Difference between revisions of "2021 AMC 12A Problems"

Latest revision as of 08:49, 25 June 2023

2021 AMC 12A (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 test 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

What is the value of $2^{1+2+3}-(2^1+2^2+2^3)?$ $\textbf{(A) }0 \qquad \textbf{(B) }50 \qquad \textbf{(C) }52 \qquad \textbf{(D) }54 \qquad \textbf{(E) }57$

Solution

Problem 2

Under what conditions is $\sqrt{a^2+b^2}=a+b$ true, where $a$ and $b$ are real numbers?

$\textbf{(A) }$ It is never true.

$\textbf{(B) }$ It is true if and only if $ab=0$ .

$\textbf{(C) }$ It is true if and only if $a+b\ge 0$ .

$\textbf{(D) }$ It is true if and only if $ab=0$ and $a+b\ge 0$ .

$\textbf{(E) }$ It is always true.

Solution

Problem 3

The sum of two natural numbers is $17{,}402$ . One of the two numbers is divisible by $10$ . If the units digit of that number is erased, the other number is obtained. What is the difference of these two numbers?

$\textbf{(A)} ~10{,}272\qquad\textbf{(B)} ~11{,}700\qquad\textbf{(C)} ~13{,}362\qquad\textbf{(D)} ~14{,}238\qquad\textbf{(E)} ~15{,}426$

Solution

Problem 4

Tom has a collection of $13$ snakes, $4$ of which are purple and $5$ of which are happy. He observes that

all of his happy snakes can add,

none of his purple snakes can subtract, and

all of his snakes that can't subtract also can't add.

Which of these conclusions can be drawn about Tom's snakes?

$\textbf{(A) }$ Purple snakes can add.

$\textbf{(B) }$ Purple snakes are happy.

$\textbf{(C) }$ Snakes that can add are purple.

$\textbf{(D) }$ Happy snakes are not purple.

$\textbf{(E) }$ Happy snakes can't subtract.

Solution

Problem 5

When a student multiplied the number $66$ by the repeating decimal, $\underline{1}.\underline{a} \ \underline{b} \ \underline{a} \ \underline{b}\ldots=\underline{1}.\overline{\underline{a} \ \underline{b}},$ where $a$ and $b$ are digits, he did not notice the notation and just multiplied $66$ times $\underline{1}.\underline{a} \ \underline{b}.$ Later he found that his answer is $0.5$ less than the correct answer. What is the $2$ -digit number $\underline{a} \ \underline{b}?$

$\textbf{(A) }15 \qquad \textbf{(B) }30 \qquad \textbf{(C) }45 \qquad \textbf{(D) }60 \qquad \textbf{(E) }75$

Solution

Problem 6

A deck of cards has only red cards and black cards. The probability of a randomly chosen card being red is $\frac13$ . When $4$ black cards are added to the deck, the probability of choosing red becomes $\frac14$ . How many cards were in the deck originally?

$\textbf{(A) }6 \qquad \textbf{(B) }9 \qquad \textbf{(C) }12 \qquad \textbf{(D) }15 \qquad \textbf{(E) }18$

Solution

Problem 7

What is the least possible value of $(xy-1)^2+(x+y)^2$ for all real numbers $x$ and $y?$

$\textbf{(A) }0 \qquad \textbf{(B) }\frac14 \qquad \textbf{(C) }\frac12 \qquad \textbf{(D) }1 \qquad \textbf{(E) }2$

Solution

Problem 8

A sequence of numbers is defined by $D_0=0,D_1=0,D_2=1$ and $D_n=D_{n-1}+D_{n-3}$ for $n\ge 3$ . What are the parities (evenness or oddness) of the triple of numbers $(D_{2021},D_{2022},D_{2023})$ , where $E$ denotes even and $O$ denotes odd?

$\textbf{(A) }(O,E,O) \qquad \textbf{(B) }(E,E,O) \qquad \textbf{(C) }(E,O,E) \qquad \textbf{(D) }(O,O,E) \qquad \textbf{(E) }(O,O,O)$

Solution

Problem 9

Which of the following is equivalent to $(2+3)(2^2+3^2)(2^4+3^4)(2^8+3^8)(2^{16}+3^{16})(2^{32}+3^{32})(2^{64}+3^{64})?$ $\textbf{(A) }3^{127}+2^{127} \qquad \textbf{(B) }3^{127}+2^{127}+2\cdot 3^{63}+3\cdot 2^{63} \qquad \textbf{(C) }3^{128}-2^{128} \qquad \textbf{(D) }3^{128}+2^{128} \qquad \textbf{(E) }5^{127}$

Solution

Problem 10

Two right circular cones with vertices facing down as shown in the figure below contain the same amount of liquid. The radii of the tops of the liquid surfaces are $3$ cm and $6$ cm. Into each cone is dropped a spherical marble of radius $1$ cm, which sinks to the bottom and is completely submerged without spilling any liquid. What is the ratio of the rise of the liquid level in the narrow cone to the rise of the liquid level in the wide cone?

$[asy] size(350); defaultpen(linewidth(0.8)); real h1 = 10, r = 3.1, s=0.75; pair P = (r,h1), Q = (-r,h1), Pp = s * P, Qp = s * Q; path e = ellipse((0,h1),r,0.9), ep = ellipse((0,h1*s),r*s,0.9); draw(ellipse(origin,r*(s-0.1),0.8)); fill(ep,gray(0.8)); fill(origin--Pp--Qp--cycle,gray(0.8)); draw((-r,h1)--(0,0)--(r,h1)^^e); draw(subpath(ep,0,reltime(ep,0.5)),linetype("4 4")); draw(subpath(ep,reltime(ep,0.5),reltime(ep,1))); draw(Qp--(0,Qp.y),Arrows(size=8)); draw(origin--(0,12),linetype("4 4")); draw(origin--(r*(s-0.1),0)); label("$3$",(-0.9,h1*s),N,fontsize(10)); real h2 = 7.5, r = 6, s=0.6, d = 14; pair P = (d+r-0.05,h2-0.15), Q = (d-r+0.05,h2-0.15), Pp = s * P + (1-s)*(d,0), Qp = s * Q + (1-s)*(d,0); path e = ellipse((d,h2),r,1), ep = ellipse((d,h2*s+0.09),r*s,1); draw(ellipse((d,0),r*(s-0.1),0.8)); fill(ep,gray(0.8)); fill((d,0)--Pp--Qp--cycle,gray(0.8)); draw(P--(d,0)--Q^^e); draw(subpath(ep,0,reltime(ep,0.5)),linetype("4 4")); draw(subpath(ep,reltime(ep,0.5),reltime(ep,1))); draw(Qp--(d,Qp.y),Arrows(size=8)); draw((d,0)--(d,10),linetype("4 4")); draw((d,0)--(d+r*(s-0.1),0)); label("$6$",(d-r/4,h2*s-0.06),N,fontsize(10)); [/asy]$

$\textbf{(A) }1:1 \qquad \textbf{(B) }47:43 \qquad \textbf{(C) }2:1 \qquad \textbf{(D) }40:13 \qquad \textbf{(E) }4:1$

Solution

Problem 11

A laser is placed at the point $(3,5)$ . The laser beam travels in a straight line. Larry wants the beam to hit and bounce off the $y$ -axis, then hit and bounce off the $x$ -axis, then hit the point $(7,5)$ . What is the total distance the beam will travel along this path?

$\textbf{(A) }2\sqrt{10} \qquad \textbf{(B) }5\sqrt2 \qquad \textbf{(C) }10\sqrt2 \qquad \textbf{(D) }15\sqrt2 \qquad \textbf{(E) }10\sqrt5$

Solution

Problem 12

All the roots of the polynomial $z^6-10z^5+Az^4+Bz^3+Cz^2+Dz+16$ are positive integers, possibly repeated. What is the value of $B$ ?

$\textbf{(A) }{-}88 \qquad \textbf{(B) }{-}80 \qquad \textbf{(C) }{-}64 \qquad \textbf{(D) }{-}41\qquad \textbf{(E) }{-}40$

Solution

Problem 13

Of the following complex numbers $z$ , which one has the property that $z^5$ has the greatest real part?

$\textbf{(A) }{-}2 \qquad \textbf{(B) }{-}\sqrt3+i \qquad \textbf{(C) }{-}\sqrt2+\sqrt2 i \qquad \textbf{(D) }{-}1+\sqrt3 i\qquad \textbf{(E) }2i$

Solution

Problem 14

What is the value of $\left(\sum_{k=1}^{20} \log_{5^k} 3^{k^2}\right)\cdot\left(\sum_{k=1}^{100} \log_{9^k} 25^k\right)?$ $\textbf{(A) }21 \qquad \textbf{(B) }100\log_5 3 \qquad \textbf{(C) }200\log_3 5 \qquad \textbf{(D) }2{,}200\qquad \textbf{(E) }21{,}000$

Solution

Problem 15

A choir director must select a group of singers from among his $6$ tenors and $8$ basses. The only requirements are that the difference between the numbers of tenors and basses must be a multiple of $4$ , and the group must have at least one singer. Let $N$ be the number of different groups that could be selected. What is the remainder when $N$ is divided by $100$ ?

$\textbf{(A) } 47\qquad\textbf{(B) } 48\qquad\textbf{(C) } 83\qquad\textbf{(D) } 95\qquad\textbf{(E) } 96\qquad$

Solution

Problem 16

In the following list of numbers, the integer $n$ appears $n$ times in the list for $1\le n \le 200$ . $1,2,2,3,3,3,4,4,4,\ldots,200,200,\ldots,200$ What is the median of the numbers in this list?

$\textbf{(A) }100.5 \qquad \textbf{(B) }134 \qquad \textbf{(C) }142 \qquad \textbf{(D) }150.5\qquad \textbf{(E) }167$

Solution

Problem 17

Trapezoid $ABCD$ has $\overline{AB}\parallel\overline{CD},BC=CD=43$ , and $\overline{AD}\perp\overline{BD}$ . Let $O$ be the intersection of the diagonals $\overline{AC}$ and $\overline{BD}$ , and let $P$ be the midpoint of $\overline{BD}$ . Given that $OP=11$ , the length of $AD$ can be written in the form $m\sqrt{n}$ , where $m$ and $n$ are positive integers and $n$ is not divisible by the square of any prime. What is $m+n$ ?

$\textbf{(A) }65 \qquad \textbf{(B) }132 \qquad \textbf{(C) }157 \qquad \textbf{(D) }194\qquad \textbf{(E) }215$

Solution

Problem 18

Let $f$ be a function defined on the set of positive rational numbers with the property that $f(a\cdot b)=f(a)+f(b)$ for all positive rational numbers $a$ and $b$ . Suppose that $f$ also has the property that $f(p)=p$ for every prime number $p$ . For which of the following numbers $x$ is $f(x)<0$ ?

$\textbf{(A) }\frac{17}{32} \qquad \textbf{(B) }\frac{11}{16} \qquad \textbf{(C) }\frac79 \qquad \textbf{(D) }\frac76\qquad \textbf{(E) }\frac{25}{11}$

Solution

Problem 19

How many solutions does the equation $\sin \left( \frac{\pi}2 \cos x\right)=\cos \left( \frac{\pi}2 \sin x\right)$ have in the closed interval $[0,\pi]$ ?

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

Solution

Problem 20

Suppose that on a parabola with vertex $V$ and a focus $F$ there exists a point $A$ such that $AF=20$ and $AV=21$ . What is the sum of all possible values of the length $FV?$

$\textbf{(A) }13 \qquad \textbf{(B) }\frac{40}3 \qquad \textbf{(C) }\frac{41}3 \qquad \textbf{(D) }14\qquad \textbf{(E) }\frac{43}3$

Solution

Problem 21

The five solutions to the equation $(z-1)(z^2+2z+4)(z^2+4z+6)=0$ may be written in the form $x_k+y_ki$ for $1\le k\le 5,$ where $x_k$ and $y_k$ are real. Let $\mathcal E$ be the unique ellipse that passes through the points $(x_1,y_1),(x_2,y_2),(x_3,y_3),(x_4,y_4),$ and $(x_5,y_5)$ . The eccentricity of $\mathcal E$ can be written in the form $\sqrt{\frac mn}$ , where $m$ and $n$ are relatively prime positive integers. What is $m+n$ ? (Recall that the eccentricity of an ellipse $\mathcal E$ is the ratio $\frac ca$ , where $2a$ is the length of the major axis of $\mathcal E$ and $2c$ is the is the distance between its two foci.)

$\textbf{(A) }7 \qquad \textbf{(B) }9 \qquad \textbf{(C) }11 \qquad \textbf{(D) }13\qquad \textbf{(E) }15$

Solution

Problem 22

Suppose that the roots of the polynomial $P(x)=x^3+ax^2+bx+c$ are $\cos \frac{2\pi}7,\cos \frac{4\pi}7,$ and $\cos \frac{6\pi}7$ , where angles are in radians. What is $abc$ ?

$\textbf{(A) }{-}\frac{3}{49} \qquad \textbf{(B) }{-}\frac{1}{28} \qquad \textbf{(C) }\frac{\sqrt[3]7}{64} \qquad \textbf{(D) }\frac{1}{32}\qquad \textbf{(E) }\frac{1}{28}$

Solution

Problem 23

Frieda the frog begins a sequence of hops on a $3\times3$ grid of squares, moving one square on each hop and choosing at random the direction of each hop up, down, left, or right. She does not hop diagonally. When the direction of a hop would take Frieda off the grid, she "wraps around" and jumps to the opposite edge. For example if Frieda begins in the center square and makes two hops "up", the first hop would place her in the top row middle square, and the second hop would cause Frieda to jump to the opposite edge, landing in the bottom row middle square. Suppose Frieda starts from the center square, makes at most four hops at random, and stops hopping if she lands on a corner square. What is the probability that she reaches a corner square on one of the four hops?

$\textbf{(A) }\frac{9}{16} \qquad \textbf{(B) }\frac{5}{8} \qquad \textbf{(C) }\frac34 \qquad \textbf{(D) }\frac{25}{32}\qquad \textbf{(E) }\frac{13}{16}$

Solution

Problem 24

Semicircle $\Gamma$ has diameter $\overline{AB}$ of length $14$ . Circle $\Omega$ lies tangent to $\overline{AB}$ at a point $P$ and intersects $\Gamma$ at points $Q$ and $R$ . If $QR=3\sqrt3$ and $\angle QPR=60^\circ$ , then the area of $\triangle PQR$ equals $\tfrac{a\sqrt{b}}{c}$ , where $a$ and $c$ are relatively prime positive integers, and $b$ is a positive integer not divisible by the square of any prime. What is $a+b+c$ ?

$\textbf{(A) }110 \qquad \textbf{(B) }114 \qquad \textbf{(C) }118 \qquad \textbf{(D) }122\qquad \textbf{(E) }126$

Solution

Problem 25

Let $d(n)$ denote the number of positive integers that divide $n$ , including $1$ and $n$ . For example, $d(1)=1,d(2)=2,$ and $d(12)=6$ . (This function is known as the divisor function.) Let $f(n)=\frac{d(n)}{\sqrt [3]n}.$ There is a unique positive integer $N$ such that $f(N)>f(n)$ for all positive integers $n\ne N$ . What is the sum of the digits of $N?$

$\textbf{(A) }5 \qquad \textbf{(B) }6 \qquad \textbf{(C) }7 \qquad \textbf{(D) }8\qquad \textbf{(E) }9$

Solution

@@ Line 11: / Line 11: @@
 <math>\textbf{(A) }</math> It is never true.
 <math>\textbf{(B) }</math> It is true if and only if <math>ab=0</math>.
 <math>\textbf{(C) }</math> It is true if and only if <math>a+b\ge 0</math>.
 <math>\textbf{(D) }</math> It is true if and only if <math>ab=0</math> and <math>a+b\ge 0</math>.
 <math>\textbf{(E) }</math> It is always true.
@@ Line 19: / Line 23: @@
 ==Problem 3==
-The sum of two natural numbers is <math>17,402</math>. One of the two numbers is divisible by <math>10</math>. If the units digit of that number is erased, the other number is obtained. What is the difference of these two numbers?
+The sum of two natural numbers is <math>17{,}402</math>. One of the two numbers is divisible by <math>10</math>. If the units digit of that number is erased, the other number is obtained. What is the difference of these two numbers?
-<math>\textbf{(A) }10,272 \qquad \textbf{(B) }11,700 \qquad \textbf{(C) }13,362 \qquad \textbf{(D) }14,238 \qquad \textbf{(E) }15,462</math>
+<math>\textbf{(A)} ~10{,}272\qquad\textbf{(B)} ~11{,}700\qquad\textbf{(C)} ~13{,}362\qquad\textbf{(D)} ~14{,}238\qquad\textbf{(E)} ~15{,}426</math>
 [[2021 AMC 12A Problems/Problem 3|Solution]]
@@ Line 27: / Line 31: @@
 ==Problem 4==
 Tom has a collection of <math>13</math> snakes, <math>4</math> of which are purple and <math>5</math> of which are happy. He observes that
-all of his happy snakes can add,
-none of his purple snakes can subtract, and
+* all of his happy snakes can add,
-all of his snakes that can't subtract also can't add.
+* none of his purple snakes can subtract, and
+* all of his snakes that can't subtract also can't add.
 Which of these conclusions can be drawn about Tom's snakes?
+<math>\textbf{(A) }</math> Purple snakes can add.
-<math>\textbf{(A) }</math> Purple snakes can add.
 <math>\textbf{(B) }</math> Purple snakes are happy.
 <math>\textbf{(C) }</math> Snakes that can add are purple.
 <math>\textbf{(D) }</math> Happy snakes are not purple.
 <math>\textbf{(E) }</math> Happy snakes can't subtract.
@@ Line 43: / Line 53: @@
 ==Problem 5==
-When a student multiplied the number <math>66</math> by the repeating decimal<cmath>\underline{1}.\underline{a}\underline{b}\underline{a}\underline{b}...=\underline{1}.\overline{\underline{a}\underline{b}},</cmath>where <math>a</math> and <math>b</math> are digits, he did not notice the notation and just multiplied <math>66</math> times <math>\underline{1}.\underline{a}\underline{b}</math>. Later he found that his answer is <math>0.5</math> less than the correct answer. What is the <math>2</math>-digit number <math>\underline{a}\underline{b}?</math>
+When a student multiplied the number <math>66</math> by the repeating decimal,
+<cmath>\underline{1}.\underline{a} \ \underline{b} \ \underline{a} \ \underline{b}\ldots=\underline{1}.\overline{\underline{a} \ \underline{b}},</cmath>
+where <math>a</math> and <math>b</math> are digits, he did not notice the notation and just multiplied <math>66</math> times <math>\underline{1}.\underline{a} \ \underline{b}.</math> Later he found that his answer is <math>0.5</math> less than the correct answer. What is the <math>2</math>-digit number <math>\underline{a} \ \underline{b}?</math>
 <math>\textbf{(A) }15 \qquad \textbf{(B) }30 \qquad \textbf{(C) }45 \qquad \textbf{(D) }60 \qquad \textbf{(E) }75</math>
 [[2021 AMC 12A Problems/Problem 5|Solution]]
@@ Line 65: / Line 76: @@
 ==Problem 8==
-A sequence of numbers is defined by <math>D_0=0,D_0=1,D_2=1</math> and <math>D_n=D_{n-1}+D_{n-3}</math> for <math>n\ge 3</math>. What are the parities (evenness or oddness) of the triple of numbers <math>(D_{2021},D_{2022},D_{2023})</math>, where <math>E</math> denotes even and <math>O</math> denotes odd?
+A sequence of numbers is defined by <math>D_0=0,D_1=0,D_2=1</math> and <math>D_n=D_{n-1}+D_{n-3}</math> for <math>n\ge 3</math>. What are the parities (evenness or oddness) of the triple of numbers <math>(D_{2021},D_{2022},D_{2023})</math>, where <math>E</math> denotes even and <math>O</math> denotes odd?
 <math>\textbf{(A) }(O,E,O) \qquad \textbf{(B) }(E,E,O) \qquad \textbf{(C) }(E,O,E) \qquad \textbf{(D) }(O,O,E) \qquad \textbf{(E) }(O,O,O)</math>
 [[2021 AMC 12A Problems/Problem 8|Solution]]
 ==Problem 9==
-Which of the following is equilvalent to<cmath>(2+3)(2^2+3^3)(2^4+3^4)(2^8+3^8)(2^{16}+3^{16})(2^{32}+3^{32})(2^{64}+3^{64})?</cmath>
+Which of the following is equivalent to<cmath>(2+3)(2^2+3^2)(2^4+3^4)(2^8+3^8)(2^{16}+3^{16})(2^{32}+3^{32})(2^{64}+3^{64})?</cmath>
 <math>\textbf{(A) }3^{127}+2^{127} \qquad \textbf{(B) }3^{127}+2^{127}+2\cdot 3^{63}+3\cdot 2^{63} \qquad \textbf{(C) }3^{128}-2^{128} \qquad \textbf{(D) }3^{128}+2^{128} \qquad \textbf{(E) }5^{127}</math>
@@ Line 79: / Line 89: @@
 ==Problem 10==
-These problems will not be posted until the 2021 AMC12A is released on Thursday, February 4, 2021.
+Two right circular cones with vertices facing down as shown in the figure below contain the same amount of liquid. The radii of the tops of the liquid surfaces are <math>3</math> cm and <math>6</math> cm. Into each cone is dropped a spherical marble of radius <math>1</math> cm, which sinks to the bottom and is completely submerged without spilling any liquid. What is the ratio of the rise of the liquid level in the narrow cone to the rise of the liquid level in the wide cone?
+<asy>
+size(350);
+defaultpen(linewidth(0.8));
+real h1 = 10, r = 3.1, s=0.75;
+pair P = (r,h1), Q = (-r,h1), Pp = s * P, Qp = s * Q;
+path e = ellipse((0,h1),r,0.9), ep = ellipse((0,h1*s),r*s,0.9);
+draw(ellipse(origin,r*(s-0.1),0.8));
+fill(ep,gray(0.8));
+fill(origin--Pp--Qp--cycle,gray(0.8));
+draw((-r,h1)--(0,0)--(r,h1)^^e);
+draw(subpath(ep,0,reltime(ep,0.5)),linetype("4 4"));
+draw(subpath(ep,reltime(ep,0.5),reltime(ep,1)));
+draw(Qp--(0,Qp.y),Arrows(size=8));
+draw(origin--(0,12),linetype("4 4"));
+draw(origin--(r*(s-0.1),0));
+label("$3$",(-0.9,h1*s),N,fontsize(10));
+real h2 = 7.5, r = 6, s=0.6, d = 14;
+pair P = (d+r-0.05,h2-0.15), Q = (d-r+0.05,h2-0.15), Pp = s * P + (1-s)*(d,0), Qp = s * Q + (1-s)*(d,0);
+path e = ellipse((d,h2),r,1), ep = ellipse((d,h2*s+0.09),r*s,1);
+draw(ellipse((d,0),r*(s-0.1),0.8));
+fill(ep,gray(0.8));
+fill((d,0)--Pp--Qp--cycle,gray(0.8));
+draw(P--(d,0)--Q^^e);
+draw(subpath(ep,0,reltime(ep,0.5)),linetype("4 4"));
+draw(subpath(ep,reltime(ep,0.5),reltime(ep,1)));
+draw(Qp--(d,Qp.y),Arrows(size=8));
+draw((d,0)--(d,10),linetype("4 4"));
+draw((d,0)--(d+r*(s-0.1),0));
+label("$6$",(d-r/4,h2*s-0.06),N,fontsize(10));
+</asy>
+<math>\textbf{(A) }1:1 \qquad \textbf{(B) }47:43 \qquad \textbf{(C) }2:1 \qquad \textbf{(D) }40:13 \qquad \textbf{(E) }4:1</math>
 [[2021 AMC 12A Problems/Problem 10|Solution]]
 ==Problem 11==
-A laser is placed at the point <math>(3,5)</math>. The laser bean travels in a straight line. Larry wants the beam to hit and bounce off the <math>y</math>-axis, then hit and bounce off the <math>x</math>-axis, then hit the point <math>(7,5)</math>. What is the total distance the beam will travel along this path?
+A laser is placed at the point <math>(3,5)</math>. The laser beam travels in a straight line. Larry wants the beam to hit and bounce off the <math>y</math>-axis, then hit and bounce off the <math>x</math>-axis, then hit the point <math>(7,5)</math>. What is the total distance the beam will travel along this path?
 <math>\textbf{(A) }2\sqrt{10} \qquad \textbf{(B) }5\sqrt2 \qquad \textbf{(C) }10\sqrt2 \qquad \textbf{(D) }15\sqrt2 \qquad \textbf{(E) }10\sqrt5</math>
@@ Line 93: / Line 137: @@
 All the roots of the polynomial <math>z^6-10z^5+Az^4+Bz^3+Cz^2+Dz+16</math> are positive integers, possibly repeated. What is the value of <math>B</math>?
-<math>\textbf{(A) }-88 \qquad \textbf{(B) }-80 \qquad \textbf{(C) }-64 \qquad \textbf{(D) }-41\qquad \textbf{(E) }-40</math>
+<math>\textbf{(A) }{-}88 \qquad \textbf{(B) }{-}80 \qquad \textbf{(C) }{-}64 \qquad \textbf{(D) }{-}41\qquad \textbf{(E) }{-}40</math>
 [[2021 AMC 12A Problems/Problem 12|Solution]]
@@ Line 100: / Line 144: @@
 Of the following complex numbers <math>z</math>, which one has the property that <math>z^5</math> has the greatest real part?
-<math>\textbf{(A) }-2 \qquad \textbf{(B) }-\sqrt3+i \qquad \textbf{(C) }-\sqrt2+\sqrt2 i \qquad \textbf{(D) }-1+\sqrt3 i\qquad \textbf{(E) }2i</math>
+<math>\textbf{(A) }{-}2 \qquad \textbf{(B) }{-}\sqrt3+i \qquad \textbf{(C) }{-}\sqrt2+\sqrt2 i \qquad \textbf{(D) }{-}1+\sqrt3 i\qquad \textbf{(E) }2i</math>
 [[2021 AMC 12A Problems/Problem 13|Solution]]
@@ Line 106: / Line 150: @@
 ==Problem 14==
 What is the value of<cmath>\left(\sum_{k=1}^{20} \log_{5^k} 3^{k^2}\right)\cdot\left(\sum_{k=1}^{100} \log_{9^k} 25^k\right)?</cmath>
-<math>\textbf{(A) }21 \qquad \textbf{(B) }100\log_5 3 \qquad \textbf{(C) }200\log_3 5 \qquad \textbf{(D) }2,200\qquad \textbf{(E) }21,000</math>
+<math>\textbf{(A) }21 \qquad \textbf{(B) }100\log_5 3 \qquad \textbf{(C) }200\log_3 5 \qquad \textbf{(D) }2{,}200\qquad \textbf{(E) }21{,}000</math>
 [[2021 AMC 12A Problems/Problem 14|Solution]]
@@ Line 113: / Line 157: @@
 A choir director must select a group of singers from among his <math>6</math> tenors and <math>8</math> basses. The only
 requirements are that the difference between the numbers of tenors and basses must be a multiple
-of <math>4</math>, and the group must have at least one singer. Let <math>N</math> be the number of groups that could be
+of <math>4</math>, and the group must have at least one singer. Let <math>N</math> be the number of different groups that could be
 selected. What is the remainder when <math>N</math> is divided by <math>100</math>?
@@ Line 121: / Line 165: @@
 ==Problem 16==
-In the following list of numbers, the integer <math>n</math> appears <math>n</math> times in the list for <math>1\le n \le 200</math>.<cmath>1,2,2,3,3,3,4,4,4,...,200,200,...,200</cmath>What is the median of the numbers in this list?
+In the following list of numbers, the integer <math>n</math> appears <math>n</math> times in the list for <math>1\le n \le 200</math>.<cmath>1,2,2,3,3,3,4,4,4,\ldots,200,200,\ldots,200</cmath>What is the median of the numbers in this list?
 <math>\textbf{(A) }100.5 \qquad \textbf{(B) }134 \qquad \textbf{(C) }142 \qquad \textbf{(D) }150.5\qquad \textbf{(E) }167</math>
@@ Line 128: / Line 172: @@
 ==Problem 17==
-Trapezoid <math>ABCD</math> has <math>\overline{AB}\parallel\overline{CD},BC=CD=43</math>, and <math>\overline{AD}\perp\overline{BD}</math>. Let <math>O</math> be the intersection of the diagonals <math>\overline{AC}</math> and <math>\overline{BD}</math>, and let <math>P</math> be the midpoint of <math>\overline{BD}</math>. Given that <math>OP=11</math>, the length of <math>AD</math> can be written in the form <math>m\sqrt{n}</math>, where <math>m</math> and <math>n</math> are positive integers and <math>n</math> is not divisible by the quare of any prime. What is <math>m+n</math>?
+Trapezoid <math>ABCD</math> has <math>\overline{AB}\parallel\overline{CD},BC=CD=43</math>, and <math>\overline{AD}\perp\overline{BD}</math>. Let <math>O</math> be the intersection of the diagonals <math>\overline{AC}</math> and <math>\overline{BD}</math>, and let <math>P</math> be the midpoint of <math>\overline{BD}</math>. Given that <math>OP=11</math>, the length of <math>AD</math> can be written in the form <math>m\sqrt{n}</math>, where <math>m</math> and <math>n</math> are positive integers and <math>n</math> is not divisible by the square of any prime. What is <math>m+n</math>?
 <math>\textbf{(A) }65 \qquad \textbf{(B) }132 \qquad \textbf{(C) }157 \qquad \textbf{(D) }194\qquad \textbf{(E) }215</math>
@@ Line 142: / Line 186: @@
 ==Problem 19==
-These problems will not be posted until the 2021 AMC12A is released on Thursday, February 4, 2021.
+How many solutions does the equation <math>\sin \left( \frac{\pi}2 \cos x\right)=\cos \left( \frac{\pi}2 \sin x\right)</math> have in the closed interval <math>[0,\pi]</math>?
+<math>\textbf{(A) }0 \qquad \textbf{(B) }1 \qquad \textbf{(C) }2 \qquad \textbf{(D) }3\qquad \textbf{(E) }4</math>
 [[2021 AMC 12A Problems/Problem 19|Solution]]
 ==Problem 20==
-These problems will not be posted until the 2021 AMC12A is released on Thursday, February 4, 2021.
+Suppose that on a parabola with vertex <math>V</math> and a focus <math>F</math> there exists a point <math>A</math> such that <math>AF=20</math> and <math>AV=21</math>. What is the sum of all possible values of the length <math>FV?</math>
+<math>\textbf{(A) }13 \qquad \textbf{(B) }\frac{40}3 \qquad \textbf{(C) }\frac{41}3 \qquad \textbf{(D) }14\qquad \textbf{(E) }\frac{43}3</math>
 [[2021 AMC 12A Problems/Problem 20|Solution]]
 ==Problem 21==
-These problems will not be posted until the 2021 AMC12A is released on Thursday, February 4, 2021.
+The five solutions to the equation<cmath>(z-1)(z^2+2z+4)(z^2+4z+6)=0</cmath> may be written in the form <math>x_k+y_ki</math> for <math>1\le k\le 5,</math> where <math>x_k</math> and <math>y_k</math> are real. Let <math>\mathcal E</math> be the unique ellipse that passes through the points <math>(x_1,y_1),(x_2,y_2),(x_3,y_3),(x_4,y_4),</math> and <math>(x_5,y_5)</math>. The eccentricity of <math>\mathcal E</math> can be written in the form <math>\sqrt{\frac mn}</math>, where <math>m</math> and <math>n</math> are relatively prime positive integers. What is <math>m+n</math>? (Recall that the eccentricity of an ellipse <math>\mathcal E</math> is the ratio <math>\frac ca</math>, where <math>2a</math> is the length of the major axis of <math>\mathcal E</math> and <math>2c</math> is the is the distance between its two foci.)
+<math>\textbf{(A) }7 \qquad \textbf{(B) }9 \qquad \textbf{(C) }11 \qquad \textbf{(D) }13\qquad \textbf{(E) }15</math>
 [[2021 AMC 12A Problems/Problem 21|Solution]]
 ==Problem 22==
-These problems will not be posted until the 2021 AMC12A is released on Thursday, February 4, 2021.
+Suppose that the roots of the polynomial <math>P(x)=x^3+ax^2+bx+c</math> are <math>\cos \frac{2\pi}7,\cos \frac{4\pi}7,</math> and <math>\cos \frac{6\pi}7</math>, where angles are in radians. What is <math>abc</math>?
+<math>\textbf{(A) }{-}\frac{3}{49} \qquad \textbf{(B) }{-}\frac{1}{28} \qquad \textbf{(C) }\frac{\sqrt[3]7}{64} \qquad \textbf{(D) }\frac{1}{32}\qquad \textbf{(E) }\frac{1}{28}</math>
 [[2021 AMC 12A Problems/Problem 22|Solution]]
 ==Problem 23==
-These problems will not be posted until the 2021 AMC12A is released on Thursday, February 4, 2021.
+Frieda the frog begins a sequence of hops on a <math>3\times3</math> grid of squares, moving one square on each hop and choosing at random the direction of each hop up, down, left, or right. She does not hop diagonally. When the direction of a hop would take Frieda off the grid, she "wraps around" and jumps to the opposite edge. For example if Frieda begins in the center square and makes two hops "up", the first hop would place her in the top row middle square, and the second hop would cause Frieda to jump to the opposite edge, landing in the bottom row middle square. Suppose Frieda starts from the center square, makes at most four hops at random, and stops hopping if she lands on a corner square. What is the probability that she reaches a corner square on one of the four hops?
+<math>\textbf{(A) }\frac{9}{16} \qquad \textbf{(B) }\frac{5}{8} \qquad \textbf{(C) }\frac34 \qquad \textbf{(D) }\frac{25}{32}\qquad \textbf{(E) }\frac{13}{16}</math>
 [[2021 AMC 12A Problems/Problem 23|Solution]]
 ==Problem 24==
-These problems will not be posted until the 2021 AMC12A is released on Thursday, February 4, 2021.
+Semicircle <math>\Gamma</math> has diameter <math>\overline{AB}</math> of length <math>14</math>. Circle <math>\Omega</math> lies tangent to <math>\overline{AB}</math> at a point <math>P</math> and intersects <math>\Gamma</math> at points <math>Q</math> and <math>R</math>. If <math>QR=3\sqrt3</math> and <math>\angle QPR=60^\circ</math>, then the area of <math>\triangle PQR</math> equals <math>\tfrac{a\sqrt{b}}{c}</math>, where <math>a</math> and <math>c</math> are relatively prime positive integers, and <math>b</math> is a positive integer not divisible by the square of any prime. What is <math>a+b+c</math>?
+<math>\textbf{(A) }110 \qquad \textbf{(B) }114 \qquad \textbf{(C) }118 \qquad \textbf{(D) }122\qquad \textbf{(E) }126</math>
 [[2021 AMC 12A Problems/Problem 24|Solution]]

2021 AMC 12A (Problems • Answer Key • Resources)
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Difference between revisions of "2021 AMC 12A Problems"

Latest revision as of 08:49, 25 June 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