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Difference between revisions of "2020 AMC 10B Problems"
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==Problem 21==
 
==Problem 21==
  
−
These problems will not be available until the 2020 AMC 10B contest is released on Wednesday, February 5, 2020.
+
In square <math>ABCD</math>, points <math>E</math> and <math>H</math> lie on <math>\overline{AB}</math> and <math>\overline{DA}</math>, respectively, so that <math>AE=AH.</math> Points <math>F</math> and <math>G</math> lie on <math>\overline{BC}</math> and <math>\overline{CD}</math>, respectively, and points <math>I</math> and <math>J</math> lie on <math>\overline{EH}</math> so that <math>\overline{FI} \perp \overline{EH}</math> and <math>\overline{GJ} \perp \overline{EH}</math>. See the figure below. Triangle <math>AEH</math>, quadrilateral <math>BFIE</math>, quadrilateral <math>DHJG</math>, and pentagon <math>FCGJI</math> each has area <math>1.</math> What is <math>FI^2</math>?
 +
<asy>
 +
real x=2sqrt(2);
 +
real y=2sqrt(16-8sqrt(2))-4+2sqrt(2);
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real z=2sqrt(8-4sqrt(2));
 +
pair A, B, C, D, E, F, G, H, I, J;
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A = (0,0);
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B = (4,0);
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C = (4,4);
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D = (0,4);
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E = (x,0);
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F = (4,y);
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G = (y,4);
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H = (0,x);
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I = F + z * dir(225);
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J = G + z * dir(225);
 +
 
 +
draw(A--B--C--D--A);
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draw(H--E);
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draw(J--G^^F--I);
 +
draw(rightanglemark(G, J, I), linewidth(.5));
 +
draw(rightanglemark(F, I, E), linewidth(.5));
 +
 
 +
dot("$A$", A, S);
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dot("$B$", B, S);
 +
dot("$C$", C, dir(90));
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dot("$D$", D, dir(90));
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dot("$E$", E, S);
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dot("$F$", F, dir(0));
 +
dot("$G$", G, N);
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dot("$H$", H, W);
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dot("$I$", I, SW);
 +
dot("$J$", J, SW);
 +
 
 +
</asy>
 +
<math>\textbf{(A) } \frac{7}{3} \qquad \textbf{(B) } 8-4\sqrt2 \qquad \textbf{(C) } 1+\sqrt2 \qquad \textbf{(D) } \frac{7}{4}\sqrt2 \qquad \textbf{(E) } 2\sqrt2</math>
  
 
[[2020 AMC 10B Problems/Problem 21|Solution]]
 
[[2020 AMC 10B Problems/Problem 21|Solution]]
Line 129: Line 164:
 
==Problem 22==
 
==Problem 22==
  
−
These problems will not be available until the 2020 AMC 10B contest is released on Wednesday, February 5, 2020.
+
What is the remainder when <math>2^{202} +202</math> is divided by <math>2^{101}+2^{51}+1</math>?
 +
 
 +
<math>\textbf{(A) } 100 \qquad\textbf{(B) } 101 \qquad\textbf{(C) } 200 \qquad\textbf{(D) } 201 \qquad\textbf{(E) } 202</math>
  
 
[[2020 AMC 10B Problems/Problem 22|Solution]]
 
[[2020 AMC 10B Problems/Problem 22|Solution]]
Line 135: Line 172:
 
==Problem 23==
 
==Problem 23==
  
−
These problems will not be available until the 2020 AMC 10B contest is released on Wednesday, February 5, 2020.
+
Square <math>ABCD</math> in the coordinate plane has vertices at the points <math>A(1,1), B(-1,1), C(-1,-1),</math> and <math>D(1,-1).</math> Consider the following four transformations:
 +
<math>L,</math> a rotation of <math>90^{\circ}</math> counterclockwise around the origin;
 +
<math>R,</math> a rotation of <math>90^{\circ}</math> clockwise around the origin;
 +
<math>H,</math> a reflection across the <math>x</math>-axis; and
 +
<math>V,</math> a reflection across the <math>y</math>-axis.
  
 +
Each of these transformations maps the squares onto itself, but the positions of the labeled vertices will change. For example, applying <math>R</math> and then <math>V</math> would send the vertex <math>A</math> at <math>(1,1)</math> to <math>(-1,-1)</math> and would send the vertex <math>B</math> at <math>(-1,1)</math> to itself. How many sequences of <math>20</math> transformations chosen from <math>\{L, R, H, V\}</math> will send all of the labeled vertices back to their original positions? (For example, <math>R, R, V, H</math> is one sequence of <math>4</math> transformations that will send the vertices back to their original positions.)
 +
 +
<math>\textbf{(A)}\ 2^{37} \qquad\textbf{(B)}\ 3\cdot 2^{36} \qquad\textbf{(C)}\  2^{38} \qquad\textbf{(D)}\ 3\cdot 2^{37} \qquad\textbf{(E)}\ 2^{39}</math>
 
[[2020 AMC 10B Problems/Problem 23|Solution]]
 
[[2020 AMC 10B Problems/Problem 23|Solution]]
  
 
==Problem 24==
 
==Problem 24==
  
−
These problems will not be available until the 2020 AMC 10B contest is released on Wednesday, February 5, 2020.
+
How many positive integers <math>n</math> satisfy<cmath>\dfrac{n+1000}{70} = \lfloor \sqrt{n} \rfloor?</cmath>(Recall that <math>\lfloor x\rfloor</math> is the greatest integer not exceeding <math>x</math>.)
 +
 
 +
<math>\textbf{(A) } 2 \qquad\textbf{(B) } 4 \qquad\textbf{(C) } 6 \qquad\textbf{(D) } 30 \qquad\textbf{(E) } 32</math>
  
 
[[2020 AMC 10B Problems/Problem 24|Solution]]
 
[[2020 AMC 10B Problems/Problem 24|Solution]]
Line 147: Line 193:
 
==Problem 25==
 
==Problem 25==
  
−
These problems will not be available until the 2020 AMC 10B contest is released on Wednesday, February 5, 2020.
+
Let <math>D(n)</math> denote the number of ways of writing the positive integer <math>n</math> as a product<cmath>n = f_1\cdot f_2\cdots f_k,</cmath>where <math>k\ge1</math>, the <math>f_i</math> are integers strictly greater than <math>1</math>, and the order in which the factors are listed matters (that is, two representations that differ only in the order of the factors are counted as distinct). For example, the number <math>6</math> can be written as <math>6</math>, <math>2\cdot 3</math>, and <math>3\cdot2</math>, so <math>D(6) = 3</math>. What is <math>D(96)</math>?
 +
 
 +
<math>\textbf{(A) } 112 \qquad\textbf{(B) } 128 \qquad\textbf{(C) } 144 \qquad\textbf{(D) } 172 \qquad\textbf{(E) } 184</math>
  
 
[[2020 AMC 10B Problems/Problem 25|Solution]]
 
[[2020 AMC 10B Problems/Problem 25|Solution]]

Revision as of 15:10, 7 February 2020

2020 AMC 10B (Answer Key)
Printable versions: Wiki • AoPS Resources • PDF

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

These problems will not be available until the 2020 AMC 10B contest is released on Wednesday, February 5, 2020.

Solution

Problem 2

These problems will not be available until the 2020 AMC 10B contest is released on Wednesday, February 5, 2020.

Solution

Problem 3

These problems will not be available until the 2020 AMC 10B contest is released on Wednesday, February 5, 2020.

Solution

Problem 4

These problems will not be available until the 2020 AMC 10B contest is released on Wednesday, February 5, 2020.

Solution

Problem 5

These problems will not be available until the 2020 AMC 10B contest is released on Wednesday, February 5, 2020.

Solution

Problem 6

These problems will not be available until the 2020 AMC 10B contest is released on Wednesday, February 5, 2020.

Solution

Problem 7

These problems will not be available until the 2020 AMC 10B contest is released on Wednesday, February 5, 2020.

Solution

Problem 8

These problems will not be available until the 2020 AMC 10B contest is released on Wednesday, February 5, 2020.

Solution

Problem 9

These problems will not be available until the 2020 AMC 10B contest is released on Wednesday, February 5, 2020.

Solution

Problem 10

These problems will not be available until the 2020 AMC 10B contest is released on Wednesday, February 5, 2020.

Solution

Problem 11

These problems will not be available until the 2020 AMC 10B contest is released on Wednesday, February 5, 2020.

Solution

Problem 12

These problems will not be available until the 2020 AMC 10B contest is released on Wednesday, February 5, 2020.

Solution

Problem 13

These problems will not be available until the 2020 AMC 10B contest is released on Wednesday, February 5, 2020.

Solution

Problem 14

These problems will not be available until the 2020 AMC 10B contest is released on Wednesday, February 5, 2020.

Solution

Problem 15

These problems will not be available until the 2020 AMC 10B contest is released on Wednesday, February 5, 2020.

Solution

Problem 16

These problems will not be available until the 2020 AMC 10B contest is released on Wednesday, February 5, 2020.

Solution

Problem 17

These problems will not be available until the 2020 AMC 10B contest is released on Wednesday, February 5, 2020.

Solution

Problem 18

These problems will not be available until the 2020 AMC 10B contest is released on Wednesday, February 5, 2020.

Solution

Problem 19

These problems will not be available until the 2020 AMC 10B contest is released on Wednesday, February 5, 2020.

Solution

Problem 20

These problems will not be available until the 2020 AMC 10B contest is released on Wednesday, February 5, 2020.

Solution

Problem 21

In square $ABCD$, points $E$ and $H$ lie on $\overline{AB}$ and $\overline{DA}$, respectively, so that $AE=AH.$ Points $F$ and $G$ lie on $\overline{BC}$ and $\overline{CD}$, respectively, and points $I$ and $J$ lie on $\overline{EH}$ so that $\overline{FI} \perp \overline{EH}$ and $\overline{GJ} \perp \overline{EH}$. See the figure below. Triangle $AEH$, quadrilateral $BFIE$, quadrilateral $DHJG$, and pentagon $FCGJI$ each has area $1.$ What is $FI^2$? [asy] real x=2sqrt(2); real y=2sqrt(16-8sqrt(2))-4+2sqrt(2); real z=2sqrt(8-4sqrt(2)); pair A, B, C, D, E, F, G, H, I, J; A = (0,0); B = (4,0); C = (4,4); D = (0,4); E = (x,0); F = (4,y); G = (y,4); H = (0,x); I = F + z * dir(225); J = G + z * dir(225); draw(A--B--C--D--A); draw(H--E); draw(J--G^^F--I); draw(rightanglemark(G, J, I), linewidth(.5)); draw(rightanglemark(F, I, E), linewidth(.5)); dot("$A$", A, S); dot("$B$", B, S); dot("$C$", C, dir(90)); dot("$D$", D, dir(90)); dot("$E$", E, S); dot("$F$", F, dir(0)); dot("$G$", G, N); dot("$H$", H, W); dot("$I$", I, SW); dot("$J$", J, SW); [/asy] $\textbf{(A) } \frac{7}{3} \qquad \textbf{(B) } 8-4\sqrt2 \qquad \textbf{(C) } 1+\sqrt2 \qquad \textbf{(D) } \frac{7}{4}\sqrt2 \qquad \textbf{(E) } 2\sqrt2$

Solution

Problem 22

What is the remainder when $2^{202} +202$ is divided by $2^{101}+2^{51}+1$?

$\textbf{(A) } 100 \qquad\textbf{(B) } 101 \qquad\textbf{(C) } 200 \qquad\textbf{(D) } 201 \qquad\textbf{(E) } 202$

Solution

Problem 23

Square $ABCD$ in the coordinate plane has vertices at the points $A(1,1), B(-1,1), C(-1,-1),$ and $D(1,-1).$ Consider the following four transformations: $L,$ a rotation of $90^{\circ}$ counterclockwise around the origin; $R,$ a rotation of $90^{\circ}$ clockwise around the origin; $H,$ a reflection across the $x$-axis; and $V,$ a reflection across the $y$-axis.

Each of these transformations maps the squares onto itself, but the positions of the labeled vertices will change. For example, applying $R$ and then $V$ would send the vertex $A$ at $(1,1)$ to $(-1,-1)$ and would send the vertex $B$ at $(-1,1)$ to itself. How many sequences of $20$ transformations chosen from $\{L, R, H, V\}$ will send all of the labeled vertices back to their original positions? (For example, $R, R, V, H$ is one sequence of $4$ transformations that will send the vertices back to their original positions.)

$\textbf{(A)}\ 2^{37} \qquad\textbf{(B)}\ 3\cdot 2^{36} \qquad\textbf{(C)}\ 2^{38} \qquad\textbf{(D)}\ 3\cdot 2^{37} \qquad\textbf{(E)}\ 2^{39}$ Solution

Problem 24

How many positive integers $n$ satisfy\[\dfrac{n+1000}{70} = \lfloor \sqrt{n} \rfloor?\](Recall that $\lfloor x\rfloor$ is the greatest integer not exceeding $x$.)

$\textbf{(A) } 2 \qquad\textbf{(B) } 4 \qquad\textbf{(C) } 6 \qquad\textbf{(D) } 30 \qquad\textbf{(E) } 32$

Solution

Problem 25

Let $D(n)$ denote the number of ways of writing the positive integer $n$ as a product\[n = f_1\cdot f_2\cdots f_k,\]where $k\ge1$, the $f_i$ are integers strictly greater than $1$, and the order in which the factors are listed matters (that is, two representations that differ only in the order of the factors are counted as distinct). For example, the number $6$ can be written as $6$, $2\cdot 3$, and $3\cdot2$, so $D(6) = 3$. What is $D(96)$?

$\textbf{(A) } 112 \qquad\textbf{(B) } 128 \qquad\textbf{(C) } 144 \qquad\textbf{(D) } 172 \qquad\textbf{(E) } 184$

Solution

See also

2020 AMC 10B (Problems • Answer Key • Resources)
Preceded by
2020 AMC 10A Problems 		Followed by
2021 AMC 10A Problems
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

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