Difference between revisions of "2018 AMC 10B Problems"

Revision as of 22:08, 15 November 2023

2018 AMC 10B (Answer Key) Printable versions: Wiki • AoPS Resources • PDF
Instructions 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. 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. No aids are permitted other than scratch paper, graph paper, ruler, compass, protractor and erasers (and calculators that are accepted for use on the SAT if before 2006. No problems on the test will require the use of a calculator). Figures are not necessarily drawn to scale. 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

Kate bakes a $20$ -inch by $18$ -inch pan of cornbread. The cornbread is cut into pieces that measure $2$ inches by $2$ inches. How many pieces of cornbread does the pan contain?

$\textbf{(A) } 90 \qquad \textbf{(B) } 100 \qquad \textbf{(C) } 180 \qquad \textbf{(D) } 200 \qquad \textbf{(E) } 360$

Solution

Problem 2

Sam drove $96$ miles in $90$ minutes. His average speed during the first $30$ minutes was $60$ mph (miles per hour), and his average speed during the second $30$ minutes was $65$ mph. What was his average speed, in mph, during the last $30$ minutes?

$\textbf{(A) } 64 \qquad \textbf{(B) } 65 \qquad \textbf{(C) } 66 \qquad \textbf{(D) } 67 \qquad \textbf{(E) } 68$

Solution

Problem 3

In the expression $\left(\underline{\qquad}\times\underline{\qquad}\right)+\left(\underline{\qquad}\times\underline{\qquad}\right)$ each blank is to be filled in with one of the digits $1,2,3,$ or $4,$ with each digit being used once. How many different values can be obtained?

$\textbf{(A) }2 \qquad \textbf{(B) }3 \qquad \textbf{(C) }4 \qquad \textbf{(D) }6 \qquad \textbf{(E) }24 \qquad$

Solution

Problem 4

A three-dimensional rectangular box with dimensions $X$ , $Y$ , and $Z$ has faces whose surface areas are $24, 24, 48, 48, 72,$ and $72$ square units. What is $X+Y+Z$ ?

$\textbf{(A) }18 \qquad \textbf{(B) }22 \qquad \textbf{(C) }24 \qquad \textbf{(D) }30 \qquad \textbf{(E) }36 \qquad$

Solution

Problem 5

How many subsets of $\{2,3,4,5,6,7,8,9\}$ contain at least one prime number?

$\textbf{(A) }128 \qquad \textbf{(B) }192 \qquad \textbf{(C) }224 \qquad \textbf{(D) }240 \qquad \textbf{(E) }256 \qquad$

Solution

Problem 6

A box contains $5$ chips, numbered $1, 2, 3, 4,$ and $5$ . Chips are drawn randomly one at a time without replacement until the sum of the values drawn exceeds $4$ . What is the probability that $3$ draws are required?

$\textbf{(A) }\frac{1}{15} \qquad \textbf{(B) }\frac{1}{10} \qquad \textbf{(C) }\frac{1}{6} \qquad \textbf{(D) }\frac{1}{5} \qquad \textbf{(E) }\frac{1}{4} \qquad$

Solution

Problem 7

In the figure below, $N$ congruent semicircles are drawn along a diameter of a large semicircle, with their diameters covering the diameter of the large semicircle with no overlap. Let $A$ be the combined area of the small semicircles and $B$ be the area of the region inside the large semicircle but outside the small semicircles. The ratio $A:B$ is $1:18$ . What is $N$ ?

$[asy] draw((0,0)--(18,0)); draw(arc((9,0),9,0,180)); filldraw(arc((1,0),1,0,180)--cycle,gray(0.8)); filldraw(arc((3,0),1,0,180)--cycle,gray(0.8)); filldraw(arc((5,0),1,0,180)--cycle,gray(0.8)); filldraw(arc((7,0),1,0,180)--cycle,gray(0.8)); label("...",(9,0.5)); filldraw(arc((11,0),1,0,180)--cycle,gray(0.8)); filldraw(arc((13,0),1,0,180)--cycle,gray(0.8)); filldraw(arc((15,0),1,0,180)--cycle,gray(0.8)); filldraw(arc((17,0),1,0,180)--cycle,gray(0.8)); [/asy]$

$\textbf{(A) }16 \qquad \textbf{(B) }17 \qquad \textbf{(C) }18 \qquad \textbf{(D) }19 \qquad \textbf{(E) }36 \qquad$

Solution

Problem 8

Sara makes a staircase out of toothpicks as shown:

$[asy] size(150); defaultpen(linewidth(0.8)); path h = ellipse((0.5,0),0.45,0.015), v = ellipse((0,0.5),0.015,0.45); for(int i=0;i<=2;i=i+1) { for(int j=0;j<=3-i;j=j+1) { filldraw(shift((i,j))*h,black); filldraw(shift((j,i))*v,black); } } [/asy]$

This is a $3$ -step staircase and uses $18$ toothpicks. How many steps would be in a staircase that used $180$ toothpicks?

$\textbf{(A) }10 \qquad \textbf{(B) }11 \qquad \textbf{(C) }12 \qquad \textbf{(D) }24 \qquad \textbf{(E) }30 \qquad$

Solution

Problem 9

The faces of each of $7$ standard dice are labeled with the integers from $1$ to $6$ . Let $p$ be the probability that when all $7$ dice are rolled, the sum of the numbers on the top faces is $10$ . What other sum occurs with the same probability $p$ ?

$\textbf{(A) }13 \qquad \textbf{(B) }26 \qquad \textbf{(C) }32 \qquad \textbf{(D) }39 \qquad \textbf{(E) }42 \qquad$

Solution

Problem 10

In the rectangular parallelepiped shown, $AB=3$ , $BC=1$ , and $CG=2$ . Point $M$ is the midpoint of $\overline{FG}$ . What is the volume of the rectangular pyramid with base $BCHE$ and apex $M$ ?

$[asy] size(250); defaultpen(fontsize(10pt)); pair A =origin; pair B = (4.75,0); pair E1=(0,3); pair F = (4.75,3); pair G = (5.95,4.2); pair C = (5.95,1.2); pair D = (1.2,1.2); pair H= (1.2,4.2); pair M = ((4.75+5.95)/2,3.6); draw(E1--M--H--E1--A--B--E1--F--B--M--C--G--H); draw(B--C); draw(F--G); draw(A--D--H--C--D,dashed); label("$A$",A,SW); label("$B$",B,SE); label("$C$",C,E); label("$D$",D,W); label("$E$",E1,W); label("$F$",F,SW); label("$G$",G,NE); label("$H$",H,NW); label("$M$",M,N); dot(A); dot(B); dot(E1); dot(F); dot(G); dot(C); dot(D); dot(H); dot(M); label("3",A/2+B/2,S); label("2",C/2+G/2,E); label("1",C/2+B/2,SE); [/asy]$

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

Solution

Problem 11

Which of the following expressions is never a prime number when $p$ is a prime number?

$\textbf{(A) }p^2+16 \qquad \textbf{(B) }p^2+24 \qquad \textbf{(C) }p^2+26 \qquad \textbf{(D) }p^2+46 \qquad \textbf{(E) }p^2+96 \qquad$

Solution

Problem 12

Line segment $\overline{AB}$ is a diameter of a circle with $AB=24$ . Point $C$ , not equal to $A$ or $B$ , lies on the circle. As point $C$ moves around the circle, the centroid (center of mass) of $\triangle{ABC}$ traces out a closed curve missing two points. To the nearest positive integer, what is the area of the region bounded by this curve?

$\textbf{(A) }25 \qquad \textbf{(B) }38 \qquad \textbf{(C) }50 \qquad \textbf{(D) }63 \qquad \textbf{(E) }75 \qquad$

Solution

Problem 13

How many of the first $2018$ numbers in the sequence $101, 1001, 10001, 100001, \dots$ are divisible by $101$ ?

$\textbf{(A) }253 \qquad \textbf{(B) }504 \qquad \textbf{(C) }505 \qquad \textbf{(D) }506 \qquad \textbf{(E) }1009 \qquad$

Solution

Problem 14

A list of $2018$ positive integers has a unique mode, which occurs exactly $10$ times. What is the least number of distinct values that can occur in the list?

$\textbf{(A) }202 \qquad \textbf{(B) }223 \qquad \textbf{(C) }224 \qquad \textbf{(D) }225 \qquad \textbf{(E) }234 \qquad$

Solution

Problem 15

A closed box with a square base is to be wrapped with a square sheet of wrapping paper. The box is centered on the wrapping paper with the vertices of the base lying on the midlines of the square sheet of paper, as shown in the figure on the left. The four corners of the wrapping paper are to be folded up over the sides and brought together to meet at the center of the top of the box, point $A$ in the figure on the right. The box has base length $w$ and height $h$ . What is the area of the sheet of wrapping paper?

$[asy] size(270pt); defaultpen(fontsize(10pt)); filldraw(((3,3)--(-3,3)--(-3,-3)--(3,-3)--cycle),lightgrey); dot((-3,3)); label("$A$",(-3,3),NW); draw((1,3)--(-3,-1),dashed+linewidth(.5)); draw((-1,3)--(3,-1),dashed+linewidth(.5)); draw((-1,-3)--(3,1),dashed+linewidth(.5)); draw((1,-3)--(-3,1),dashed+linewidth(.5)); draw((0,2)--(2,0)--(0,-2)--(-2,0)--cycle,linewidth(.5)); draw((0,3)--(0,-3),linetype("2.5 2.5")+linewidth(.5)); draw((3,0)--(-3,0),linetype("2.5 2.5")+linewidth(.5)); label('$w$',(-1,-1),SW); label('$w$',(1,-1),SE); draw((4.5,0)--(6.5,2)--(8.5,0)--(6.5,-2)--cycle); draw((4.5,0)--(8.5,0)); draw((6.5,2)--(6.5,-2)); label("$A$",(6.5,0),NW); dot((6.5,0)); [/asy]$

$\textbf{(A) } 2(w+h)^2 \qquad \textbf{(B) } \frac{(w+h)^2}2 \qquad \textbf{(C) } 2w^2+4wh \qquad \textbf{(D) } 2w^2 \qquad \textbf{(E) } w^2h$

Solution

Problem 16

Let $a_1,a_2,\dots,a_{2018}$ be a strictly increasing sequence of positive integers such that $a_1+a_2+\cdots+a_{2018}=2018^{2018}.$ What is the remainder when $a_1^3+a_2^3+\cdots+a_{2018}^3$ is divided by $6$ ?

$\textbf{(A) }0 \qquad \textbf{(B) }1 \qquad \textbf{(C) }2 \qquad \textbf{(D) }3 \qquad \textbf{(E) }4 \qquad$

Solution

Problem 17

In rectangle $PQRS$ , $PQ=8$ and $QR=6$ . Points $A$ and $B$ lie on $\overline{PQ}$ , points $C$ and $D$ lie on $\overline{QR}$ , points $E$ and $F$ lie on $\overline{RS}$ , and points $G$ and $H$ lie on $\overline{SP}$ so that $AP=BQ<4$ and the convex octagon $ABCDEFGH$ is equilateral. The length of a side of this octagon can be expressed in the form $k+m\sqrt{n}$ , where $k$ , $m$ , and $n$ are integers and $n$ is not divisible by the square of any prime. What is $k+m+n$ ?

$\textbf{(A) }1 \qquad \textbf{(B) }7 \qquad \textbf{(C) }21 \qquad \textbf{(D) }92 \qquad \textbf{(E) }106 \qquad$

Solution

Problem 18

Three young brother-sister pairs from different families need to take a trip in a van. These six children will occupy the second and third rows in the van, each of which has three seats. To avoid disruptions, siblings may not sit right next to each other in the same row, and no child may sit directly in front of his or her sibling. How many seating arrangements are possible for this trip?

$\textbf{(A) }60 \qquad \textbf{(B) }72 \qquad \textbf{(C) }92 \qquad \textbf{(D) }96 \qquad \textbf{(E) }120 \qquad$

Solution

Problem 19

Joey and Chloe and their daughter Zoe all have the same birthday. Joey is $1$ year older than Chloe, and Zoe is exactly $1$ year old today. Today is the first of the $9$ birthdays on which Chloe's age will be an integral multiple of Zoe's age. What will be the sum of the two digits of Joey's age the next time his age is a multiple of Zoe's age?

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

Solution

Problem 20

A function $f$ is defined recursively by $f(1)=f(2)=1$ and $f(n)=f(n-1)-f(n-2)+n$ for all integers $n \geq 3$ . What is $f(2018)$ ?

$\textbf{(A) } 2016 \qquad \textbf{(B) } 2017 \qquad \textbf{(C) } 2018 \qquad \textbf{(D) } 2019 \qquad \textbf{(E) } 2020$

Solution

Problem 21

Mary chose an even $4$ -digit number $n$ . She wrote down all the divisors of $n$ in increasing order from left to right: $1,2,\ldots,\dfrac{n}{2},n$ . At some moment Mary wrote $323$ as a divisor of $n$ . What is the smallest possible value of the next divisor written to the right of $323$ ?

$\textbf{(A) }324 \qquad \textbf{(B) }330 \qquad \textbf{(C) }340 \qquad \textbf{(D) }361 \qquad \textbf{(E) }646 \qquad$

Solution

Problem 22

Real numbers $x$ and $y$ are chosen independently and uniformly at random from the interval $[0,1]$ . Which of the following numbers is closest to the probability that $x,y,$ and $1$ are the side lengths of an obtuse triangle?

$\textbf{(A) }0.21 \qquad \textbf{(B) }0.25 \qquad \textbf{(C) }0.29 \qquad \textbf{(D) }0.50 \qquad \textbf{(E) }0.79 \qquad$

Solution

Problem 23

How many ordered pairs $(a, b)$ of positive integers satisfy the equation $a\cdot b + 63 = 20\cdot \text{lcm}(a, b) + 12\cdot\text{gcd}(a,b),$ where $\text{gcd}(a,b)$ denotes the greatest common divisor of $a$ and $b$ , and $\text{lcm}(a,b)$ denotes their least common multiple?

$\textbf{(A)} \text{ 0} \qquad \textbf{(B)} \text{ 2} \qquad \textbf{(C)} \text{ 4} \qquad \textbf{(D)} \text{ 6} \qquad \textbf{(E)} \text{ 8}$

Solution

Problem 24

Let $ABCDEF$ be a regular hexagon with side length $1$ . Denote by $X$ , $Y$ , and $Z$ the midpoints of sides $\overline {AB}$ , $\overline{CD}$ , and $\overline{EF}$ , respectively. What is the area of the convex hexagon whose interior is the intersection of the interiors of $\triangle ACE$ and $\triangle XYZ$ ?

$\textbf{(A)}\ \frac {3}{8}\sqrt{3} \qquad \textbf{(B)}\ \frac {7}{16}\sqrt{3} \qquad \textbf{(C)}\ \frac {15}{32}\sqrt{3} \qquad \textbf{(D)}\ \frac {1}{2}\sqrt{3} \qquad \textbf{(E)}\ \frac {9}{16}\sqrt{3}$

Solution

Problem 25

Let $\lfloor x \rfloor$ denote the greatest integer less than or equal to $x$ . How many real numbers $x$ satisfy the equation $x^2 + 10,000\lfloor x \rfloor = 10,000x$ ?

$\textbf{(A) } 197 \qquad \textbf{(B) } 198 \qquad \textbf{(C) } 199 \qquad \textbf{(D) } 200 \qquad \textbf{(E) } 201$

Solution

2018 AMC 10B (Problems • Answer Key • Resources)
Preceded by 2018 AMC 10A	Followed by 2019 AMC 10A
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All AMC 10 Problems and Solutions

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Difference between revisions of "2018 AMC 10B Problems"

Revision as of 22:08, 15 November 2023

Contents

Problem 1

Problem 2

Problem 3

Problem 4

Problem 5

Problem 6

Problem 7

Problem 8

Problem 9

Problem 10

Problem 11

Problem 12

Problem 13

Problem 14

Problem 15

Problem 16

Problem 17

Problem 18

Problem 19

Problem 20

Problem 21

Problem 22

Problem 23

Problem 24

Problem 25

See also

@@ Line 1: / Line 1: @@
+{{AMC10 Problems|year=2018|ab=B}}
 == Problem 1 ==
-Kate bakes 20-inch by 18-inch pan of cornbread. The cornbread is cut into pieces that measure 2 inches by 2 inches. How many pieces of cornbread does the pan contain?
+Kate bakes a <math>20</math>-inch by <math>18</math>-inch pan of cornbread. The cornbread is cut into pieces that measure <math>2</math> inches by <math>2</math> inches. How many pieces of cornbread does the pan contain?
 <math>
@@ Line 9: / Line 10: @@
 \textbf{(E) } 360
 </math>
+[[2018 AMC 10B Problems/Problem 1|Solution]]
 == Problem 2 ==
-Sam drove 96 miles in 90 minutes. His average speed during the first 30 minutes was 60 mph (miles per hour), and his average speed during the second 30 minutes was 65 mph. What was his average speed, in mph, during the last 30 minutes?
+Sam drove <math>96</math> miles in <math>90</math> minutes. His average speed during the first <math>30</math> minutes was <math>60</math> mph (miles per hour), and his average speed during the second <math>30</math> minutes was <math>65</math> mph. What was his average speed, in mph, during the last <math>30</math> minutes?
 <math>
@@ Line 21: / Line 24: @@
 </math>
+[[2018 AMC 10B Problems/Problem 2|Solution]]
 == Problem 3 ==
@@ Line 33: / Line 37: @@
 \textbf{(E) }24 \qquad
 </math>
+[[2018 AMC 10B Problems/Problem 3|Solution]]
 == Problem 4 ==
-A three-dimensional rectangular box with dimensions <math>X</math>, <math>Y</math>, and <math>Z</math> has faces whose surface areas are 24, 24, 48, 48, 72, and 72 square units. What is <math>X+Y+Z</math>?
+A three-dimensional rectangular box with dimensions <math>X</math>, <math>Y</math>, and <math>Z</math> has faces whose surface areas are <math>24, 24, 48, 48, 72,</math> and <math>72</math> square units. What is <math>X+Y+Z</math>?
 <math>
@@ Line 45: / Line 51: @@
 \textbf{(E) }36 \qquad
 </math>
+[[2018 AMC 10B Problems/Problem 4|Solution]]
 == Problem 5 ==
@@ Line 57: / Line 65: @@
 \textbf{(E) }256 \qquad
 </math>
+[[2018 AMC 10B Problems/Problem 5|Solution]]
 == Problem 6 ==
-A box contains 5 chips, numbered 1, 2, 3, 4, and 5. Chips are drawn randomly one at a time without replacement until the sum of the values drawn exceeds 4. What is the probability that 3 draws are required?
+A box contains <math>5</math> chips, numbered <math>1, 2, 3, 4,</math> and <math>5</math>. Chips are drawn randomly one at a time without replacement until the sum of the values drawn exceeds <math>4</math>. What is the probability that <math>3</math> draws are required?
 <math>
@@ Line 69: / Line 79: @@
 \textbf{(E) }\frac{1}{4} \qquad
 </math>
+[[2018 AMC 10B Problems/Problem 6|Solution]]
 == Problem 7 ==
-In the figure below, <math>N</math> congruent semicircles are drawn along a diameter of a large semicircle, with their diameters covering the diameter of the large semicircle with no overlap. Let <math>A</math> be the combined area of the small semicircles and <math>B</math> be the area of the region inside the large semicircle but outside the small semicircles. The ratio <math>A:B</math> is 1:18. What is <math>N</math> ?
+In the figure below, <math>N</math> congruent semicircles are drawn along a diameter of a large semicircle, with their diameters covering the diameter of the large semicircle with no overlap. Let <math>A</math> be the combined area of the small semicircles and <math>B</math> be the area of the region inside the large semicircle but outside the small semicircles. The ratio <math>A:B</math> is <math>1:18</math>. What is <math>N</math>?
@@ Line 84: / Line 96: @@
 \textbf{(E) }36 \qquad
 </math>
+[[2018 AMC 10B Problems/Problem 7|Solution]]
 == Problem 8 ==
-Sara makes a staircase out of toothpicks as shown: <br>
+Sara makes a staircase out of toothpicks as shown:
-<Insert diagram><br>
-This is a 3-step staircase and uses 18 toothpicks. How many steps would be in a staircase that used 180 toothpicks?
+<asy>
+size(150);
+defaultpen(linewidth(0.8));
+path h = ellipse((0.5,0),0.45,0.015), v = ellipse((0,0.5),0.015,0.45);
+for(int i=0;i<=2;i=i+1)
+{
+for(int j=0;j<=3-i;j=j+1)
+{
+filldraw(shift((i,j))*h,black);
+filldraw(shift((j,i))*v,black);
+}
+}
+</asy>
+This is a <math>3</math>-step staircase and uses <math>18</math> toothpicks. How many steps would be in a staircase that used <math>180</math> toothpicks?
 <math>
@@ Line 97: / Line 125: @@
 \textbf{(E) }30 \qquad
 </math>
+[[2018 AMC 10B Problems/Problem 8|Solution]]
 == Problem 9 ==
-The faces of each of 7 standard dice are labeled with the integers from 1 to 6. Let <math>p</math> be the probability that when all 7 dice are rolled, the sum of the numbers on the top faces is 10. What other sum occurs with the same probability <math>p</math> ?
+The faces of each of <math>7</math> standard dice are labeled with the integers from <math>1</math> to <math>6</math>. Let <math>p</math> be the probability that when all <math>7</math> dice are rolled, the sum of the numbers on the top faces is <math>10</math>. What other sum occurs with the same probability <math>p</math>?
 <math>
@@ Line 109: / Line 139: @@
 \textbf{(E) }42 \qquad
 </math>
+[[2018 AMC 10B Problems/Problem 9|Solution]]
 == Problem 10 ==
-In the rectangular parallelepiped shown, <math>AB=3</math>, <math>BC=1</math>, and <math>CG=2</math>. Point <math>M</math> is the midpoint of <math>\overline{FG}</math>. What is the volume of the rectangular pyramid with base <math>BCHE</math> and apex <math>M</math> ? <br>
+In the rectangular parallelepiped shown, <math>AB=3</math>, <math>BC=1</math>, and <math>CG=2</math>. Point <math>M</math> is the midpoint of <math>\overline{FG}</math>. What is the volume of the rectangular pyramid with base <math>BCHE</math> and apex <math>M</math>?
-<Insert Diagram>
-<br>
+<asy>
+size(250);
+defaultpen(fontsize(10pt));
+pair A =origin;
+pair B = (4.75,0);
+pair E1=(0,3);
+pair F = (4.75,3);
+pair G = (5.95,4.2);
+pair C = (5.95,1.2);
+pair D = (1.2,1.2);
+pair H= (1.2,4.2);
+pair M = ((4.75+5.95)/2,3.6);
+draw(E1--M--H--E1--A--B--E1--F--B--M--C--G--H);
+draw(B--C);
+draw(F--G);
+draw(A--D--H--C--D,dashed);
+label("$A$",A,SW);
+label("$B$",B,SE);
+label("$C$",C,E);
+label("$D$",D,W);
+label("$E$",E1,W);
+label("$F$",F,SW);
+label("$G$",G,NE);
+label("$H$",H,NW);
+label("$M$",M,N);
+dot(A);
+dot(B);
+dot(E1);
+dot(F);
+dot(G);
+dot(C);
+dot(D);
+dot(H);
+dot(M);
+label("3",A/2+B/2,S);
+label("2",C/2+G/2,E);
+label("1",C/2+B/2,SE);
+</asy>
 <math>
@@ Line 123: / Line 192: @@
 \textbf{(E) }2 \qquad
 </math>
+[[2018 AMC 10B Problems/Problem 10|Solution]]
 == Problem 11 ==
@@ Line 135: / Line 206: @@
 \textbf{(E) }p^2+96 \qquad
 </math>
+[[2018 AMC 10B Problems/Problem 11|Solution]]
 == Problem 12 ==
@@ Line 147: / Line 220: @@
 \textbf{(E) }75 \qquad
 </math>
+[[2018 AMC 10B Problems/Problem 12|Solution]]
 == Problem 13 ==
@@ Line 159: / Line 234: @@
 \textbf{(E) }1009 \qquad
 </math>
+[[2018 AMC 10B Problems/Problem 13|Solution]]
 == Problem 14 ==
@@ Line 171: / Line 248: @@
 \textbf{(E) }234 \qquad
 </math>
+[[2018 AMC 10B Problems/Problem 14|Solution]]
 == Problem 15 ==
@@ Line 198: / Line 277: @@
 <math>\textbf{(A) } 2(w+h)^2 \qquad \textbf{(B) } \frac{(w+h)^2}2 \qquad \textbf{(C) } 2w^2+4wh \qquad \textbf{(D) } 2w^2 \qquad \textbf{(E) } w^2h </math>
+[[2018 AMC 10B Problems/Problem 15|Solution]]
 == Problem 16 ==
@@ Line 210: / Line 291: @@
 \textbf{(E) }4 \qquad
 </math>
+[[2018 AMC 10B Problems/Problem 16|Solution]]
 == Problem 17 ==
@@ Line 222: / Line 305: @@
 \textbf{(E) }106 \qquad
 </math>
+[[2018 AMC 10B Problems/Problem 17|Solution]]
 == Problem 18 ==
-Three young brother-sister pairs from different families need tot take a trip in a van. These six children will occupy the second and third rows in the van, each of which has three seats. To avoid disruptions, siblings may not sit right next to each other in the same row, and no child may sit directly in front of his or her sibling. How many seating arrangements are possible for this trip?
+Three young brother-sister pairs from different families need to take a trip in a van. These six children will occupy the second and third rows in the van, each of which has three seats. To avoid disruptions, siblings may not sit right next to each other in the same row, and no child may sit directly in front of his or her sibling. How many seating arrangements are possible for this trip?
 <math>
@@ Line 232: / Line 317: @@
 \textbf{(C) }92 \qquad
 \textbf{(D) }96 \qquad
-\textbf{(E) }160 \qquad
+\textbf{(E) }120 \qquad
 </math>
+[[2018 AMC 10B Problems/Problem 18|Solution]]
 == Problem 19 ==
-Joey and Chloe and their daughter Zoe all have the same birthday. Joey is 1 year older than Chloe, and Zoe is exactly 1 year old today. Today is the first of the 9 birthdays on which Chloe's age will be an integral multiple of Zoe's age. What will be the sum of the two digits of Joey's age the next time his age is a multiple of Zoe's age?
+Joey and Chloe and their daughter Zoe all have the same birthday. Joey is <math>1</math> year older than Chloe, and Zoe is exactly <math>1</math> year old today. Today is the first of the <math>9</math> birthdays on which Chloe's age will be an integral multiple of Zoe's age. What will be the sum of the two digits of Joey's age the next time his age is a multiple of Zoe's age?
 <math>
@@ Line 247: / Line 334: @@
 </math>
-== Problem 20 ==
+[[2018 AMC 10B Problems/Problem 19|Solution]]
+== Problem 20==
 A function <math>f</math> is defined recursively by <math>f(1)=f(2)=1</math> and <cmath>f(n)=f(n-1)-f(n-2)+n</cmath>for all integers <math>n \geq 3</math>. What is <math>f(2018)</math>?
-<math>
+<math>\textbf{(A) } 2016 \qquad \textbf{(B) } 2017 \qquad \textbf{(C) } 2018 \qquad \textbf{(D) } 2019 \qquad \textbf{(E) } 2020</math>
-\textbf{(A) }2016 \qquad
-\textbf{(B) }2017 \qquad
+[[2018 AMC 10B Problems/Problem 20|Solution]]
-\textbf{(C) }2018 \qquad
-\textbf{(D) }2019 \qquad
-\textbf{(E) }2020 \qquad
-</math>
 == Problem 21 ==
-Mary chose an even <math>4</math>-digit number <math>n</math>. She wrote down all the divisors of <math>n</math> in increasing order from left to right: <math>1,2,...,\dfrac{n}{2},n</math>. At some moment Mary wrote <math>323</math> as a divisor of <math>n</math>. What is the smallest possible value of the next divisor written to the right of <math>323</math>.
+Mary chose an even <math>4</math>-digit number <math>n</math>. She wrote down all the divisors of <math>n</math> in increasing order from left to right: <math>1,2,\ldots,\dfrac{n}{2},n</math>. At some moment Mary wrote <math>323</math> as a divisor of <math>n</math>. What is the smallest possible value of the next divisor written to the right of <math>323</math>?
 <math>
@@ Line 270: / Line 354: @@
 \textbf{(E) }646 \qquad
 </math>
+[[2018 AMC 10B Problems/Problem 21|Solution]]
 == Problem 22 ==
@@ Line 282: / Line 368: @@
 \textbf{(E) }0.79 \qquad
 </math>
+[[2018 AMC 10B Problems/Problem 22|Solution]]
 == Problem 23 ==
-How many ordered pairs <math>(a, b)</math> of positive integers satisfy the equation <cmath>a\cdot b + 63 = 20\cdot \text{lcm}(a, b) + 12\cdot\text{gcd}(a,b),</cmath> where <math>\text{gcd}(a,b)</math> denotes the greatest common divisor of <math>a</math> and <math>b</math>, and <math>\text{lcm}(a,b)</math> denotes their least common multiple?
+How many ordered pairs <math>(a, b)</math> of positive integers satisfy the equation
+<cmath>a\cdot b + 63 = 20\cdot \text{lcm}(a, b) + 12\cdot\text{gcd}(a,b),</cmath>
+where <math>\text{gcd}(a,b)</math> denotes the greatest common divisor of <math>a</math> and <math>b</math>, and <math>\text{lcm}(a,b)</math> denotes their least common multiple?
-<math>
+<math>\textbf{(A)} \text{ 0} \qquad \textbf{(B)} \text{ 2} \qquad \textbf{(C)} \text{ 4} \qquad \textbf{(D)} \text{ 6} \qquad \textbf{(E)} \text{ 8}</math>
-\textbf{(A) }0 \qquad
-\textbf{(B) }2 \qquad
+[[2018 AMC 10B Problems/Problem 23|Solution]]
-\textbf{(C) }4 \qquad
-\textbf{(D) }6 \qquad
+== Problem 24 ==
-\textbf{(E) }8 \qquad
-</math>
+Let <math>ABCDEF</math> be a regular hexagon with side length <math>1</math>. Denote by <math>X</math>, <math>Y</math>, and <math>Z</math> the midpoints of sides <math>\overline {AB}</math>, <math>\overline{CD}</math>, and <math>\overline{EF}</math>, respectively. What is the area of the convex hexagon whose interior is the intersection of the interiors of <math>\triangle ACE</math> and <math>\triangle XYZ</math>?
+<math>\textbf{(A)}\ \frac {3}{8}\sqrt{3} \qquad \textbf{(B)}\ \frac {7}{16}\sqrt{3} \qquad \textbf{(C)}\ \frac {15}{32}\sqrt{3} \qquad  \textbf{(D)}\ \frac {1}{2}\sqrt{3} \qquad \textbf{(E)}\ \frac {9}{16}\sqrt{3} </math>
+[[2018 AMC 10B Problems/Problem 24|Solution]]
+== Problem 25 ==
+Let <math>\lfloor x \rfloor</math> denote the greatest integer less than or equal to <math>x</math>. How many real numbers <math>x</math> satisfy the equation <math>x^2 + 10,000\lfloor x \rfloor = 10,000x</math>?
+<math>\textbf{(A) } 197 \qquad \textbf{(B) } 198 \qquad \textbf{(C) } 199 \qquad \textbf{(D) } 200 \qquad \textbf{(E) } 201</math>
+[[2018 AMC 10B Problems/Problem 25|Solution]]
+==See also==
+{{AMC10 box|year=2018|ab=B|before=[[2018 AMC 10A]]|after=[[2019 AMC 10A]]}}
+* [[AMC 10]]
+* [[AMC 10 Problems and Solutions]]
+* [[Mathematics competitions]]
+* [[Mathematics competition resources]]
+{{MAA Notice}}