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AMC 12A 2023
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문제별 결과
1
Competition Math
오답률 100%
오답
Cities \(A\) and \(B\) are \(45\) miles apart. Alicia lives in \(A\) and Beth
lives in \(B\). Alicia bikes towards \(B\) at 18 miles per hour. Leaving at
the same time, Beth bikes toward \(A\) at 12 miles per hour. How many
miles from City \(A\) will they be when they meet?
A
\(20\)
B
\(24\)
C
\(25\)
D
\(26\)
E
\(27\)
해설 없음
2
Competition Math
오답률 100%
오답
The weight of \(\dfrac{1}{3}\) of a large pizza together with \(3 \dfrac{1}{2}\) cups of
orange slices is the same weight of \(\dfrac{3}{4}\) of a large pizza together
with \(\dfrac{1}{2}\) cups of orange slices. A cup of orange slices weigh \(\dfrac{1}{4}\)
of a pound. What is the weight, in pounds, of a large pizza?
A
\(1 \dfrac{4}{5}\)
B
\(2\)
C
\(2 \dfrac{2}{5}\)
D
\(3\)
E
\(3 \dfrac{3}{5}\)
해설 없음
3
Competition Math
오답률 100%
오답
How many positive perfect squares less than \(2023\) are divisible by \(5\)?
A
\(8\)
B
\(9\)
C
\(10\)
D
\(11\)
E
\(12\)
해설 없음
4
Competition Math
오답률 100%
오답
How many digits are in the base-ten representation of
\(8^5 \cdot 5^10 \cdot 15^5\)?
A
\( 14\)
B
\( 15\)
C
\( 16\)
D
\( 17\)
E
\( 18\)
해설 없음
5
Competition Math
오답률 100%
오답
Janet rolls a standard \(6\)-sided die \(4\) times and keeps a running total
of the numbers she rolls. What is the probability that at some point,
her running total will equal \(3 ?\)
A
\(\dfrac{2}{9}\)
B
\(\dfrac{49}{216}\)
C
\(\dfrac{25}{108}\)
D
\(\dfrac{17}{72}\)
E
\(\dfrac{13}{54}\)
해설 없음
6
Competition Math
오답률 100%
오답
Points \(A\) and \(B\) lie on the graph of \(y = \log_2 x\). The midpoint of
\(\overline{A B}\) is \(( 6 , 2 )\). What is the positive difference
between the \(x\)-coordinates of \(A\) and \(B\)?
A
\( 2 \sqrt{11}\)
B
\( 4 \sqrt{3}\)
C
\( 8\)
D
\( 4 \sqrt{5}\)
E
\( 9\)
해설 없음
7
Competition Math
오답률 100%
오답
A digital display shows the current date as an \(8\)-digit integer
consisting of a \(4\)-digit year, followed by a \(2\)-digit month, followed
by a \(2\)-digit date within the month. For example, Arbor Day this year
is displayed as \(20230428\). For how many dates in \(2023\) will each digit
appear an even number of times in the 8-digital display for that date?
A
\( 5\)
B
\( 6\)
C
\( 7\)
D
\( 8\)
E
\( 9\)
해설 없음
8
Competition Math
오답률 100%
오답
Maureen is keeping track of the mean of her quiz scores this semester.
If Maureen scores an \(11\) on the next quiz, her mean will increase by
\(1\). If she scores an \(11\) on each of the next three quizzes, her mean
will increase by \(2\). What is the mean of her quiz scores currently?
A
\(4\)
B
\(5\)
C
\(6\)
D
\(7\)
E
\(8\)
해설 없음
9
Competition Math
오답률 100%
오답
A square of area \(2\) is inscribed in a square of area \(3\), creating four
congruent triangles, as shown below. What is the ratio of the shorter
leg to the longer leg in the shaded right triangle?
A
\(\dfrac{1}{5}\)
B
\(\dfrac{1}{4}\)
C
\(2 - \sqrt{3}\)
D
\(\sqrt{3} - \sqrt{2}\)
E
\(\sqrt{2} - 1\)
해설 없음
10
Competition Math
오답률 100%
오답
Positive real numbers \(x\) and \(y\) satisfy \(y^3 = x^2\) and
\(( y - x )^2 = 4 y^2\). What is \(x + y\)?
A
\(12\)
B
\(18\)
C
\(24\)
D
\(36\)
E
\(42\)
해설 없음
11
Competition Math
오답률 100%
오답
What is the degree measure of the acute angle formed by lines with
slopes \(2\) and \(\dfrac{1}{3}\)?
A
\( 30\)
B
\( 37.5\)
C
\( 45\)
D
\( 52.5\)
E
\( 60\)
해설 없음
12
Competition Math
오답률 100%
오답
What is the value of
\( 2^3 - 1^3 + 4^3 - 3^3 + 6^3 - 5^3 + \cdots + 18^3 - 17^3 ? \)
A
\(2023\)
B
\(2679\)
C
\(2941\)
D
\(3159\)
E
\(3235\)
해설 없음
13
Competition Math
오답률 100%
오답
In a table tennis tournament every participant played every other
participant exactly once. Although there were twice as many right-handed
players as left-handed players, the number of games won by left-handed
players was \(40 %\) more than the number of games won by right-handed
players. (There were no ties and no ambidextrous players.) What is the
total number of games played?
A
\(15\)
B
\(36\)
C
\(45\)
D
\(48\)
E
\(66\)
해설 없음
14
Competition Math
오답률 100%
오답
How many complex numbers satisfy the equation \(z^5 = \overline{z}\), where
\(\overline{z}\) is the conjugate of the complex number \(z\)?
A
\( 2\)
B
\( 3\)
C
\( 5\)
D
\( 6\)
E
\( 7\)
해설 없음
15
Competition Math
오답률 100%
오답
Usain is walking for exercise by zigzagging across a \(100\)-meter by
\(30\)-meter rectangular field, beginning at point \(A\) and ending on the
segment \(\overline{B C}\). He wants to increase the distance walked by
zigzagging as shown in the figure below \(( A P Q R S )\). What angle
\(\theta\)\(\angle P A B = \angle Q P C = \angle R Q B = \cdots.c\) will produce
a length that is \(120\) meters? (This figure is not drawn to scale. Do
not assume that the zigzag path has exactly four segments as shown;
there could be more or fewer.)
A
\( \arccos \dfrac{5}{6}\)
B
\( \arccos \dfrac{4}{5}\)
C
\( \arccos \dfrac{3}{10}\)
D
\( \arcsin \dfrac{4}{5}\)
E
\( \arcsin \dfrac{5}{6}\)
해설 없음
16
Competition Math
오답률 100%
오답
Consider the set of complex numbers \(z\) satisfying
\(\| 1 + z + z^2 \| = 4\). The maximum value of the imaginary part of \(z\)
can be written in the form \(\dfrac{\sqrt{m}}{n}\), where \(m\) and \(n\) are
relatively prime positive integers. What is \(m + n\)?
A
\( 20\)
B
\( 21\)
C
\( 22\)
D
\( 23\)
E
\( 24\)
해설 없음
17
Competition Math
오답률 100%
오답
Flora the frog starts at \(0\) on the number line and makes a sequence of
jumps to the right. In any one jump, independent of previous jumps,
Flora leaps a positive integer distance \(m\) with probability \(1 / 2^m\).
What is the probability that Flora will eventually land at \(10\)?
A
\(\dfrac{5}{512}\)
B
\(\dfrac{45}{1024}\)
C
\(\dfrac{127}{1024}\)
D
\(\dfrac{511}{1024}\)
E
\(\dfrac{1}{2}\)
해설 없음
18
Competition Math
오답률 100%
오답
Circle \(C_1\) and \(C_2\) each have radius \(1\), and the distance between
their centers is \(\dfrac{1}{2}\). Circle \(C_3\) is the largest circle internally
tangent to both \(C_1\) and \(C_2\). Circle \(C_4\) is internally tangent to
both \(C_1\) and \(C_2\) and externally tangent to \(C_3\). What is the radius
of \(C_4\)?
A
\(\dfrac{1}{14}\)
B
\(\dfrac{1}{12}\)
C
\(\dfrac{1}{10}\)
D
\(\dfrac{3}{28}\)
E
\(\dfrac{1}{9}\)
해설 없음
19
Competition Math
오답률 100%
오답
What is the product of all the solutions to the equation
\( \log_{7 x} 2023 \cdot \log_{289 x} 2023 = \log_{2023 x} 2023 ? \)
A
\(( \log_2023 7 \cdot \log_2023 289 )^2\)
B
\(\log_2023 7 \cdot \log_2023 289\)
C
\(1 \)
D
\(\log_7 2023 \cdot \log_289 2023\)
E
\(( \log_7 2023 \cdot \log_289 2023 )^2\)
해설 없음
20
Competition Math
오답률 100%
오답
Rows 1, 2, 3, 4, and 5 of a triangular array of integers are shown
below: Each
row after the first row is formed by placing a 1 at each end of the row,
and each interior entry is 1 greater than the sum of the two numbers
diagonally above it in the previous row. What is the units digit of the
sum of the 2023 numbers in the 2023rd row?
A
\(1\)
B
\(3\)
C
\(5\)
D
\(7\)
E
\(9\)
해설 없음
21
Competition Math
오답률 100%
오답
If \(A\) and \(B\) are vertices of a polyhedron, define the distance
\(d ( A , B )\) to be the minimum number of edges of the polyhedron one
must traverse in order to connect \(A\) and \(B\). For example, if
\(\overline{A B}\) is an edge of the polyhedron, then \(d ( A , B ) = 1\),
but if \(\overline{A C}\) and \(\overline{C B}\) are edges and \(\overline{A B}\)
is not an edge, then \(d ( A , B ) = 2\). Let \(Q\), \(R\), and \(S\) be
randomly chosen distinct vertices of a regular icosahedron (regular
polyhedron made up of 20 equilateral triangles). What is the probability
that \(d ( Q , R ) > d ( R , S )\)?
A
\( \dfrac{7}{22}\)
B
\( \dfrac{1}{3}\)
C
\( \dfrac{3}{8}\)
D
\( \dfrac{5}{12}\)
E
\( \dfrac{1}{2}\)
해설 없음
22
Competition Math
오답률 100%
오답
Let \(f\) be the unique function defined on the positive integers such
that \( \displaystyle\sum_{d | n} d \cdot f \left(\dfrac{n}{d}\right) = 1 \) for all positive
integers \(n\), where the sum is taken over all positive divisors of \(n\).
What is \(f ( 2023 )\)?
A
\( - 1536\)
B
\( 96\)
C
\( 108\)
D
\( 116\)
E
\( 144\)
해설 없음
23
Competition Math
오답률 100%
오답
How many ordered pairs of positive real numbers \(( a , b )\) satisfy
the equation \( ( 1 + 2 a ) ( 2 + 2 b ) ( 2 a + b ) = 32 a b ? \)
A
\(0\)
B
\(1\)
C
\(2\)
D
\(3\)
E
\(\text{an infinite number}\)
해설 없음
24
Competition Math
오답률 100%
오답
Let \(K\) be the number of sequences \(A_1\), \(A_2\), \(\cdots\), \(A_n\) such
that \(n\) is a positive integer less than or equal to \(10\), each \(A_i\) is
a subset of \({ 1 , 2 , 3 , \cdots , 10 }\), and \(A_{i - 1}\) is a subset
of \(A_i\) for each \(i\) between \(2\) and \(n\), inclusive. For example,
\({ }\), \({ 5 , 7 }\), \({ 2 , 5 , 7 }\), \({ 2 , 5 , 7 }\),
\({ 2 , 5 , 6 , 7 , 9 }\) is one such sequence, with \(n = 5\).What is the
remainder when \(K\) is divided by \(10\)?
A
\(1\)
B
\(3\)
C
\(5\)
D
\(7\)
E
\(9\)
해설 없음
25
Competition Math
오답률 100%
오답
There is a unique sequence of integers \(a_1 , a_2 , \cdots.c a_2023\)
such that
\( \tan 2023 x = \dfrac{a_1 \tan x + a_3 \tan^3 x + a_5 \tan^5 x + \cdots.c + a_2023 \tan^2023 x}{1 + a_2 \tan^2 x + a_4 \tan^4 x \cdots.c + a_2022 \tan^2022 x} \)
whenever \(\tan 2023 x\) is defined. What is \(a_2023 ?\)
A
\(- 2023\)
B
\(- 2022\)
C
\(- 1\)
D
\(1\)
E
\(2023\)
해설 없음