E)}\ 6</math>
+
<math> \textbf{(A)}\ 2\qquad\textbf{(B)}\ 3\qquad\textbf{(C)}\ 4\qquad\textbf{(D)}\ 5\qquad\textbf{(E)}\ 6</math>
  
−
[[2014 AMC 12A Problems/Problem 3|Solution]]
+
[[2014 AMC 10A Problems/Problem 4|Solution]]
  
 
==Problem 4==
 
==Problem 4==
 
Suppose that <math>a</math> cows give <math>b</math> gallons of milk in <math>c</math> days. At this rate, how many gallons of milk will <math>d</math> cows give in <math>e</math> days?
 
Suppose that <math>a</math> cows give <math>b</math> gallons of milk in <math>c</math> days. At this rate, how many gallons of milk will <math>d</math> cows give in <math>e</math> days?
  
−
<math> \textbf{(A)}\ \frac{bde}{ac}\qquad\textbf{(B)}\ \frac{ac}{bde}\qquad\textbf{(C)}\ \frac{abde}{c}\qquad\textbf{(D)}}\ \frac{bcde}{a}\qquad\textbf{(E)}\ \frac{abc}{de}</math>
+
<math> \textbf{(A)}\ \frac{bde}{ac}\qquad\textbf{(B)}\ \frac{ac}{bde}\qquad\textbf{(C)}\ \frac{abde}{c}\qquad\textbf{(D)}\ \frac{bcde}{a}\qquad\textbf{(E)}\ \frac{abc}{de}</math>
  
−
[[2014 AMC 12A Problems/Problem 4|Solution]]
+
[[2014 AMC 10A Problems/Problem 6|Solution]]
  
 
==Problem 5==
 
==Problem 5==
 
On an algebra quiz, <math>10\%</math> of the students scored <math>70</math> points, <math>35\%</math> scored <math>80</math> points, <math>30\%</math> scored <math>90</math> points, and the rest scored <math>100</math> points. What is the difference between the mean and median score of the students' scores on this quiz?
 
On an algebra quiz, <math>10\%</math> of the students scored <math>70</math> points, <math>35\%</math> scored <math>80</math> points, <math>30\%</math> scored <math>90</math> points, and the rest scored <math>100</math> points. What is the difference between the mean and median score of the students' scores on this quiz?
  
−
<math> \textbf{(A)}\ 1\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ 3\qquad\textbf{(D)}}\ 4\qquad\textbf{(E)}\ 5</math>
+
<math> \textbf{(A)}\ 1\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ 3\qquad\textbf{(D)}\ 4\qquad\textbf{(E)}\ 5</math>
  
−
[[2014 AMC 12A Problems/Problem 5|Solution]]
+
[[2014 AMC 10A Problems/Problem 5|Solution]]
  
 
==Problem 6==
 
==Problem 6==
Line 46: Line 49:
 
\textbf{(D) }99\qquad
 
\textbf{(D) }99\qquad
 
\textbf{(E) }110</math>
 
\textbf{(E) }110</math>
 +
 +
[[2014 AMC 12A  Problems/Problem 6|Solution]]
  
 
==Problem 7==
 
==Problem 7==
Line 55: Line 60:
 
\textbf{(D) }\sqrt[9]3\qquad
 
\textbf{(D) }\sqrt[9]3\qquad
 
\textbf{(E) }\sqrt[10]3\qquad</math>
 
\textbf{(E) }\sqrt[10]3\qquad</math>
 +
 +
[[2014 AMC 12A  Problems/Problem 7|Solution]]
  
 
==Problem 8==
 
==Problem 8==
 
A customer who intends to purchase an appliance has three coupons, only one of which may be used:
 
A customer who intends to purchase an appliance has three coupons, only one of which may be used:
  
−
Coupon 1: <math>10\%</math> off the listed price if the listed price is at least <math>50</math>
+
Coupon 1: <math>10\%</math> off the listed price if the listed price is at least <math>\$50</math>
  
−
Coupon 2: <math>20</math> dollars off the listed price if the listed price is at least <math>100</math>
+
Coupon 2: <math>\$20</math> off the listed price if the listed price is at least <math>\$100</math>
  
−
Coupon 3: <math>18\%</math> off the amount by which the listed price exceeds <math>100</math>
+
Coupon 3: <math>18\%</math> off the amount by which the listed price exceeds <math>\$100</math>
  
 
For which of the following listed prices will coupon <math>1</math> offer a greater price reduction than either coupon <math>2</math> or coupon <math>3</math>?
 
For which of the following listed prices will coupon <math>1</math> offer a greater price reduction than either coupon <math>2</math> or coupon <math>3</math>?
  
−
<math>\textbf{(A) }179.95\qquad
+
<math>\textbf{(A) }\$179.95\qquad
−
\textbf{(B) }199.95\qquad
+
\textbf{(B) }\$199.95\qquad
−
\textbf{(C) }219.95\qquad
+
\textbf{(C) }\$219.95\qquad
−
\textbf{(D) }239.95\qquad
+
\textbf{(D) }\$239.95\qquad
−
\textbf{(E) }259.95\qquad</math>
+
\textbf{(E) }\$259.95\qquad</math>
 +
 
 +
[[2014 AMC 10A  Problems/Problem 11|Solution]]
  
 
==Problem 9==
 
==Problem 9==
 
Five positive consecutive integers starting with <math>a</math> have average <math>b</math>. What is the average of <math>5</math> consecutive integers that start with <math>b</math>?
 
Five positive consecutive integers starting with <math>a</math> have average <math>b</math>. What is the average of <math>5</math> consecutive integers that start with <math>b</math>?
  
−
<math> \textbf{(A)}\ a+3\qquad\textbf{(B)}\ a+4\qquad\textbf{(C)}\ a+5\qquad\textbf{(D)}}\ a+6\qquad\textbf{(E)}\ a+7</math>
+
<math> \textbf{(A)}\ a+3\qquad\textbf{(B)}\ a+4\qquad\textbf{(C)}\ a+5\qquad\textbf{(D)}\ a+6\qquad\textbf{(E)}\ a+7</math>
 +
 
 +
[[2014 AMC 10A  Problems/Problem 10|Solution]]
  
 
==Problem 10==
 
==Problem 10==
Line 86: Line 97:
 
\textbf{(D) }\dfrac{\sqrt2}2\qquad
 
\textbf{(D) }\dfrac{\sqrt2}2\qquad
 
\textbf{(E) }\dfrac{\sqrt3}2</math>
 
\textbf{(E) }\dfrac{\sqrt3}2</math>
 +
 +
[[2014 AMC 12A  Problems/Problem 10|Solution]]
  
 
==Problem 11==
 
==Problem 11==
Line 95: Line 108:
 
\textbf{(D) }245\qquad
 
\textbf{(D) }245\qquad
 
\textbf{(E) }280\qquad</math>
 
\textbf{(E) }280\qquad</math>
 +
 +
[[2014 AMC 10A  Problems/Problem 15|Solution]]
  
 
==Problem 12==
 
==Problem 12==
Line 104: Line 119:
 
\textbf{(D) }2+\sqrt3\qquad
 
\textbf{(D) }2+\sqrt3\qquad
 
\textbf{(E) }4\qquad</math>
 
\textbf{(E) }4\qquad</math>
 +
 +
[[2014 AMC 12A  Problems/Problem 12|Solution]]
  
 
==Problem 13==
 
==Problem 13==
Line 113: Line 130:
 
\textbf{(D) }3120\qquad
 
\textbf{(D) }3120\qquad
 
\textbf{(E) }3125\qquad</math>
 
\textbf{(E) }3125\qquad</math>
 +
 +
[[2014 AMC 12A  Problems/Problem 13|Solution]]
  
 
==Problem 14==
 
==Problem 14==
 +
Let <math>a<b<c</math> be three integers such that <math>a,b,c</math> is an arithmetic progression and <math>a,c,b</math> is a geometric progression.  What is the smallest possible value of <math>c</math>?
 +
 +
<math>\textbf{(A) }-2\qquad
 +
\textbf{(B) }1\qquad
 +
\textbf{(C) }2\qquad
 +
\textbf{(D) }4\qquad
 +
\textbf{(E) }6\qquad</math>
 +
 +
[[2014 AMC 12A  Problems/Problem 14|Solution]]
  
 
==Problem 15==
 
==Problem 15==
 +
A five-digit palindrome is a positive integer with respective digits <math>abcba</math>, where <math>a</math> is non-zero.  Let <math>S</math> be the sum of all five-digit palindromes.  What is the sum of the digits of <math>S</math>?
 +
 +
<math>\textbf{(A) }9\qquad
 +
\textbf{(B) }18\qquad
 +
\textbf{(C) }27\qquad
 +
\textbf{(D) }36\qquad
 +
\textbf{(E) }45\qquad</math>
 +
 +
[[2014 AMC 12A  Problems/Problem 15|Solution]]
  
 
==Problem 16==
 
==Problem 16==
 +
The product <math>(8)(888\ldots 8)</math>, where the second factor has <math>k</math> digits, is an integer whose digits have a sum of <math>1000</math>.  What is <math>k</math>?
 +
 +
<math>\textbf{(A) }901\qquad
 +
\textbf{(B) }911\qquad
 +
\textbf{(C) }919\qquad
 +
\textbf{(D) }991\qquad
 +
\textbf{(E) }999\qquad</math>
 +
 +
[[2014 AMC 10A  Problems/Problem 20|Solution]]
  
 
==Problem 17==
 
==Problem 17==
 +
A <math>4\times 4\times h</math> rectangular box contains a sphere of radius <math>2</math> and eight smaller spheres of radius <math>1</math>.  The smaller spheres are each tangent to three sides of the box, and the larger sphere is tangent to each of the smaller spheres.  What is <math>h</math>?
 +
 +
<center><asy>
 +
import graph3;
 +
import solids;
 +
real h=2+2*sqrt(7);
 +
currentprojection=orthographic((0.75,-5,h/2+1),target=(2,2,h/2));
 +
currentlight=light(4,-4,4);
 +
draw((0,0,0)--(4,0,0)--(4,4,0)--(0,4,0)--(0,0,0)^^(4,0,0)--(4,0,h)--(4,4,h)--(0,4,h)--(0,4,0));
 +
draw(shift((1,3,1))*unitsphere,gray(0.85));
 +
draw(shift((3,3,1))*unitsphere,gray(0.85));
 +
draw(shift((3,1,1))*unitsphere,gray(0.85));
 +
draw(shift((1,1,1))*unitsphere,gray(0.85));
 +
draw(shift((2,2,h/2))*scale(2,2,2)*unitsphere,gray(0.85));
 +
draw(shift((1,3,h-1))*unitsphere,gray(0.85));
 +
draw(shift((3,3,h-1))*unitsphere,gray(0.85));
 +
draw(shift((3,1,h-1))*unitsphere,gray(0.85));
 +
draw(shift((1,1,h-1))*unitsphere,gray(0.85));
 +
draw((0,0,0)--(0,0,h)--(4,0,h)^^(0,0,h)--(0,4,h));
 +
</asy></center>
 +
 +
<math>\textbf{(A) }2+2\sqrt 7\qquad
 +
\textbf{(B) }3+2\sqrt 5\qquad
 +
\textbf{(C) }4+2\sqrt 7\qquad
 +
\textbf{(D) }4\sqrt 5\qquad
 +
\textbf{(E) }4\sqrt 7\qquad</math>
 +
 +
[[2014 AMC 12A  Problems/Problem 17|Solution]]
  
 
==Problem 18==
 
==Problem 18==
 +
The domain of the function <math>f(x)=\log_{\frac12}(\log_4(\log_{\frac14}(\log_{16}(\log_{\frac1{16}}x))))</math> is an interval of length <math>\tfrac mn</math>, where <math>m</math> and <math>n</math> are relatively prime positive integers.  What is <math>m+n</math>?
 +
 +
<math>\textbf{(A) }19\qquad
 +
\textbf{(B) }31\qquad
 +
\textbf{(C) }271\qquad
 +
\textbf{(D) }319\qquad
 +
\textbf{(E) }511\qquad</math>
 +
 +
[[2014 AMC 12A  Problems/Problem 18|Solution]]
  
 
==Problem 19==
 
==Problem 19==
 +
There are exactly <math>N</math> distinct rational numbers <math>k</math> such that <math>|k|<200</math> and <cmath>5x^2+kx+12=0</cmath> has at least one integer solution for <math>x</math>.  What is <math>N</math>?
 +
 +
<math>\textbf{(A) }6\qquad
 +
\textbf{(B) }12\qquad
 +
\textbf{(C) }24\qquad
 +
\textbf{(D) }48\qquad
 +
\textbf{(E) }78\qquad</math>
 +
 +
[[2014 AMC 12A  Problems/Problem 19|Solution]]
  
 
==Problem 20==
 
==Problem 20==
 +
In <math>\triangle BAC</math>, <math>\angle BAC=40^\circ</math>, <math>AB=10</math>, and <math>AC=6</math>.  Points <math>D</math> and <math>E</math> lie on <math>\overline{AB}</math> and <math>\overline{AC}</math> respectively.  What is the minimum possible value of <math>BE+DE+CD</math>?
 +
 +
<math>\textbf{(A) }6\sqrt 3+3\qquad
 +
\textbf{(B) }\dfrac{27}2\qquad
 +
\textbf{(C) }8\sqrt 3\qquad
 +
\textbf{(D) }14\qquad
 +
\textbf{(E) }3\sqrt 3+9\qquad</math>
 +
 +
[[2014 AMC 12A  Problems/Problem 20|Solution]]
  
 
==Problem 21==
 
==Problem 21==
 +
For every real number <math>x</math>, let <math>\lfloor x\rfloor</math> denote the greatest integer not exceeding <math>x</math>, and let <cmath>f(x)=\lfloor x\rfloor(2014^{x-\lfloor x\rfloor}-1).</cmath>  The set of all numbers <math>x</math> such that <math>1\leq x<2014</math> and <math>f(x)\leq 1</math> is a union of disjoint intervals.  What is the sum of the lengths of those intervals?
 +
 +
<math>\textbf{(A) }1\qquad
 +
\textbf{(B) }\dfrac{\log 2015}{\log 2014}\qquad
 +
\textbf{(C) }\dfrac{\log 2014}{\log 2013}\qquad
 +
\textbf{(D) }\dfrac{2014}{2013}\qquad
 +
\textbf{(E) }2014^{\frac1{2014}}\qquad</math>
 +
 +
[[2014 AMC 12A  Problems/Problem 21|Solution]]
  
 
==Problem 22==
 
==Problem 22==
 +
 +
The number <math>5^{867}</math> is between <math>2^{2013}</math> and <math>2^{2014}</math>.  How many pairs of integers <math>(m,n)</math> are there such that <math>1\leq m\leq 2012</math> and <cmath>5^n<2^m<2^{m+2}<5^{n+1}?</cmath>
 +
<math>\textbf{(A) }278\qquad
 +
\textbf{(B) }279\qquad
 +
\textbf{(C) }280\qquad
 +
\textbf{(D) }281\qquad
 +
\textbf{(E) }282\qquad</math>
 +
 +
[[2014 AMC 10A  Problems/Problem 25|Solution]]
  
 
==Problem 23==
 
==Problem 23==
 +
The fraction <cmath>\dfrac1{99^2}=0.\overline{b_{n-1}b_{n-2}\ldots b_2b_1b_0},</cmath> where <math>n</math> is the length of the period of the repeating decimal expansion.  What is the sum <math>b_0+b_1+\cdots+b_{n-1}</math>?
 +
 +
<math>\textbf{(A) }874\qquad
 +
\textbf{(B) }883\qquad
 +
\textbf{(C) }887\qquad
 +
\textbf{(D) }891\qquad
 +
\textbf{(E) }892\qquad</math>
 +
 +
[[2014 AMC 12A  Problems/Problem 23|Solution]]
  
 
==Problem 24==
 
==Problem 24==
 +
Let <math>f_0(x)=x+|x-100|-|x+100|</math>, and for <math>n\geq 1</math>, let <math>f_n(x)=|f_{n-1}(x)|-1</math>.  For how many values of <math>x</math> is <math>f_{100}(x)=0</math>?
 +
 +
<math>\textbf{(A) }299\qquad
 +
\textbf{(B) }300\qquad
 +
\textbf{(C) }301\qquad
 +
\textbf{(D) }302\qquad
 +
\textbf{(E) }303\qquad</math>
 +
 +
[[2014 AMC 12A  Problems/Problem 24|Solution]]
  
 
==Problem 25==
 
==Problem 25==
 +
The parabola <math>P</math> has focus <math>(0,0)</math> and goes through the points <math>(4,3)</math> and <math>(-4,-3)</math>.  For how many points <math>(x,y)\in P</math> with integer coordinates is it true that <math>|4x+3y|\leq 1000</math>?
 +
 +
<math>\textbf{(A) }38\qquad
 +
\textbf{(B) }40\qquad
 +
\textbf{(C) }42\qquad
 +
\textbf{(D) }44\qquad
 +
\textbf{(E) }46\qquad</math>
 +
 +
[[2014 AMC 12A  Problems/Problem 25|Solution]]
 +
 +
==See also==
 +
 +
{{AMC12 box|year=2014|ab=A|before=[[2013 AMC 12B Problems]]|after=[[2014 AMC 12B Problems]]}}
 +
 +
{{MAA Notice}}

Latest revision as of 18:04, 11 April 2024

2014 AMC 12A (Answer Key)
Printable versions: Wiki • AoPS Resources • PDF

Instructions

  1. This is a 25-question, multiple choice test. Each question is followed by answers marked A, B, C, D and E. Only one of these is correct.
  2. You will receive 6 points for each correct answer, 2.5 points for each problem left unanswered if the year is before 2006, 1.5 points for each problem left unanswered if the year is after 2006, and 0 points for each incorrect answer.
  3. No aids are permitted other than scratch paper, graph paper, ruler, compass, protractor and erasers (and calculators that are accepted for use on the test if before 2006. No problems on the test will require the use of a calculator).
  4. Figures are not necessarily drawn to scale.
  5. You will have 75 minutes working time to complete the test.
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25


Problem 1

What is $10 \cdot \left(\tfrac{1}{2} + \tfrac{1}{5} + \tfrac{1}{10}\right)^{-1}?$

$\textbf{(A)}\ 3\qquad\textbf{(B)}\ 8\qquad\textbf{(C)}\ \frac{25}{2}\qquad\textbf{(D)}\ \frac{170}{3}\qquad\textbf{(E)}\ 170$

Solution

Problem 2

At the theater children get in for half price. The price for $5$ adult tickets and $4$ child tickets is $$24.50$. How much would $8$ adult tickets and $6$ child tickets cost?

$\textbf{(A) }$35\qquad \textbf{(B) }$38.50\qquad \textbf{(C) }$40\qquad \textbf{(D) }$42\qquad \textbf{(E) }$42.50$

Solution

Problem 3

Walking down Jane Street, Ralph passed four houses in a row, each painted a different color. He passed the orange house before the red house, and he passed the blue house before the yellow house. The blue house was not next to the yellow house. How many orderings of the colored houses are possible?

$\textbf{(A)}\ 2\qquad\textbf{(B)}\ 3\qquad\textbf{(C)}\ 4\qquad\textbf{(D)}\ 5\qquad\textbf{(E)}\ 6$

Solution

Problem 4

Suppose that $a$ cows give $b$ gallons of milk in $c$ days. At this rate, how many gallons of milk will $d$ cows give in $e$ days?

$\textbf{(A)}\ \frac{bde}{ac}\qquad\textbf{(B)}\ \frac{ac}{bde}\qquad\textbf{(C)}\ \frac{abde}{c}\qquad\textbf{(D)}\ \frac{bcde}{a}\qquad\textbf{(E)}\ \frac{abc}{de}$

Solution

Problem 5

On an algebra quiz, $10\%$ of the students scored $70$ points, $35\%$ scored $80$ points, $30\%$ scored $90$ points, and the rest scored $100$ points. What is the difference between the mean and median score of the students' scores on this quiz?

$\textbf{(A)}\ 1\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ 3\qquad\textbf{(D)}\ 4\qquad\textbf{(E)}\ 5$

Solution

Problem 6

The difference between a two-digit number and the number obtained by reversing its digits is $5$ times the sum of the digits of either number. What is the sum of the two digit number and its reverse?

$\textbf{(A) }44\qquad \textbf{(B) }55\qquad \textbf{(C) }77\qquad \textbf{(D) }99\qquad \textbf{(E) }110$

Solution

Problem 7

The first three terms of a geometric progression are $\sqrt 3$, $\sqrt[3]3$, and $\sqrt[6]3$. What is the fourth term?

$\textbf{(A) }1\qquad \textbf{(B) }\sqrt[7]3\qquad \textbf{(C) }\sqrt[8]3\qquad \textbf{(D) }\sqrt[9]3\qquad \textbf{(E) }\sqrt[10]3\qquad$

Solution

Problem 8

A customer who intends to purchase an appliance has three coupons, only one of which may be used:

Coupon 1: $10\%$ off the listed price if the listed price is at least $$50$

Coupon 2: $$20$ off the listed price if the listed price is at least $$100$

Coupon 3: $18\%$ off the amount by which the listed price exceeds $$100$

For which of the following listed prices will coupon $1$ offer a greater price reduction than either coupon $2$ or coupon $3$?

$\textbf{(A) }$179.95\qquad \textbf{(B) }$199.95\qquad \textbf{(C) }$219.95\qquad \textbf{(D) }$239.95\qquad \textbf{(E) }$259.95\qquad$

Solution

Problem 9

Five positive consecutive integers starting with $a$ have average $b$. What is the average of $5$ consecutive integers that start with $b$?

$\textbf{(A)}\ a+3\qquad\textbf{(B)}\ a+4\qquad\textbf{(C)}\ a+5\qquad\textbf{(D)}\ a+6\qquad\textbf{(E)}\ a+7$

Solution

Problem 10

Three congruent isosceles triangles are constructed with their bases on the sides of an equilateral triangle of side length $1$. The sum of the areas of the three isosceles triangles is the same as the area of the equilateral triangle. What is the length of one of the two congruent sides of one of the isosceles triangles?

$\textbf{(A) }\dfrac{\sqrt3}4\qquad \textbf{(B) }\dfrac{\sqrt3}3\qquad \textbf{(C) }\dfrac23\qquad \textbf{(D) }\dfrac{\sqrt2}2\qquad \textbf{(E) }\dfrac{\sqrt3}2$

Solution

Problem 11

David drives from his home to the airport to catch a flight. He drives $35$ miles in the first hour, but realizes that he will be $1$ hour late if he continues at this speed. He increases his speed by $15$ miles per hour for the rest of the way to the airport and arrives $30$ minutes early. How many miles is the airport from his home?

$\textbf{(A) }140\qquad \textbf{(B) }175\qquad \textbf{(C) }210\qquad \textbf{(D) }245\qquad \textbf{(E) }280\qquad$

Solution

Problem 12

Two circles intersect at points $A$ and $B$. The minor arcs $AB$ measure $30^\circ$ on one circle and $60^\circ$ on the other circle. What is the ratio of the area of the larger circle to the area of the smaller circle?

$\textbf{(A) }2\qquad \textbf{(B) }1+\sqrt3\qquad \textbf{(C) }3\qquad \textbf{(D) }2+\sqrt3\qquad \textbf{(E) }4\qquad$

Solution

Problem 13

A fancy bed and breakfast inn has $5$ rooms, each with a distinctive color-coded decor. One day $5$ friends arrive to spend the night. There are no other guests that night. The friends can room in any combination they wish, but with no more than $2$ friends per room. In how many ways can the innkeeper assign the guests to the rooms?

$\textbf{(A) }2100\qquad \textbf{(B) }2220\qquad \textbf{(C) }3000\qquad \textbf{(D) }3120\qquad \textbf{(E) }3125\qquad$

Solution

Problem 14

Let $a<b<c$ be three integers such that $a,b,c$ is an arithmetic progression and $a,c,b$ is a geometric progression. What is the smallest possible value of $c$?

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

Solution

Problem 15

A five-digit palindrome is a positive integer with respective digits $abcba$, where $a$ is non-zero. Let $S$ be the sum of all five-digit palindromes. What is the sum of the digits of $S$?

$\textbf{(A) }9\qquad \textbf{(B) }18\qquad \textbf{(C) }27\qquad \textbf{(D) }36\qquad \textbf{(E) }45\qquad$

Solution

Problem 16

The product $(8)(888\ldots 8)$, where the second factor has $k$ digits, is an integer whose digits have a sum of $1000$. What is $k$?

$\textbf{(A) }901\qquad \textbf{(B) }911\qquad \textbf{(C) }919\qquad \textbf{(D) }991\qquad \textbf{(E) }999\qquad$

Solution

Problem 17

A $4\times 4\times h$ rectangular box contains a sphere of radius $2$ and eight smaller spheres of radius $1$. The smaller spheres are each tangent to three sides of the box, and the larger sphere is tangent to each of the smaller spheres. What is $h$?

[asy] import graph3; import solids; real h=2+2*sqrt(7); currentprojection=orthographic((0.75,-5,h/2+1),target=(2,2,h/2)); currentlight=light(4,-4,4); draw((0,0,0)--(4,0,0)--(4,4,0)--(0,4,0)--(0,0,0)^^(4,0,0)--(4,0,h)--(4,4,h)--(0,4,h)--(0,4,0)); draw(shift((1,3,1))*unitsphere,gray(0.85)); draw(shift((3,3,1))*unitsphere,gray(0.85)); draw(shift((3,1,1))*unitsphere,gray(0.85)); draw(shift((1,1,1))*unitsphere,gray(0.85)); draw(shift((2,2,h/2))*scale(2,2,2)*unitsphere,gray(0.85)); draw(shift((1,3,h-1))*unitsphere,gray(0.85)); draw(shift((3,3,h-1))*unitsphere,gray(0.85)); draw(shift((3,1,h-1))*unitsphere,gray(0.85)); draw(shift((1,1,h-1))*unitsphere,gray(0.85)); draw((0,0,0)--(0,0,h)--(4,0,h)^^(0,0,h)--(0,4,h)); [/asy]

$\textbf{(A) }2+2\sqrt 7\qquad \textbf{(B) }3+2\sqrt 5\qquad \textbf{(C) }4+2\sqrt 7\qquad \textbf{(D) }4\sqrt 5\qquad \textbf{(E) }4\sqrt 7\qquad$

Solution

Problem 18

The domain of the function $f(x)=\log_{\frac12}(\log_4(\log_{\frac14}(\log_{16}(\log_{\frac1{16}}x))))$ is an interval of length $\tfrac mn$, where $m$ and $n$ are relatively prime positive integers. What is $m+n$?

$\textbf{(A) }19\qquad \textbf{(B) }31\qquad \textbf{(C) }271\qquad \textbf{(D) }319\qquad \textbf{(E) }511\qquad$

Solution

Problem 19

There are exactly $N$ distinct rational numbers $k$ such that $|k|<200$ and \[5x^2+kx+12=0\] has at least one integer solution for $x$. What is $N$?

$\textbf{(A) }6\qquad \textbf{(B) }12\qquad \textbf{(C) }24\qquad \textbf{(D) }48\qquad \textbf{(E) }78\qquad$

Solution

Problem 20

In $\triangle BAC$, $\angle BAC=40^\circ$, $AB=10$, and $AC=6$. Points $D$ and $E$ lie on $\overline{AB}$ and $\overline{AC}$ respectively. What is the minimum possible value of $BE+DE+CD$?

$\textbf{(A) }6\sqrt 3+3\qquad \textbf{(B) }\dfrac{27}2\qquad \textbf{(C) }8\sqrt 3\qquad \textbf{(D) }14\qquad \textbf{(E) }3\sqrt 3+9\qquad$

Solution

Problem 21

For every real number $x$, let $\lfloor x\rfloor$ denote the greatest integer not exceeding $x$, and let \[f(x)=\lfloor x\rfloor(2014^{x-\lfloor x\rfloor}-1).\] The set of all numbers $x$ such that $1\leq x<2014$ and $f(x)\leq 1$ is a union of disjoint intervals. What is the sum of the lengths of those intervals?

$\textbf{(A) }1\qquad \textbf{(B) }\dfrac{\log 2015}{\log 2014}\qquad \textbf{(C) }\dfrac{\log 2014}{\log 2013}\qquad \textbf{(D) }\dfrac{2014}{2013}\qquad \textbf{(E) }2014^{\frac1{2014}}\qquad$

Solution

Problem 22

The number $5^{867}$ is between $2^{2013}$ and $2^{2014}$. How many pairs of integers $(m,n)$ are there such that $1\leq m\leq 2012$ and \[5^n<2^m<2^{m+2}<5^{n+1}?\] $\textbf{(A) }278\qquad \textbf{(B) }279\qquad \textbf{(C) }280\qquad \textbf{(D) }281\qquad \textbf{(E) }282\qquad$

Solution

Problem 23

The fraction \[\dfrac1{99^2}=0.\overline{b_{n-1}b_{n-2}\ldots b_2b_1b_0},\] where $n$ is the length of the period of the repeating decimal expansion. What is the sum $b_0+b_1+\cdots+b_{n-1}$?

$\textbf{(A) }874\qquad \textbf{(B) }883\qquad \textbf{(C) }887\qquad \textbf{(D) }891\qquad \textbf{(E) }892\qquad$

Solution

Problem 24

Let $f_0(x)=x+|x-100|-|x+100|$, and for $n\geq 1$, let $f_n(x)=|f_{n-1}(x)|-1$. For how many values of $x$ is $f_{100}(x)=0$?

$\textbf{(A) }299\qquad \textbf{(B) }300\qquad \textbf{(C) }301\qquad \textbf{(D) }302\qquad \textbf{(E) }303\qquad$

Solution

Problem 25

The parabola $P$ has focus $(0,0)$ and goes through the points $(4,3)$ and $(-4,-3)$. For how many points $(x,y)\in P$ with integer coordinates is it true that $|4x+3y|\leq 1000$?

$\textbf{(A) }38\qquad \textbf{(B) }40\qquad \textbf{(C) }42\qquad \textbf{(D) }44\qquad \textbf{(E) }46\qquad$

Solution

See also

2014 AMC 12A (Problems • Answer Key • Resources)
Preceded by
2013 AMC 12B Problems 		Followed by
2014 AMC 12B Problems
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All AMC 12 Problems and Solutions

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