Exam Complete
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AMC 10B 2018
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Topic Breakdown
Counting and Probability
Weak
0/8 · 0%
Geometry
Weak
0/6 · 0%
Number Theory
Weak
0/6 · 0%
Algebra
Weak
0/5 · 0%
Target your weak topics with focused practice.
Practice weak topics →Results by Question
1
Counting and Probability
Wrong
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?
\(90\)
Correct Answer
B
\(100\)
C
\(180\)
D
\(200\)
E
\(360\)
No explanation
2
Algebra
Wrong
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?
A
\(64\)
B
\(65\)
C
\(66\)
\(67\)
Correct Answer
E
\(68\)
No explanation
3
Counting and Probability
Wrong
In the expression
\((\underline{#h(2em)} \times \underline{#h(2em)}) + (\underline{#h(2em)} \times \underline{#h(2em)})\)
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?
A
\(2\)
\(3\)
Correct Answer
C
\(4\)
D
\(6\)
E
\(24\)
No explanation
4
Algebra
Wrong
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 \(X + Y + Z\)?
A
\(18\)
\(22\)
Correct Answer
C
\(24\)
D
\(30\)
E
\(36\)
No explanation
5
Counting and Probability
Wrong
How many subsets of \(\{ 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 \}\) contain at least
one prime number?
A
\(128\)
B
\(192\)
C
\(224\)
\(240\)
Correct Answer
E
\(256\)
No explanation
6
Counting and Probability
Wrong
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
\(\dfrac{1}{15}\)
B
\(\dfrac{1}{10}\)
C
\(\dfrac{1}{6}\)
\(\dfrac{1}{5}\)
Correct Answer
E
\(\dfrac{1}{4}\)
No explanation
7
Geometry
Wrong
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 \(N\)?
A
\(16\)
B
\(17\)
C
\(18\)
\(19\)
Correct Answer
E
\(36\)
No explanation
8
Algebra
Wrong
Sara makes a staircase out of toothpicks as shown: This is a \(3\)-step staircase
and uses \(18\) toothpicks. How many steps would be in a staircase that
used \(180\) toothpicks?
A
\(10\)
B
\(11\)
\(12\)
Correct Answer
D
\(24\)
E
\(30\)
No explanation
9
Counting and Probability
Wrong
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 \(p\)?
A
\(13\)
B
\(26\)
C
\(32\)
\(39\)
Correct Answer
E
\(42\)
No explanation
10
Geometry
Wrong
In the rectangular parallelepiped shown, \(A B = 3\), \(B C = 1\), and
\(C G = 2\). Point \(M\) is the midpoint of \(\overline{F G}\). What is the
volume of the rectangular pyramid with base \(B C H E\) and apex \(M\)?
A
\(1\)
B
\(\dfrac{4}{3}\)
C
\(\dfrac{3}{2}\)
D
\(\dfrac{5}{3}\)
\(2\)
Correct Answer
No explanation
11
Number Theory
Wrong
Which of the following expressions is never a prime number when \(p\) is a
prime number?
A
\(p^2 + 16\)
B
\(p^2 + 24\)
\(p^2 + 26\)
Correct Answer
D
\(p^2 + 46\)
E
\(p^2 + 96\)
No explanation
12
Geometry
Wrong
Line segment \(\overline{A B}\) is a diameter of a circle with \(A B = 24\).
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
\(\triangle A B C\) traces out a closed curve missing two points.
To the nearest positive integer, what is the area of the region bounded
by this curve?
A
\(25\)
B
\(38\)
\(50\)
Correct Answer
D
\(63\)
E
\(75\)
No explanation
13
Number Theory
Wrong
How many of the first \(2018\) numbers in the sequence
\(101 , 1001 , 10001 , 100001 , \cdots\) are divisible by \(101\)?
A
\(253\)
B
\(504\)
\(505\)
Correct Answer
D
\(506\)
E
\(1009\)
No explanation
14
Counting and Probability
Wrong
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?
A
\(202\)
B
\(223\)
C
\(224\)
\(225\)
Correct Answer
E
\(234\)
No explanation
15
Geometry
Wrong
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?
\(2 ( w + h )^2\)
Correct Answer
B
\(\dfrac{( w + h )^2}{2}\)
C
\(2 w^2 + 4 w h\)
D
\(2 w^2\)
E
\(w^2 h\)
No explanation
16
Number Theory
Wrong
Let \(a_1 , a_2 , \cdots , a_2018\) be a strictly increasing sequence of
positive integers such that
\( a_1 + a_2 + \cdots.c + a_2018 = 2018^2018 . \) What is the remainder
when \(a_1^3 + a_2^3 + \cdots.c + a_2018^3\) is divided by \(6\)?
A
\(0\)
B
\(1\)
C
\(2\)
D
\(3\)
\(4\)
Correct Answer
No explanation
17
Geometry
Wrong
In rectangle \(P Q R S\), \(P Q = 8\) and \(Q R = 6\). Points \(A\) and \(B\) lie
on \(\overline{P Q}\), points \(C\) and \(D\) lie on \(\overline{Q R}\), points
\(E\) and \(F\) lie on \(\overline{R S}\), and points \(G\) and \(H\) lie on
\(\overline{S P}\) so that \(A P = B Q < 4\) and the convex octagon
\(A B C D E F G H\) 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 \(k + m + n\)?
A
\(1\)
\(7\)
Correct Answer
C
\(21\)
D
\(92\)
E
\(106\)
No explanation
18
Counting and Probability
Wrong
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?
A
\(60\)
B
\(72\)
C
\(92\)
\(96\)
Correct Answer
E
\(120\)
No explanation
19
Number Theory
Wrong
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?
A
\(7\)
B
\(8\)
C
\(9\)
D
\(10\)
\(11\)
Correct Answer
No explanation
20
Algebra
Wrong
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 \(f ( 2018 )\)?
A
\(2016\)
\(2017\)
Correct Answer
C
\(2018\)
D
\(2019\)
E
\(2020\)
No explanation
21
Number Theory
Wrong
Mary chose an even \(4\)-digit number \(n\). She wrote down all the divisors
of \(n\) in increasing order from left to right:
\(1 , 2 , \cdots , \dfrac{n}{2} , n\). 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 \(323\)?
A
\(324\)
B
\(330\)
\(340\)
Correct Answer
D
\(361\)
E
\(646\)
No explanation
22
Counting and Probability
Wrong
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?
A
\(0.21\)
B
\(0.25\)
\(0.29\)
Correct Answer
D
\(0.50\)
E
\(0.79\)
No explanation
23
Number Theory
Wrong
How many ordered pairs \(( a , b )\) of positive integers satisfy the
equation
\( a \cdot b + 63 = 20 \cdot \text{\operatorname{lcm}} ( a , b ) + 12 \cdot \text{\gcd} ( a , b ) , \)
where \(\text{\gcd} ( a , b )\) denotes the greatest common divisor
of \(a\) and \(b\), and \(\text{\operatorname{lcm}} ( a , b )\) denotes their least
common multiple?
A
\(\text{ 0}\)
\(\text{ 2}\)
Correct Answer
C
\(\text{ 4}\)
D
\(\text{ 6}\)
E
\(\text{ 8}\)
No explanation
24
Geometry
Wrong
Let \(A B C D E F\) be a regular hexagon with side length \(1\). Denote by
\(X\), \(Y\), and \(Z\) the midpoints of sides \(\overline{A B}\),
\(\overline{C D}\), and \(\overline{E F}\), respectively. What is the area of
the convex hexagon whose interior is the intersection of the interiors
of \(\triangle A C E\) and \(\triangle X Y Z\)?
A
\(\dfrac{3}{8} \sqrt{3}\)
B
\(\dfrac{7}{16} \sqrt{3}\)
\(\dfrac{15}{32} \sqrt{3}\)
Correct Answer
D
\(\dfrac{1}{2} \sqrt{3}\)
E
\(\dfrac{9}{16} \sqrt{3}\)
No explanation
25
Algebra
Wrong
Let \(⌊ x ⌋\) denote the greatest integer less than or equal
to \(x\). How many real numbers \(x\) satisfy the equation
\(x^2 + 10 , 000 ⌊ x ⌋ = 10 , 000 x\)?
A
\(197\)
B
\(198\)
\(199\)
Correct Answer
D
\(200\)
E
\(201\)
No explanation
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