Difference between revisions of "2019 AMC 12B Problems"
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==Problem 1== | ==Problem 1== | ||
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+ | [[2019 AMC 12B Problems/Problem 1|Solution]] | ||
==Problem 2== | ==Problem 2== | ||
Consider the statement, "If <math>n</math> is not prime, then <math>n-2</math> is prime." Which of the following values of <math>n</math> is a counterexample to this statement. | Consider the statement, "If <math>n</math> is not prime, then <math>n-2</math> is prime." Which of the following values of <math>n</math> is a counterexample to this statement. | ||
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<math>\textbf{(A) } 11 \qquad \textbf{(B) } 15 \qquad \textbf{(C) } 19 \qquad \textbf{(D) } 21 \qquad \textbf{(E) } 27</math> | <math>\textbf{(A) } 11 \qquad \textbf{(B) } 15 \qquad \textbf{(C) } 19 \qquad \textbf{(D) } 21 \qquad \textbf{(E) } 27</math> | ||
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+ | [[2019 AMC 12B Problems/Problem 2|Solution]] | ||
==Problem 3== | ==Problem 3== | ||
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+ | [[2019 AMC 12B Problems/Problem 3|Solution]] | ||
==Problem 4== | ==Problem 4== | ||
A positive integer <math>n</math> satisfies the equation <math>(n+1)!+(n+2)!=440\cdot n!</math>. What is the sum of the digits of <math>n</math>? | A positive integer <math>n</math> satisfies the equation <math>(n+1)!+(n+2)!=440\cdot n!</math>. What is the sum of the digits of <math>n</math>? | ||
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<math>\textbf{(A) } 2 \qquad \textbf{(B) } 5 \qquad \textbf{(C) } 10\qquad \textbf{(D) } 12 \qquad \textbf{(E) } 15</math> | <math>\textbf{(A) } 2 \qquad \textbf{(B) } 5 \qquad \textbf{(C) } 10\qquad \textbf{(D) } 12 \qquad \textbf{(E) } 15</math> | ||
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+ | [[2019 AMC 12B Problems/Problem 4|Solution]] | ||
==Problem 5== | ==Problem 5== | ||
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<math>\textbf{(A) } 18 \qquad \textbf{(B) } 21 \qquad \textbf{(C) } 24\qquad \textbf{(D) } 25 \qquad \textbf{(E) } 28</math> | <math>\textbf{(A) } 18 \qquad \textbf{(B) } 21 \qquad \textbf{(C) } 24\qquad \textbf{(D) } 25 \qquad \textbf{(E) } 28</math> | ||
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+ | [[2019 AMC 12B Problems/Problem 5|Solution]] | ||
==Problem 6== | ==Problem 6== | ||
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<math>\textbf{(A) }0\qquad\textbf{(B) }2\qquad\textbf{(C) }4\qquad\textbf{(D) }8\qquad\textbf{(E) }\text{infinitely many}</math> | <math>\textbf{(A) }0\qquad\textbf{(B) }2\qquad\textbf{(C) }4\qquad\textbf{(D) }8\qquad\textbf{(E) }\text{infinitely many}</math> | ||
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+ | [[2019 AMC 12B Problems/Problem 6|Solution]] | ||
==Problem 7== | ==Problem 7== | ||
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<math>\textbf{(A) } -5 \qquad\textbf{(B) } 0 \qquad\textbf{(C) } 5 \qquad\textbf{(D) } \frac{15}{4} \qquad\textbf{(E) } \frac{35}{4}</math> | <math>\textbf{(A) } -5 \qquad\textbf{(B) } 0 \qquad\textbf{(C) } 5 \qquad\textbf{(D) } \frac{15}{4} \qquad\textbf{(E) } \frac{35}{4}</math> | ||
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+ | [[2019 AMC 12B Problems/Problem 7|Solution]] | ||
==Problem 8== | ==Problem 8== | ||
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+ | [[2019 AMC 12B Problems/Problem 8|Solution]] | ||
==Problem 9== | ==Problem 9== | ||
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+ | [[2019 AMC 12B Problems/Problem 9|Solution]] | ||
==Problem 10== | ==Problem 10== | ||
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+ | [[2019 AMC 12B Problems/Problem 10|Solution]] | ||
==Problem 11== | ==Problem 11== | ||
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<math>\textbf{(A) } 12 \qquad \textbf{(B) } 28 \qquad \textbf{(C) } 36\qquad \textbf{(D) } 42 \qquad \textbf{(E) } 66</math> | <math>\textbf{(A) } 12 \qquad \textbf{(B) } 28 \qquad \textbf{(C) } 36\qquad \textbf{(D) } 42 \qquad \textbf{(E) } 66</math> | ||
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+ | [[2019 AMC 12B Problems/Problem 11|Solution]] | ||
==Problem 12== | ==Problem 12== | ||
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+ | [[2019 AMC 12B Problems/Problem 12|Solution]] | ||
==Problem 13== | ==Problem 13== | ||
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+ | [[2019 AMC 12B Problems/Problem 13|Solution]] | ||
==Problem 14== | ==Problem 14== | ||
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<math>\textbf{(A) }98\qquad\textbf{(B) }100\qquad\textbf{(C) }117\qquad\textbf{(D) }119\qquad\textbf{(E) }121</math> | <math>\textbf{(A) }98\qquad\textbf{(B) }100\qquad\textbf{(C) }117\qquad\textbf{(D) }119\qquad\textbf{(E) }121</math> | ||
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+ | [[2019 AMC 12B Problems/Problem 14|Solution]] | ||
==Problem 15== | ==Problem 15== | ||
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+ | [[2019 AMC 12B Problems/Problem 15|Solution]] | ||
==Problem 16== | ==Problem 16== | ||
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<math>\textbf{(A) } \frac{15}{256} \qquad \textbf{(B) } \frac{1}{16} \qquad \textbf{(C) } \frac{15}{128}\qquad \textbf{(D) } \frac{1}{8} \qquad \textbf{(E) } \frac14</math> | <math>\textbf{(A) } \frac{15}{256} \qquad \textbf{(B) } \frac{1}{16} \qquad \textbf{(C) } \frac{15}{128}\qquad \textbf{(D) } \frac{1}{8} \qquad \textbf{(E) } \frac14</math> | ||
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+ | [[2019 AMC 12B Problems/Problem 16|Solution]] | ||
==Problem 17== | ==Problem 17== | ||
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<math>\textbf{(A) }0\qquad\textbf{(B) }1\qquad\textbf{(C) }2\qquad\textbf{(D) }4\qquad\textbf{(E) }\text{infinitely many}</math> | <math>\textbf{(A) }0\qquad\textbf{(B) }1\qquad\textbf{(C) }2\qquad\textbf{(D) }4\qquad\textbf{(E) }\text{infinitely many}</math> | ||
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+ | [[2019 AMC 12B Problems/Problem 17|Solution]] | ||
==Problem 18== | ==Problem 18== | ||
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<math>\textbf{(A) } \frac{3\sqrt2}{2} \qquad\textbf{(B) } \frac{3\sqrt3}{2} \qquad\textbf{(C) } 2\sqrt2 \qquad\textbf{(D) } 2\sqrt3 \qquad\textbf{(E) } 3\sqrt2</math> | <math>\textbf{(A) } \frac{3\sqrt2}{2} \qquad\textbf{(B) } \frac{3\sqrt3}{2} \qquad\textbf{(C) } 2\sqrt2 \qquad\textbf{(D) } 2\sqrt3 \qquad\textbf{(E) } 3\sqrt2</math> | ||
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+ | [[2019 AMC 12B Problems/Problem 18|Solution]] | ||
==Problem 19== | ==Problem 19== | ||
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+ | [[2019 AMC 12B Problems/Problem 19|Solution]] | ||
==Problem 20== | ==Problem 20== | ||
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<math>\textbf{(A) }\frac{83\pi}{8}\qquad\textbf{(B) }\frac{21\pi}{2}\qquad\textbf{(C) } | <math>\textbf{(A) }\frac{83\pi}{8}\qquad\textbf{(B) }\frac{21\pi}{2}\qquad\textbf{(C) } | ||
\frac{85\pi}{8}\qquad\textbf{(D) }\frac{43\pi}{4}\qquad\textbf{(E) }\frac{87\pi}{8}</math> | \frac{85\pi}{8}\qquad\textbf{(D) }\frac{43\pi}{4}\qquad\textbf{(E) }\frac{87\pi}{8}</math> | ||
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+ | [[2019 AMC 12B Problems/Problem 20|Solution]] | ||
==Problem 21== | ==Problem 21== | ||
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<math>\textbf{(A) } 3 \qquad\textbf{(B) } 4 \qquad\textbf{(C) } 5 \qquad\textbf{(D) } 6 \qquad\textbf{(E) } \text{infinitely many}</math> | <math>\textbf{(A) } 3 \qquad\textbf{(B) } 4 \qquad\textbf{(C) } 5 \qquad\textbf{(D) } 6 \qquad\textbf{(E) } \text{infinitely many}</math> | ||
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+ | [[2019 AMC 12B Problems/Problem 21|Solution]] | ||
==Problem 22== | ==Problem 22== | ||
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<math>\textbf{(A) } [9,26] \qquad\textbf{(B) } [27,80] \qquad\textbf{(C) } [81,242]\qquad\textbf{(D) } [243,728] \qquad\textbf{(E) } [729,\infty]</math> | <math>\textbf{(A) } [9,26] \qquad\textbf{(B) } [27,80] \qquad\textbf{(C) } [81,242]\qquad\textbf{(D) } [243,728] \qquad\textbf{(E) } [729,\infty]</math> | ||
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+ | [[2019 AMC 12B Problems/Problem 22|Solution]] | ||
==Problem 23== | ==Problem 23== | ||
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+ | How many sequences of <math>0</math>s and <math>1</math>s of length <math>19</math> are there that begin with a <math>0</math>, end with a <math>0</math>, contain no two consecutive <math>0</math>s, and contain no three consecutive <math>1</math>s? | ||
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+ | <math>\textbf{(A) }55\qquad\textbf{(B) }60\qquad\textbf{(C) }65\qquad\textbf{(D) }70\qquad\textbf{(E) }75</math> | ||
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+ | [[2019 AMC 12B Problems/Problem 23|Solution]] | ||
==Problem 24== | ==Problem 24== | ||
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Let <math>\omega=-\tfrac{1}{2}+\tfrac{1}{2}i\sqrt3.</math> Let <math>S</math> denote all points in the complex plane of the form <math>a+b\omega+c\omega^2,</math> where <math>0\leq a \leq 1,0\leq b\leq 1,</math> and <math>0\leq c\leq 1.</math> What is the area of <math>S</math>? | Let <math>\omega=-\tfrac{1}{2}+\tfrac{1}{2}i\sqrt3.</math> Let <math>S</math> denote all points in the complex plane of the form <math>a+b\omega+c\omega^2,</math> where <math>0\leq a \leq 1,0\leq b\leq 1,</math> and <math>0\leq c\leq 1.</math> What is the area of <math>S</math>? | ||
<math>\textbf{(A) } \frac{1}{2}\sqrt3 \qquad\textbf{(B) } \frac{3}{4}\sqrt3 \qquad\textbf{(C) } \frac{3}{2}\sqrt3\qquad\textbf{(D) } \frac{1}{2}\pi\sqrt3 \qquad\textbf{(E) } \pi</math> | <math>\textbf{(A) } \frac{1}{2}\sqrt3 \qquad\textbf{(B) } \frac{3}{4}\sqrt3 \qquad\textbf{(C) } \frac{3}{2}\sqrt3\qquad\textbf{(D) } \frac{1}{2}\pi\sqrt3 \qquad\textbf{(E) } \pi</math> | ||
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+ | [[2019 AMC 12B Problems/Problem 24|Solution]] | ||
==Problem 25== | ==Problem 25== | ||
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+ | Let <math>ABCD</math> be a convex quadrilateral with <math>BC=2</math> and <math>CD=6.</math> Suppose that the centroids of <math>\triangle ABC,\triangle BCD,</math> and <math>\triangle ACD</math> form the vertices of an equilateral triangle. What is the maximum possible value of <math>ABCD</math>? | ||
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+ | <math>\textbf{(A) } 27 \qquad\textbf{(B) } 16\sqrt3 \qquad\textbf{(C) } 12+10\sqrt3 \qquad\textbf{(D) } 9+12\sqrt3 \qquad\textbf{(E) } 30</math> | ||
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+ | [[2019 AMC 12B Problems/Problem 25|Solution]] |
Revision as of 13:23, 14 February 2019
Contents
- 1 Problem 1
- 2 Problem 2
- 3 Problem 3
- 4 Problem 4
- 5 Problem 5
- 6 Problem 6
- 7 Problem 7
- 8 Problem 8
- 9 Problem 9
- 10 Problem 10
- 11 Problem 11
- 12 Problem 12
- 13 Problem 13
- 14 Problem 14
- 15 Problem 15
- 16 Problem 16
- 17 Problem 17
- 18 Problem 18
- 19 Problem 19
- 20 Problem 20
- 21 Problem 21
- 22 Problem 22
- 23 Problem 23
- 24 Problem 24
- 25 Problem 25
Problem 1
Problem 2
Consider the statement, "If is not prime, then
is prime." Which of the following values of
is a counterexample to this statement.
Problem 3
Problem 4
A positive integer satisfies the equation
. What is the sum of the digits of
?
Problem 5
Each piece of candy in a store costs a whole number of cents. Casper has exactly enough money to buy either 12 pieces of red candy, 14 pieces of green candy, 15 pieces of blue candy, or pieces of purple candy. A piece of purple candy costs 20 cents. What is the smallest possible value of
?
Problem 6
In a given plane, points and
are
units apart. How many points
are there in the plane such that the perimeter of
is
units and the area of
is
square units?
Problem 7
What is the sum of all real numbers for which the median of the numbers
and
is equal to the mean of those five numbers?
Problem 8
Problem 9
Problem 10
Problem 11
How many unordered pairs of edges of a given cube determine a plane?
Problem 12
Problem 13
Problem 14
Let be the set of all positive integer divisors of
How many numbers are the product of two distinct elements of
Problem 15
Problem 16
There are lily pads in a row numbered 0 to 11, in that order. There are predators on lily pads 3 and 6, and a morsel of food on lily pad 10. Fiona the frog starts on pad 0, and from any given lily pad, has a chance to hop to the next pad, and an equal chance to jump 2 pads. What is the probability that Fiona reaches pad 10 without landing on either pad 3 or pad 6?
Problem 17
How many nonzero complex numbers have the property that
and
when represented by points in the complex plane, are the three distinct vertices of an equilateral triangle?
Problem 18
Square pyramid has base
which measures
cm on a side, and altitude
perpendicular to the base
which measures
cm. Point
lies on
one third of the way from
to
point
lies on
one third of the way from
to
and point
lies on
two thirds of the way from
to
What is the area, in square centimeters, of
Problem 19
Problem 20
Points and
lie on circle
in the plane. Suppose that the tangent lines to
at
and
intersect at a point on the
-axis. What is the area of
?
Problem 21
How many quadratic polynomials with real coefficients are there such that the set of roots equals the set of coefficients? (For clarification: If the polynomial is and the roots are
and
then the requirement is that
.)
Problem 22
Define a sequence recursively by and
for all nonnegative integers
Let
be the least positive integer such that
In which of the following intervals does
lie?
Problem 23
How many sequences of s and
s of length
are there that begin with a
, end with a
, contain no two consecutive
s, and contain no three consecutive
s?
Problem 24
Let Let
denote all points in the complex plane of the form
where
and
What is the area of
?
Problem 25
Let be a convex quadrilateral with
and
Suppose that the centroids of
and
form the vertices of an equilateral triangle. What is the maximum possible value of
?