AMC 10 CLASSES
AoPS has trained thousands of the top scorers on AMC tests over the last 20 years in our online AMC 10 Problem Series course.
CHECK SCHEDULE

Difference between revisions of "2021 AMC 10A Problems"

m
(Problem 3)
(35 intermediate revisions by 11 users not shown)
Line 27: Line 27:
  
 
==Problem 5==
 
==Problem 5==
The quiz scores of a class with <math>k > 12</math> students have a mean of <math>8</math>. The mean of a collection of <math>12</math> of these quiz scores is <math>14</math>. What is the mean of the remaining quiz scores of terms of <math>k</math>?
+
The quiz scores of a class with <math>k > 12</math> students have a mean of <math>8</math>. The mean of a collection of <math>12</math> of these quiz scores is <math>14</math>. What is the mean of the remaining quiz scores in terms of <math>k</math>?
  
 
<math>\textbf{(A)} ~\frac{14-8}{k-12} \qquad\textbf{(B)} ~\frac{8k-168}{k-12} \qquad\textbf{(C)} ~\frac{14}{12} - \frac{8}{k} \qquad\textbf{(D)} ~\frac{14(k-12)}{k^2} \qquad\textbf{(E)} ~\frac{14(k-12)}{8k}</math>
 
<math>\textbf{(A)} ~\frac{14-8}{k-12} \qquad\textbf{(B)} ~\frac{8k-168}{k-12} \qquad\textbf{(C)} ~\frac{14}{12} - \frac{8}{k} \qquad\textbf{(D)} ~\frac{14(k-12)}{k^2} \qquad\textbf{(E)} ~\frac{14(k-12)}{8k}</math>
Line 41: Line 41:
 
[[2021 AMC 10A Problems/Problem 6|Solution]]
 
[[2021 AMC 10A Problems/Problem 6|Solution]]
 
==Problem 7==
 
==Problem 7==
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 knows that:
+
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
+
* 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?
 
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{(B) }</math> Purple snakes are happy.
<math>\textbf{(D)}</math> Happy snakes are not purple.
+
 
<math>\textbf{(E)}</math> Happy snakes can't subtract.
+
<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.
  
 
[[2021 AMC 10A Problems/Problem 7|Solution]]
 
[[2021 AMC 10A Problems/Problem 7|Solution]]
 +
 
==Problem 8==
 
==Problem 8==
 
When a student multiplied the number <math>66</math> by the repeating decimal
 
When a student multiplied the number <math>66</math> by the repeating decimal
Line 74: Line 82:
 
<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>
 
<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} + 3^{128} \qquad\textbf{(E)} ~5^{127}</math>
 
<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} + 3^{128} \qquad\textbf{(E)} ~5^{127}</math>
 +
 +
[[2021 AMC 10A Problems/Problem 10|Solution]]
  
 
==Problem 11==
 
==Problem 11==
Line 83: Line 93:
  
 
==Problem 12==
 
==Problem 12==
 +
 +
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 \text{ cm}</math> and <math>6 \text{ cm}</math>. Into each cone is dropped a spherical marble of radius <math>1 \text{ cm}</math>, 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 10A Problems/Problem 12|Solution]]
 +
 
==Problem 13==
 
==Problem 13==
 
What is the volume of tetrahedron <math>ABCD</math> with edge lengths <math>AB = 2</math>, <math>AC = 3</math>, <math>AD = 4</math>, <math>BC = \sqrt{13}</math>, <math>BD = 2\sqrt{5}</math>, and <math>CD = 5</math> ?
 
What is the volume of tetrahedron <math>ABCD</math> with edge lengths <math>AB = 2</math>, <math>AC = 3</math>, <math>AD = 4</math>, <math>BC = \sqrt{13}</math>, <math>BD = 2\sqrt{5}</math>, and <math>CD = 5</math> ?
Line 90: Line 139:
 
[[2021 AMC 10A Problems/Problem 13|Solution]]
 
[[2021 AMC 10A Problems/Problem 13|Solution]]
 
==Problem 14==
 
==Problem 14==
All the roots of 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>?
+
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 10A Problems/Problem 14|Solution]]
  
 
==Problem 15==
 
==Problem 15==
Line 105: Line 156:
  
 
<math>\textbf{(A)} ~100.5\qquad\textbf{(B)} ~134\qquad\textbf{(C)} ~142\qquad\textbf{(D)} ~150.5\qquad\textbf{(E)} ~167</math>
 
<math>\textbf{(A)} ~100.5\qquad\textbf{(B)} ~134\qquad\textbf{(C)} ~142\qquad\textbf{(D)} ~150.5\qquad\textbf{(E)} ~167</math>
 +
 +
[[2021 AMC 10A Problems/Problem 16|Solution]]
  
 
==Problem 17==
 
==Problem 17==
A choir direction 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 number 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 different groups that could be selected. What is the remainder when <math>N</math> is divided by <math>100</math>?
+
Trapezoid <math>ABCD</math> has <math>\overline{AB} \parallel \overline{CD}</math>, <math>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 <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) } 47\qquad\textbf{(B) } 48\qquad\textbf{(C) } 83\qquad\textbf{(D) } 95\qquad\textbf{(E) } 96\qquad</math>
+
<math>\textbf{(A)} ~65\qquad\textbf{(B)} ~132\qquad\textbf{(C)} ~157\qquad\textbf{(D)} ~194\qquad\textbf{(E)} ~215</math>
  
 
[[2021 AMC 10A Problems/Problem 17|Solution]]
 
[[2021 AMC 10A Problems/Problem 17|Solution]]
  
 
==Problem 18==
 
==Problem 18==
Let <math>f</math> be a function defined on the set of positive rational numbers with the property that <math>f(a \cdot b) = f(a) + f(b)</math> for all positive rational numbers <math>a</math> and <math>b</math>. Suppose that <math>f</math> also have the property that <math>f(p) = p</math> for every prime number <math>p</math>. For which of the following numbers <math>x</math> is <math>f(x) < 0</math>?
+
Let <math>f</math> be a function defined on the set of positive rational numbers with the property that <math>f(a\cdot b)=f(a)+f(b)</math> for all positive rational numbers <math>a</math> and <math>b</math>. Furthermore, suppose that <math>f</math> also has the property that <math>f(p)=p</math> for every prime number <math>p</math>. For which of the following numbers <math>x</math> is <math>f(x)<0</math>?
  
 
<math>\textbf{(A)} ~\frac{17}{32}\qquad\textbf{(B)} ~\frac{11}{16}\qquad\textbf{(C)} ~\frac{7}{9}\qquad\textbf{(D)} ~\frac{7}{6} \qquad\textbf{(E)} ~\frac{25}{11}</math>
 
<math>\textbf{(A)} ~\frac{17}{32}\qquad\textbf{(B)} ~\frac{11}{16}\qquad\textbf{(C)} ~\frac{7}{9}\qquad\textbf{(D)} ~\frac{7}{6} \qquad\textbf{(E)} ~\frac{25}{11}</math>
 +
 +
[[2021 AMC 10A Problems/Problem 18|Solution]]
  
 
==Problem 19==
 
==Problem 19==
Line 129: Line 184:
  
 
<math>\textbf{(A)} ~10\qquad\textbf{(B)} ~18\qquad\textbf{(C)} ~24\qquad\textbf{(D)} ~32\qquad\textbf{(E)} ~44</math>
 
<math>\textbf{(A)} ~10\qquad\textbf{(B)} ~18\qquad\textbf{(C)} ~24\qquad\textbf{(D)} ~32\qquad\textbf{(E)} ~44</math>
 +
 +
[[2021 AMC 10A Problems/Problem 20|Solution]]
  
 
==Problem 21==
 
==Problem 21==
Line 134: Line 191:
  
 
<math>\textbf{(A)} ~47\qquad\textbf{(B)} ~52\qquad\textbf{(C)} ~55\qquad\textbf{(D)} ~58\qquad\textbf{(E)} ~63</math>
 
<math>\textbf{(A)} ~47\qquad\textbf{(B)} ~52\qquad\textbf{(C)} ~55\qquad\textbf{(D)} ~58\qquad\textbf{(E)} ~63</math>
 +
 +
[[2021 AMC 10A Problems/Problem 21|Solution]]
  
 
==Problem 22==
 
==Problem 22==
Line 139: Line 198:
  
 
<math>\textbf{(A)} ~10\qquad\textbf{(B)} ~13\qquad\textbf{(C)} ~15\qquad\textbf{(D)} ~17\qquad\textbf{(E)} ~20</math>
 
<math>\textbf{(A)} ~10\qquad\textbf{(B)} ~13\qquad\textbf{(C)} ~15\qquad\textbf{(D)} ~17\qquad\textbf{(E)} ~20</math>
 +
 +
[[2021 AMC 10A Problems/Problem 22|Solution]]
  
 
==Problem 23==
 
==Problem 23==
Line 151: Line 212:
  
 
<math>\textbf{(A)} ~\frac{8a^2}{(a+1)^2}\qquad\textbf{(B)} ~\frac{4a}{a+1}\qquad\textbf{(C)} ~\frac{8a}{a+1}\qquad\textbf{(D)} ~\frac{8a^2}{a^2+1}\qquad\textbf{(E)} ~\frac{8a}{a^2+1}</math>
 
<math>\textbf{(A)} ~\frac{8a^2}{(a+1)^2}\qquad\textbf{(B)} ~\frac{4a}{a+1}\qquad\textbf{(C)} ~\frac{8a}{a+1}\qquad\textbf{(D)} ~\frac{8a^2}{a^2+1}\qquad\textbf{(E)} ~\frac{8a}{a^2+1}</math>
 +
 +
[[2021 AMC 10A Problems/Problem 24|Solution]]
  
 
==Problem 25==
 
==Problem 25==
Line 157: Line 220:
 
<math>\textbf{(A)} ~12\qquad\textbf{(B)} ~18\qquad\textbf{(C)} ~24\qquad\textbf{(D)} ~30\qquad\textbf{(E)} ~36</math>
 
<math>\textbf{(A)} ~12\qquad\textbf{(B)} ~18\qquad\textbf{(C)} ~24\qquad\textbf{(D)} ~30\qquad\textbf{(E)} ~36</math>
  
 +
[[2021 AMC 10A Problems/Problem 25|Solution]]
  
 
==See also==
 
==See also==
{{AMC12 box|year=2021|ab=A|num-b=15|num-a=17}}
+
{{AMC10 box|year=2021|ab=A|before=[[2020 AMC 10B]]|after=[[2021 AMC 10B]]}}
 +
* [[AMC 10]]
 +
* [[AMC 10 Problems and Solutions]]
 +
* [[Mathematics competitions]]
 +
* [[Mathematics competition resources]]
 
{{MAA Notice}}
 
{{MAA Notice}}

Revision as of 16:01, 27 April 2021

2021 AMC 10A (Answer Key)
Printable versions: WikiAoPS ResourcesPDF

Instructions

  1. 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.
  2. 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.
  3. 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).
  4. Figures are not necessarily drawn to scale.
  5. 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\[(2^2-2)-(3^2-3)+(4^2-4)?\]$\textbf{(A)} ~1 \qquad\textbf{(B)} ~2 \qquad\textbf{(C)} ~5 \qquad\textbf{(D)} ~8 \qquad\textbf{(E)} ~12$

Solution

Problem 2

Portia's high school has $3$ times as many students as Lara's high school. The two high schools have a total of $2600$ students. How many students does Portia's high school have?

$\textbf{(A)} ~600 \qquad\textbf{(B)} ~650 \qquad\textbf{(C)} ~1950 \qquad\textbf{(D)} ~2000\qquad\textbf{(E)} ~2050$

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

A cart rolls down a hill, travelling $5$ inches the first second and accelerating so that during each successive $1$-second time interval, it travels $7$ inches more than during the previous $1$-second interval. The cart takes $30$ seconds to reach the bottom of the hill. How far, in inches, does it travel?

$\textbf{(A)} ~215 \qquad\textbf{(B)} ~360\qquad\textbf{(C)} ~2992\qquad\textbf{(D)} ~3195\qquad\textbf{(E)} ~3242$

Solution

Problem 5

The quiz scores of a class with $k > 12$ students have a mean of $8$. The mean of a collection of $12$ of these quiz scores is $14$. What is the mean of the remaining quiz scores in terms of $k$?

$\textbf{(A)} ~\frac{14-8}{k-12} \qquad\textbf{(B)} ~\frac{8k-168}{k-12} \qquad\textbf{(C)} ~\frac{14}{12} - \frac{8}{k} \qquad\textbf{(D)} ~\frac{14(k-12)}{k^2} \qquad\textbf{(E)} ~\frac{14(k-12)}{8k}$

Solution

Problem 6

Chantal and Jean start hiking from a trailhead toward a fire tower. Jean is wearing a heavy backpack and walks slower. Chantal starts walking at $4$ miles per hour. Halfway to the tower, the trail becomes really steep, and Chantal slows down to $2$ miles per hour. After reaching the tower, she immediately turns around and descends the steep part of the trail at $3$ miles per hour. She meets Jean at the halfway point. What was Jean's average speed, in miles per hour, until they meet?

$\textbf{(A)} ~\frac{12}{13} \qquad\textbf{(B)} ~1 \qquad\textbf{(C)} ~\frac{13}{12} \qquad\textbf{(D)}   ~\frac{24}{13} \qquad\textbf{(E)} ~2$

Solution

Problem 7

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 8

When a student multiplied the number $66$ by the repeating decimal \[\underline{1}.\underline{a}    \underline{b} \underline{a} \underline{b} \cdots =   \underline{1}.\overline{\underline{ab}}\]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 integer $\underline{ab}$?

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

Solution

Problem 9

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

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

Solution

Problem 10

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

Solution

Problem 11

For which of the following integers $b$ is the base-$b$ number $2021_b - 221_b$ not divisible by $3$?

$\textbf{(A)} ~3 \qquad\textbf{(B)} ~4\qquad\textbf{(C)} ~6\qquad\textbf{(D)} ~7\qquad\textbf{(E)} ~8$

Solution

Problem 12

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 $3 \text{ cm}$ and $6 \text{ cm}$. Into each cone is dropped a spherical marble of radius $1 \text{ cm}$, 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 13

What is the volume of tetrahedron $ABCD$ with edge lengths $AB = 2$, $AC = 3$, $AD = 4$, $BC = \sqrt{13}$, $BD = 2\sqrt{5}$, and $CD = 5$ ?

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

Solution

Problem 14

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 15

Values for $A,B,C,$ and $D$ are to be selected from $\{1, 2, 3, 4, 5, 6\}$ without replacement (i.e. no two letters have the same value). How many ways are there to make such choices so that the two curves $y=Ax^2+B$ and $y=Cx^2+D$ intersect? (The order in which the curves are listed does not matter; for example, the choices $A=3, B=2, C=4, D=1$ is considered the same as the choices $A=4, B=1, C=3, D=2.$)

$\textbf{(A) }30 \qquad \textbf{(B) }60 \qquad \textbf{(C) }90 \qquad \textbf{(D) }180 \qquad \textbf{(E) }360$

Solution

Problem 16

In the following list of numbers, the integer $n$ appears $n$ times in the list for $1 \leq n \leq 200$.\[1, 2, 2, 3, 3, 3, 4, 4, 4, 4, \dots, 200, 200, \dots , 200\]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 $AD$ 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 $b$. Furthermore, suppose that $f$ also has the property that $f(p)=p$ for every prime number $p$. For which of the following numbers $x$ is $f(x)<0$?

$\textbf{(A)} ~\frac{17}{32}\qquad\textbf{(B)} ~\frac{11}{16}\qquad\textbf{(C)} ~\frac{7}{9}\qquad\textbf{(D)} ~\frac{7}{6} \qquad\textbf{(E)} ~\frac{25}{11}$

Solution

Problem 19

The area of the region bounded by the graph of \[x^2+y^2 = 3|x-y| + 3|x+y|\]is $m+n\pi$, where $m$ and $n$ are integers. What is $m + n$?

$\textbf{(A)} ~18\qquad\textbf{(B)} ~27\qquad\textbf{(C)} ~36\qquad\textbf{(D)} ~45\qquad\textbf{(E)} ~54$

Solution

Problem 20

In how many ways can the sequence $1, 2, 3, 4, 5$ be rearranged so that no three consecutive terms are increasing and no three consecutive terms are decreasing?

$\textbf{(A)} ~10\qquad\textbf{(B)} ~18\qquad\textbf{(C)} ~24\qquad\textbf{(D)} ~32\qquad\textbf{(E)} ~44$

Solution

Problem 21

Let $ABCDEF$ be an equiangular hexagon. The lines $AB, CD,$ and $EF$ determine a triangle with area $192\sqrt{3}$, and the lines $BC, DE,$ and $FA$ determine a triangle with area $324\sqrt{3}$. The perimeter of hexagon $ABCDEF$ can be expressed as $m +n\sqrt{p}$, where $m, n,$ and $p$ are positive integers and $p$ is not divisible by the square of any prime. What is $m + n + p$?

$\textbf{(A)} ~47\qquad\textbf{(B)} ~52\qquad\textbf{(C)} ~55\qquad\textbf{(D)} ~58\qquad\textbf{(E)} ~63$

Solution

Problem 22

Hiram's algebra notes are $50$ pages long and are printed on $25$ sheets of paper; the first sheet contains pages $1$ and $2$, the second sheet contains pages $3$ and $4$, and so on. One day he leaves his notes on the table before leaving for lunch, and his roommate decides to borrow some pages from the middle of the notes. When Hiram comes back, he discovers that his roommate has taken a consecutive set of sheets from the notes and that the average (mean) of the page numbers on all remaining sheets is exactly $19$. How many sheets were borrowed?

$\textbf{(A)} ~10\qquad\textbf{(B)} ~13\qquad\textbf{(C)} ~15\qquad\textbf{(D)} ~17\qquad\textbf{(E)} ~20$

Solution

Problem 23

Frieda the frog begins a sequence of hops on a $3 \times 3$ 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?

$\textbf{(A)} ~\frac{9}{16}\qquad\textbf{(B)} ~\frac{5}{8}\qquad\textbf{(C)} ~\frac{3}{4}\qquad\textbf{(D)} ~\frac{25}{32}\qquad\textbf{(E)} ~\frac{13}{16}$

Solution

Problem 24

The interior of a quadrilateral is bounded by the graphs of $(x+ay)^2 = 4a^2$ and $(ax-y)^2 = a^2$, where $a$ a positive real number. What is the area of this region in terms of $a$, valid for all $a > 0$?

$\textbf{(A)} ~\frac{8a^2}{(a+1)^2}\qquad\textbf{(B)} ~\frac{4a}{a+1}\qquad\textbf{(C)} ~\frac{8a}{a+1}\qquad\textbf{(D)} ~\frac{8a^2}{a^2+1}\qquad\textbf{(E)} ~\frac{8a}{a^2+1}$

Solution

Problem 25

How many ways are there to place $3$ indistinguishable red chips, $3$ indistinguishable blue chips, and $3$ indistinguishable green chips in the squares of a $3 \times 3$ grid so that no two chips of the same color are directly adjacent to each other, either vertically or horizontally.

$\textbf{(A)} ~12\qquad\textbf{(B)} ~18\qquad\textbf{(C)} ~24\qquad\textbf{(D)} ~30\qquad\textbf{(E)} ~36$

Solution

See also

2021 AMC 10A (ProblemsAnswer KeyResources)
Preceded by
2020 AMC 10B
Followed by
2021 AMC 10B
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
All AMC 10 Problems and Solutions

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. AMC logo.png