Difference between revisions of "2021 AMC 12A Problems/Problem 11"

m (Diagram)
m (Solution 1 (Geometry))
 
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==Problem==
 
==Problem==
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>
 
<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>
  
 
==Diagram==
 
==Diagram==
[[File:2021 AMC 12A Problem 11 (1) LaTeX Revised.png|center]]
+
<asy>
Graph in Desmos: https://www.desmos.com/calculator/bsiulzrjrn
+
/* Made by MRENTHUSIASM */
 +
size(200);
  
 +
int xMin = -3;
 +
int xMax = 9;
 +
int yMin = -3;
 +
int yMax = 7;
 +
 +
draw((xMin,0)--(xMax,0),black+linewidth(1.5),EndArrow(5));
 +
draw((0,yMin)--(0,yMax),black+linewidth(1.5),EndArrow(5));
 +
label("$x$",(xMax,0),(2,0));
 +
label("$y$",(0,yMax),(0,2));
 +
 +
pair A = (3,5);
 +
pair B = (0,2);
 +
pair C = (2,0);
 +
pair D = (7,5);
 +
 +
draw(A--B--C--D,red);
 +
dot(A,linewidth(3.5));
 +
dot(B,linewidth(3.5));
 +
dot(C,linewidth(3.5));
 +
dot(D,linewidth(3.5));
 +
label("$(3,5)$",A,(0,2));
 +
label("$(7,5)$",D,(0,2));
 +
</asy>
 
~MRENTHUSIASM
 
~MRENTHUSIASM
  
==Solution 1==
+
==Solution 1 (Geometry)==
Every time the laser bounces off a wall, instead we can imagine it going straight by reflecting it about the wall. Thus, the laser starts at <math>(3, 5)</math> and ends at <math>(-7, -5)</math>, so the path's length is <math>\sqrt{10^2+10^2}=\boxed{\textbf{(C)} 10\sqrt{2}}</math>
+
Let <math>A=(3,5), D=(7,5), B</math> be the point where the beam hits and bounces off the <math>y</math>-axis, and <math>C</math> be the point where the beam hits and bounces off the <math>x</math>-axis.
~JHawk0224
 
  
==Solution 2 (Detailed Explanation of Solution 1)==
+
First, we reflect <math>\overline{BC}</math> about the <math>y</math>-axis to get <math>\overline{BC'}.</math> Then, we reflect <math>\overline{CD}</math> about the <math>y</math>-axis to get <math>\overline{C'D'}.</math> Finally, we reflect <math>\overline{C'D'}</math> about the <math>x</math>-axis to get <math>\overline{C'D''},</math> as shown below.
Let <math>A=(3,5), D=(7,5), B</math> be the point where the beam hits the <math>y</math>-axis, and <math>C</math> be the point where the beam hits the <math>x</math>-axis.
+
<asy>
 +
/* Made by MRENTHUSIASM */
 +
size(225);
  
Reflecting <math>\overline{BC}</math> about the <math>y</math>-axis gives <math>\overline{BC'}.</math> Then, reflecting <math>\overline{CD}</math> about the <math>y</math>-axis gives <math>\overline{C'D'}.</math> Finally, reflecting <math>\overline{C'D'}</math> about the <math>x</math>-axis gives <math>\overline{C'D''},</math> as shown below.
+
int xMin = -9;
 +
int xMax = 9;
 +
int yMin = -7;
 +
int yMax = 7;
  
[[File:2021 AMC 12A Problem 11 (2) LaTeX.png|center]]
+
draw((xMin,0)--(xMax,0),black+linewidth(1.5),EndArrow(5));
 +
draw((0,yMin)--(0,yMax),black+linewidth(1.5),EndArrow(5));
 +
label("$x$",(xMax,0),(2,0));
 +
label("$y$",(0,yMax),(0,2));
  
Graph in Desmos: https://www.desmos.com/calculator/lxjt0ewbou
+
pair A = (3,5);
 +
pair B = (0,2);
 +
pair C = (2,0);
 +
pair D = (7,5);
 +
pair E = (-2,0);
 +
pair F = (-7,5);
 +
pair G = (-7,-5);
  
It follows that <math>D''=(-7,-5).</math> The total distance that the beam will travel is  
+
draw(A--B--C--D,red);
 +
draw(B--E,heavygreen+dashed);
 +
draw(E--F,heavygreen+dashed);
 +
draw(E--G,heavygreen+dashed);
 +
dot(A,linewidth(3.5));
 +
dot(B,linewidth(3.5));
 +
dot(C,linewidth(3.5));
 +
dot(D,linewidth(3.5));
 +
dot(E,linewidth(3.5));
 +
dot(F,linewidth(3.5));
 +
dot(G,linewidth(3.5));
 +
label("$A(3,5)$",A,(0,2));
 +
label("$B$",B,(-2,0));
 +
label("$C$",C,(0,-2));
 +
label("$D(7,5)$",D,(0,2));
 +
label("$C'$",E,(0,-2));
 +
label("$D'$",F,(0,2));
 +
label("$D''$",G,(0,-2));
 +
</asy>
 +
It follows that <math>D'=(-7,5)</math> and <math>D''=(-7,-5).</math> The total distance that the beam will travel is  
 
<cmath>\begin{align*}
 
<cmath>\begin{align*}
 
AB+BC+CD&=AB+BC'+C'D' \\
 
AB+BC+CD&=AB+BC'+C'D' \\
Line 29: Line 85:
 
&=AD'' \\
 
&=AD'' \\
 
&=\sqrt{((3-(-7))^2+(5-(-5))^2} \\
 
&=\sqrt{((3-(-7))^2+(5-(-5))^2} \\
&=\sqrt{200} \\
 
 
&=\boxed{\textbf{(C) }10\sqrt2}.
 
&=\boxed{\textbf{(C) }10\sqrt2}.
 
\end{align*}</cmath>
 
\end{align*}</cmath>
 +
~MRENTHUSIASM (Solution)
 +
 +
~JHawk0224 (Proposal)
 +
 +
==Solution 2 (Algebra)==
 +
Define points <math>A,B,C,</math> and <math>D</math> as Solution 1 does.
 +
 +
When a straight line hits and bounces off a coordinate axis at point <math>P,</math> the ray entering <math>P</math> and the ray leaving <math>P</math> have negative slopes. Let <math>\ell</math> be the line containing <math>P</math> and perpendicular to that coordinate axis. <i><b>Geometrically, these two rays coincide when reflected about <math>\boldsymbol{\ell}.</math></b></i>
 +
 +
Let the slope of <math>\overline{AB}</math> be <math>m.</math> It follows that the slope of <math>\overline{BC}</math> is <math>-m,</math> and the slope of <math>\overline{CD}</math> is <math>m.</math> Here, we conclude that <math>\overline{AB}\parallel\overline{CD}.</math>
  
 +
Next, we locate <math>E</math> on <math>\overline{CD}</math> such that <math>\overline{BE}\parallel\overline{AD}.</math> We obtain parallelogram <math>ABED,</math> as shown below.
 +
<asy>
 +
/* Made by MRENTHUSIASM */
 +
size(200);
 +
 +
int xMin = -3;
 +
int xMax = 9;
 +
int yMin = -3;
 +
int yMax = 7;
 +
 +
draw((xMin,0)--(xMax,0),black+linewidth(1.5),EndArrow(5));
 +
draw((0,yMin)--(0,yMax),black+linewidth(1.5),EndArrow(5));
 +
label("$x$",(xMax,0),(2,0));
 +
label("$y$",(0,yMax),(0,2));
 +
 +
pair A = (3,5);
 +
pair B = (0,2);
 +
pair C = (2,0);
 +
pair D = (7,5);
 +
pair E = (4,2);
 +
 +
draw(A--B--C--D,red);
 +
draw(A--D,heavygreen+dashed);
 +
draw(B--E,heavygreen+dashed);
 +
dot(A,linewidth(3.5));
 +
dot(B,linewidth(3.5));
 +
dot(C,linewidth(3.5));
 +
dot(D,linewidth(3.5));
 +
dot(E,linewidth(3.5));
 +
label("$A(3,5)$",A,(0,2));
 +
label("$B$",B,(-2,0));
 +
label("$C$",C,(0,-2));
 +
label("$D(7,5)$",D,(0,2));
 +
label("$E$",E,(2,0));
 +
</asy>
 +
Let <math>B=(0,b).</math> In parallelogram <math>ABED,</math> we get <math>E=(4,b).</math> By symmetry, we get <math>C=(2,0).</math>
 +
 +
Applying the slope formula to <math>\overline{AB}</math> and <math>\overline{DC}</math> gives <cmath>m=\frac{5-b}{3-0}=\frac{5-0}{7-2}.</cmath> Equating the last two expressions gives <math>b=2.</math>
 +
 +
By the Distance Formula, we have <math>AB=3\sqrt2,BC=2\sqrt2,</math> and <math>CD=5\sqrt2.</math> The total distance that the beam will travel is <cmath>AB+BC+CD=\boxed{\textbf{(C) }10\sqrt2}.</cmath>
 
~MRENTHUSIASM
 
~MRENTHUSIASM
  
==Solution 3 (Slopes and Parallelogram)==
+
==Solution 3 (Answer Choices and Educated Guesses)==
Define points <math>A,B,C,</math> and <math>D</math> as Solution 2 does.
+
Define points <math>A,B,C,</math> and <math>D</math> as Solution 1 does.
  
When a line segment hits and bounces off a coordinate axis at point <math>P,</math> the ray entering <math>P</math> and the ray leaving <math>P</math> have negative slopes. <i><b>Geometrically, these two rays coincide when reflected about the line perpendicular to that coordinate axis, creating line symmetry.</b></i> Let the slope of <math>\overline{AB}</math> be <math>m.</math> It follows that the slope of <math>\overline{BC}</math> is <math>-m,</math> and the slope of <math>\overline{CD}</math> is <math>m.</math> Here, we conclude that <math>\overline{AB}\parallel\overline{CD}.</math>
+
Since choices <math>\textbf{(B)}, \textbf{(C)},</math> and <math>\textbf{(D)}</math> all involve <math>\sqrt2,</math> we suspect that one of them is the correct answer. We take a guess in faith that <math>\overline{AB},\overline{BC},</math> and <math>\overline{CD}</math> all form <math>45^\circ</math> angles with the coordinate axes, from which <math>B=(0,2)</math> and <math>C=(2,0).</math> The given condition <math>D=(7,5)</math> verifies our guess, as shown below.
 +
<asy>
 +
/* Made by MRENTHUSIASM */
 +
size(200);
  
Next, we locate <math>E</math> on <math>\overline{CD}</math> such that <math>\overline{BE}\parallel\overline{AD},</math> from which <math>ABED</math> is a parallelogram, as shown below.
+
int xMin = -3;
[[File:2021 AMC 12A Problem 11 (3) LaTeX.png|center]]
+
int xMax = 9;
Graph in Desmos: https://www.desmos.com/calculator/lgfiiqgqc2
+
int yMin = -3;
 +
int yMax = 7;
  
Let <math>B=(0,b).</math> By the property of slopes, we get <math>E=(4,b).</math> By symmetry, we obtain <math>C=(2,0).</math>
+
//Draws the horizontal gridlines
 +
void horizontalLines()
 +
{
 +
  for (int i = yMin+1; i < yMax; ++i)
 +
  {
 +
    draw((xMin,i)--(xMax,i), mediumgray+linewidth(0.4));
 +
  }
 +
}
  
Applying the slope formula on <math>\overline{AB}</math> and <math>\overline{DC}</math> gives <cmath>m=\frac{5-b}{3-0}=\frac{5-0}{7-2}.</cmath> Equating the last two expressions gives <math>b=2.</math>
+
//Draws the vertical gridlines
 +
void verticalLines()
 +
{
 +
  for (int i = xMin+1; i < xMax; ++i)
 +
  {
 +
    draw((i,yMin)--(i,yMax), mediumgray+linewidth(0.4));
 +
  }
 +
}
  
By the Distance Formula, <math>AB=3\sqrt2,BC=2\sqrt2,</math> and <math>CD=5\sqrt2.</math> The total distance that the beam will travel is <cmath>AB+BC+CD=\boxed{\textbf{(C) }10\sqrt2}.</cmath>
+
//Draws the horizontal ticks
 +
void horizontalTicks()
 +
{
 +
  for (int i = yMin+1; i < yMax; ++i)
 +
  {
 +
    draw((-3/16,i)--(3/16,i), black+linewidth(1));
 +
  }
 +
}
  
~MRENTHUSIASM
+
//Draws the vertical ticks
 +
void verticalTicks()
 +
{
 +
  for (int i = xMin+1; i < xMax; ++i)
 +
  {
 +
    draw((i,-3/16)--(i,3/16), black+linewidth(1));
 +
  }
 +
}
 +
 
 +
horizontalLines();
 +
verticalLines();
 +
horizontalTicks();
 +
verticalTicks();
 +
draw((xMin,0)--(xMax,0),black+linewidth(1.5),EndArrow(5));
 +
draw((0,yMin)--(0,yMax),black+linewidth(1.5),EndArrow(5));
 +
label("$x$",(xMax,0),(2,0));
 +
label("$y$",(0,yMax),(0,2));
  
==Solution 4 (Answer Choices and Educated Guesses)==
+
pair A = (3,5);
Define points <math>A,B,C,</math> and <math>D</math> as Solution 2 does.
+
pair B = (0,2);
 +
pair C = (2,0);
 +
pair D = (7,5);
  
Since choices <math>\textbf{(B)}, \textbf{(C)},</math> and <math>\textbf{(D)}</math> all involve <math>\sqrt2,</math> we suspect that one of them is the correct answer. We take a guess in faith that <math>\overline{AB},\overline{BC},</math> and <math>\overline{CD}</math> all form <math>45^\circ</math> angles with the coordinate axes, from which <math>B=(0,2)</math> and <math>C=(2,0).</math> The given condition <math>D=(7,5)</math> verifies our guess. Following the penultimate paragraph of Solution 3 gives the answer <math>\boxed{\textbf{(C) }10\sqrt2}.</math>
+
draw(A--B--C--D,red);
 +
dot(A,linewidth(3.5));
 +
dot(B,linewidth(3.5));
 +
dot(C,linewidth(3.5));
 +
dot(D,linewidth(3.5));
 +
label("$A(3,5)$",A,(0,2),UnFill);
 +
label("$B$",B,(-2,0),UnFill);
 +
label("$C$",C,(0,-2),UnFill);
 +
label("$D(7,5)$",D,(0,2),UnFill);
 +
</asy>
 +
Following the last paragraph of Solution 2 gives the answer <math>\boxed{\textbf{(C) }10\sqrt2}.</math>
  
 
~MRENTHUSIASM
 
~MRENTHUSIASM

Latest revision as of 12:49, 25 September 2021

Problem

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?

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

Diagram

[asy] /* Made by MRENTHUSIASM */ size(200);   int xMin = -3; int xMax = 9; int yMin = -3; int yMax = 7;  draw((xMin,0)--(xMax,0),black+linewidth(1.5),EndArrow(5)); draw((0,yMin)--(0,yMax),black+linewidth(1.5),EndArrow(5)); label("$x$",(xMax,0),(2,0)); label("$y$",(0,yMax),(0,2));  pair A = (3,5); pair B = (0,2); pair C = (2,0); pair D = (7,5);  draw(A--B--C--D,red); dot(A,linewidth(3.5)); dot(B,linewidth(3.5)); dot(C,linewidth(3.5)); dot(D,linewidth(3.5)); label("$(3,5)$",A,(0,2)); label("$(7,5)$",D,(0,2)); [/asy] ~MRENTHUSIASM

Solution 1 (Geometry)

Let $A=(3,5), D=(7,5), B$ be the point where the beam hits and bounces off the $y$-axis, and $C$ be the point where the beam hits and bounces off the $x$-axis.

First, we reflect $\overline{BC}$ about the $y$-axis to get $\overline{BC'}.$ Then, we reflect $\overline{CD}$ about the $y$-axis to get $\overline{C'D'}.$ Finally, we reflect $\overline{C'D'}$ about the $x$-axis to get $\overline{C'D''},$ as shown below. [asy] /* Made by MRENTHUSIASM */ size(225);   int xMin = -9; int xMax = 9; int yMin = -7; int yMax = 7;  draw((xMin,0)--(xMax,0),black+linewidth(1.5),EndArrow(5)); draw((0,yMin)--(0,yMax),black+linewidth(1.5),EndArrow(5)); label("$x$",(xMax,0),(2,0)); label("$y$",(0,yMax),(0,2));  pair A = (3,5); pair B = (0,2); pair C = (2,0); pair D = (7,5); pair E = (-2,0); pair F = (-7,5); pair G = (-7,-5);  draw(A--B--C--D,red); draw(B--E,heavygreen+dashed); draw(E--F,heavygreen+dashed); draw(E--G,heavygreen+dashed); dot(A,linewidth(3.5)); dot(B,linewidth(3.5)); dot(C,linewidth(3.5)); dot(D,linewidth(3.5)); dot(E,linewidth(3.5)); dot(F,linewidth(3.5)); dot(G,linewidth(3.5)); label("$A(3,5)$",A,(0,2)); label("$B$",B,(-2,0)); label("$C$",C,(0,-2)); label("$D(7,5)$",D,(0,2)); label("$C'$",E,(0,-2)); label("$D'$",F,(0,2)); label("$D''$",G,(0,-2)); [/asy] It follows that $D'=(-7,5)$ and $D''=(-7,-5).$ The total distance that the beam will travel is \begin{align*} AB+BC+CD&=AB+BC'+C'D' \\ &=AB+BC'+C'D'' \\ &=AD'' \\ &=\sqrt{((3-(-7))^2+(5-(-5))^2} \\ &=\boxed{\textbf{(C) }10\sqrt2}. \end{align*} ~MRENTHUSIASM (Solution)

~JHawk0224 (Proposal)

Solution 2 (Algebra)

Define points $A,B,C,$ and $D$ as Solution 1 does.

When a straight line hits and bounces off a coordinate axis at point $P,$ the ray entering $P$ and the ray leaving $P$ have negative slopes. Let $\ell$ be the line containing $P$ and perpendicular to that coordinate axis. Geometrically, these two rays coincide when reflected about $\boldsymbol{\ell}.$

Let the slope of $\overline{AB}$ be $m.$ It follows that the slope of $\overline{BC}$ is $-m,$ and the slope of $\overline{CD}$ is $m.$ Here, we conclude that $\overline{AB}\parallel\overline{CD}.$

Next, we locate $E$ on $\overline{CD}$ such that $\overline{BE}\parallel\overline{AD}.$ We obtain parallelogram $ABED,$ as shown below. [asy] /* Made by MRENTHUSIASM */ size(200);   int xMin = -3; int xMax = 9; int yMin = -3; int yMax = 7;  draw((xMin,0)--(xMax,0),black+linewidth(1.5),EndArrow(5)); draw((0,yMin)--(0,yMax),black+linewidth(1.5),EndArrow(5)); label("$x$",(xMax,0),(2,0)); label("$y$",(0,yMax),(0,2));  pair A = (3,5); pair B = (0,2); pair C = (2,0); pair D = (7,5); pair E = (4,2);  draw(A--B--C--D,red); draw(A--D,heavygreen+dashed); draw(B--E,heavygreen+dashed); dot(A,linewidth(3.5)); dot(B,linewidth(3.5)); dot(C,linewidth(3.5)); dot(D,linewidth(3.5)); dot(E,linewidth(3.5)); label("$A(3,5)$",A,(0,2)); label("$B$",B,(-2,0)); label("$C$",C,(0,-2)); label("$D(7,5)$",D,(0,2)); label("$E$",E,(2,0)); [/asy] Let $B=(0,b).$ In parallelogram $ABED,$ we get $E=(4,b).$ By symmetry, we get $C=(2,0).$

Applying the slope formula to $\overline{AB}$ and $\overline{DC}$ gives \[m=\frac{5-b}{3-0}=\frac{5-0}{7-2}.\] Equating the last two expressions gives $b=2.$

By the Distance Formula, we have $AB=3\sqrt2,BC=2\sqrt2,$ and $CD=5\sqrt2.$ The total distance that the beam will travel is \[AB+BC+CD=\boxed{\textbf{(C) }10\sqrt2}.\] ~MRENTHUSIASM

Solution 3 (Answer Choices and Educated Guesses)

Define points $A,B,C,$ and $D$ as Solution 1 does.

Since choices $\textbf{(B)}, \textbf{(C)},$ and $\textbf{(D)}$ all involve $\sqrt2,$ we suspect that one of them is the correct answer. We take a guess in faith that $\overline{AB},\overline{BC},$ and $\overline{CD}$ all form $45^\circ$ angles with the coordinate axes, from which $B=(0,2)$ and $C=(2,0).$ The given condition $D=(7,5)$ verifies our guess, as shown below. [asy] /* Made by MRENTHUSIASM */ size(200);   int xMin = -3; int xMax = 9; int yMin = -3; int yMax = 7;  //Draws the horizontal gridlines void horizontalLines() {   for (int i = yMin+1; i < yMax; ++i)   {     draw((xMin,i)--(xMax,i), mediumgray+linewidth(0.4));   } }  //Draws the vertical gridlines void verticalLines() {   for (int i = xMin+1; i < xMax; ++i)   {     draw((i,yMin)--(i,yMax), mediumgray+linewidth(0.4));   } }  //Draws the horizontal ticks void horizontalTicks() {   for (int i = yMin+1; i < yMax; ++i)   {     draw((-3/16,i)--(3/16,i), black+linewidth(1));   } }  //Draws the vertical ticks void verticalTicks() {   for (int i = xMin+1; i < xMax; ++i)   {     draw((i,-3/16)--(i,3/16), black+linewidth(1));   } }  horizontalLines(); verticalLines(); horizontalTicks(); verticalTicks(); draw((xMin,0)--(xMax,0),black+linewidth(1.5),EndArrow(5)); draw((0,yMin)--(0,yMax),black+linewidth(1.5),EndArrow(5)); label("$x$",(xMax,0),(2,0)); label("$y$",(0,yMax),(0,2));  pair A = (3,5); pair B = (0,2); pair C = (2,0); pair D = (7,5);  draw(A--B--C--D,red); dot(A,linewidth(3.5)); dot(B,linewidth(3.5)); dot(C,linewidth(3.5)); dot(D,linewidth(3.5)); label("$A(3,5)$",A,(0,2),UnFill); label("$B$",B,(-2,0),UnFill); label("$C$",C,(0,-2),UnFill); label("$D(7,5)$",D,(0,2),UnFill); [/asy] Following the last paragraph of Solution 2 gives the answer $\boxed{\textbf{(C) }10\sqrt2}.$

~MRENTHUSIASM

Video Solution by OmegaLearn (Using Reflections and Distance Formula)

https://youtu.be/e7tNtd-fgeo

~ pi_is_3.14

Video Solution by Hawk Math

https://www.youtube.com/watch?v=AjQARBvdZ20

Video Solution by TheBeautyofMath

https://youtu.be/ySWSHyY9TwI

~IceMatrix

See also

2021 AMC 12A (ProblemsAnswer KeyResources)
Preceded by
Problem 10
Followed by
Problem 12
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 12 Problems and Solutions

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

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