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Difference between revisions of "2020 AMC 12B Problems/Problem 10"

(Deleted Sol 6. Intersecting Chords will work, but the current version is not completed. I will complete it when I have time.)
(1. Grouped the solutions with similar ideas together. 2. Prioritize solutions that have diagrams. Let me know if you are unhappy with this change.)
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~MRENTHUSIASM
 
~MRENTHUSIASM
  
−
==Solution 1 (Angle Chasing/Trig)==
+
==Solution 1 (Inscribed Angle Theorem and Pythagorean Theorem)==
−
Let <math>O</math> be the center of the circle and the point of tangency between <math>\omega</math> and <math>\overline{AD}</math> be represented by <math>K</math>. We know that <math>\overline{AK} = \overline{KD} = \overline{DM} = \frac{1}{2}</math>. Consider the right triangle <math>\bigtriangleup ADM</math>. Let <math>\measuredangle AMD = \theta</math>.  
+
Let <math>N</math> be the midpoint of <math>\overline{AB},</math> from which <math>\angle ANM=90^\circ.</math> Note that <math>\angle NPM=90^\circ</math> by the Inscribed Angle Theorem.
  
−
Since <math>\omega</math> is tangent to <math>\overline{DC}</math> at <math>M</math>, <math>\measuredangle PMO = 90 - \theta</math>. Now, consider <math>\bigtriangleup POM</math>. This triangle is iscoceles because <math>\overline{PO}</math> and <math>\overline{OM}</math> are both radii of <math>\omega</math>. Therefore, <math>\measuredangle POM = 180 - 2(90 - \theta) = 2\theta</math>.
+
We have the following diagram:
  
−
We can now use Law of Cosines on <math>\angle{POM}</math> to find the length of <math>{PM}</math> and subtract it from the length of <math>{AM}</math> to find <math>{AP}</math>. Since <math>\cos{\theta} = \frac{1}{\sqrt{5}}</math> and <math>\sin{\theta} = \frac{2}{\sqrt{5}}</math>, the double angle formula tells us that <math>\cos{2\theta} = -\frac{3}{5}</math>. We have
+
[[File:2020 AMC 12B Problem 10 Solution.png|center]]
−
<cmath>
 
−
PM^2 = \frac{1}{2} - \frac{1}{2}\cos{2\theta} \implies PM = \frac{2\sqrt{5}}{5}.
 
−
</cmath>
 
−
By Pythagorean theorem, we find that <math>AM = \frac{\sqrt{5}}{2} \implies AP=\boxed{\textbf{(B) } \frac{\sqrt5}{10}}</math>.
 
  
−
~awesome1st
+
Since <math>AN=\frac12</math> and <math>NM=1,</math> we get <math>AM=\frac{\sqrt5}{2}</math> by the Pythagorean Theorem.
  
−
==Solution 2 (Coordinate Geometry)==
+
Let <math>AP=x.</math> It follows that <math>PM=\frac{\sqrt5}{2}-x.</math> Applying the Pythagorean Theorem to right <math>\triangle ANP</math> gives <math>NP^2=\left(\frac12\right)^2-x^2,</math> and applying the Pythagorean Theorem to right <math>\triangle MNP</math> gives <math>NP^2=1^2-\left(\frac{\sqrt5}{2}-x\right)^2.</math> Equating the expressions for <math>NP^2</math> produces
 +
<cmath>\begin{align*}
 +
\left(\frac12\right)^2-x^2&=1^2-\left(\frac{\sqrt5}{2}-x\right)^2 \\
 +
\frac14-x^2&=1-\frac54+\sqrt5x-x^2 \\
 +
\frac12&=\sqrt5x.
 +
\end{align*}</cmath>
 +
Finally, dividing both sides by <math>\sqrt5</math> and then rationalizing the denominator, we obtain <math>x=\frac{1}{2\sqrt5}=\boxed{\textbf{(B) } \frac{\sqrt5}{10}}.</math>
  
−
Place circle <math>\omega</math> in the Cartesian plane such that the center lies on the origin. Then we can easily find the equation for <math>\omega</math> as <math>x^2+y^2=\frac{1}{4}</math>, because it is not translated and the radius is <math>\frac{1}{2}</math>.
+
~MRENTHUSIASM
−
 
 
−
We have <math>A=\left(-\frac{1}{2}, \frac{1}{2}\right)</math> and <math>M=\left(0, -\frac{1}{2}\right)</math>. The slope of the line passing through these two points is <math>\frac{\frac{1}{2}+\frac{1}{2}}{-\frac{1}{2}-0}=\frac{1}{-\frac{1}{2}}=-2</math>, and the <math>y</math>-intercept is simply <math>M</math>. This gives us the line passing through both points as <math>y=-2x-\frac{1}{2}</math>.
 
−
 
 
−
We substitute this into the equation for the circle to get <math>x^2+\left(-2x-\frac{1}{2}\right)^2=\frac{1}{4}</math>, or <math>x^2+4x^2+2x+\frac{1}{4}=\frac{1}{4}</math>. Simplifying gives <math>x(5x+2)=0</math>. The roots of this quadratic are <math>x=0</math> and <math>x=-\frac{2}{5}</math>, but if <math>x=0</math> we get point <math>M</math>, so we only want <math>x=-\frac{2}{5}</math>.
 
−
 
 
−
We plug this back into the linear equation to find <math>y=\frac{3}{10}</math>, and so <math>P=\left(-\frac{2}{5}, \frac{3}{10}\right)</math>. Finally, we use distance formula on <math>A</math> and <math>P</math> to get <math>AP=\sqrt{\left(-\frac{5}{10}+\frac{4}{10}\right)^2+\left(\frac{5}{10}-\frac{3}{10}\right)^2}=\sqrt{\frac{1}{100}+\frac{4}{100}}=\boxed{\textbf{(B) } \frac{\sqrt5}{10}}</math>.
 
−
 
 
−
~Argonauts16
 
−
 
 
−
==Solution 3 (Power of a Point)==
 
−
Let circle <math>\omega</math> intersect <math>\overline{AB}</math> at point <math>N</math>. By Power of a Point, we have <math>AN^2=AP\cdot AM</math>. We know <math>AN=\frac{1}{2}</math> because <math>N</math> is the midpoint of <math>\overline{AB}</math>, and we can easily find <math>AM</math> by the Pythagorean Theorem, which gives us <math>AM=\sqrt{1^2+\left(\frac{1}{2}\right)^2}=\frac{\sqrt{5}}{2}</math>. Our equation is now <math>\frac{1}{4}=AP\cdot \frac{\sqrt{5}}{2}</math>, or <math>AP=\frac{2}{4\sqrt{5}}=\frac{1}{2\sqrt{5}}=\frac{\sqrt{5}}{2\cdot 5}</math>, thus our answer is
 
−
<math>\boxed{\textbf{(B) } \frac{\sqrt5}{10}}.</math>
 
−
 
 
−
~Argonauts16
 
−
 
 
−
==Solution 4==
 
−
Take <math>O</math> as the center and draw segment <math>ON</math> perpendicular to <math>AM</math>, <math>ON\cap AM=N</math>, link <math>OM</math>. Then we have <math>OM\parallel AD</math>. So <math>\angle DAM=\angle OMA</math>. Since <math>AD=2AM=2OM=1</math>, we have <math>\cos\angle DAM=\cos\angle OMP=\frac{2}{\sqrt{5}}</math>. As a result, <math>NM=OM\cos\angle OMP=\frac{1}{2}\cdot \frac{2}{\sqrt{5}}=\frac{1}{\sqrt{5}}.</math> Thus <math>PM=2NM=\frac{2}{\sqrt{5}}=\frac{2\sqrt{5}}{5}</math>. Since <math>AM=\frac{\sqrt{5}}{2}</math>, we have <math>AP=AM-PM=\boxed{\textbf{(B) } \frac{\sqrt5}{10}}</math>.
 
−
 
 
−
~FANYUCHEN20020715
 
  
−
==Solution 5 (Similar Triangles)==
+
==Solution 2 (Similar Triangles)==
 
Call the midpoint of <math>\overline{AB}</math> point <math>N</math>. Draw in <math>\overline{NM}</math> and <math>\overline{NP}</math>. Note that <math>\angle{NPM}=90^{\circ}</math> due to Thales's Theorem.
 
Call the midpoint of <math>\overline{AB}</math> point <math>N</math>. Draw in <math>\overline{NM}</math> and <math>\overline{NP}</math>. Note that <math>\angle{NPM}=90^{\circ}</math> due to Thales's Theorem.
  
Line 87: Line 70:
 
~QIDb602
 
~QIDb602
  
−
==Solution 7 (Inscribed Angle Theorem and Pythagorean Theorem)==
+
==Solution 3 (Power of a Point)==
−
Let <math>N</math> be the midpoint of <math>\overline{AB},</math> from which <math>\angle ANM=90^\circ.</math> Note that <math>\angle NPM=90^\circ</math> by the Inscribed Angle Theorem.
+
Let circle <math>\omega</math> intersect <math>\overline{AB}</math> at point <math>N</math>. By Power of a Point, we have <math>AN^2=AP\cdot AM</math>. We know <math>AN=\frac{1}{2}</math> because <math>N</math> is the midpoint of <math>\overline{AB}</math>, and we can easily find <math>AM</math> by the Pythagorean Theorem, which gives us <math>AM=\sqrt{1^2+\left(\frac{1}{2}\right)^2}=\frac{\sqrt{5}}{2}</math>. Our equation is now <math>\frac{1}{4}=AP\cdot \frac{\sqrt{5}}{2}</math>, or <math>AP=\frac{2}{4\sqrt{5}}=\frac{1}{2\sqrt{5}}=\frac{\sqrt{5}}{2\cdot 5}</math>, thus our answer is
 +
<math>\boxed{\textbf{(B) } \frac{\sqrt5}{10}}.</math>
 +
 
 +
~Argonauts16
 +
 
 +
==Solution 4 (Trigonometry)==
 +
Let <math>O</math> be the center of the circle and the point of tangency between <math>\omega</math> and <math>\overline{AD}</math> be represented by <math>K</math>. We know that <math>\overline{AK} = \overline{KD} = \overline{DM} = \frac{1}{2}</math>. Consider the right triangle <math>\bigtriangleup ADM</math>. Let <math>\measuredangle AMD = \theta</math>.
 +
 
 +
Since <math>\omega</math> is tangent to <math>\overline{DC}</math> at <math>M</math>, <math>\measuredangle PMO = 90 - \theta</math>. Now, consider <math>\bigtriangleup POM</math>. This triangle is iscoceles because <math>\overline{PO}</math> and <math>\overline{OM}</math> are both radii of <math>\omega</math>. Therefore, <math>\measuredangle POM = 180 - 2(90 - \theta) = 2\theta</math>.
 +
 
 +
We can now use Law of Cosines on <math>\angle{POM}</math> to find the length of <math>{PM}</math> and subtract it from the length of <math>{AM}</math> to find <math>{AP}</math>. Since <math>\cos{\theta} = \frac{1}{\sqrt{5}}</math> and <math>\sin{\theta} = \frac{2}{\sqrt{5}}</math>, the double angle formula tells us that <math>\cos{2\theta} = -\frac{3}{5}</math>. We have
 +
<cmath>
 +
PM^2 = \frac{1}{2} - \frac{1}{2}\cos{2\theta} \implies PM = \frac{2\sqrt{5}}{5}.
 +
</cmath>
 +
By Pythagorean theorem, we find that <math>AM = \frac{\sqrt{5}}{2} \implies AP=\boxed{\textbf{(B) } \frac{\sqrt5}{10}}</math>.
 +
 
 +
~awesome1st
 +
 
 +
==Solution 5 (Trigonometry)==
 +
Take <math>O</math> as the center and draw segment <math>ON</math> perpendicular to <math>AM</math>, <math>ON\cap AM=N</math>, link <math>OM</math>. Then we have <math>OM\parallel AD</math>. So <math>\angle DAM=\angle OMA</math>. Since <math>AD=2AM=2OM=1</math>, we have <math>\cos\angle DAM=\cos\angle OMP=\frac{2}{\sqrt{5}}</math>. As a result, <math>NM=OM\cos\angle OMP=\frac{1}{2}\cdot \frac{2}{\sqrt{5}}=\frac{1}{\sqrt{5}}.</math> Thus <math>PM=2NM=\frac{2}{\sqrt{5}}=\frac{2\sqrt{5}}{5}</math>. Since <math>AM=\frac{\sqrt{5}}{2}</math>, we have <math>AP=AM-PM=\boxed{\textbf{(B) } \frac{\sqrt5}{10}}</math>.
 +
 
 +
~FANYUCHEN20020715
 +
 
 +
==Solution 6 (Coordinate Geometry)==
  
−
We have the following diagram:
+
Place circle <math>\omega</math> in the Cartesian plane such that the center lies on the origin. Then we can easily find the equation for <math>\omega</math> as <math>x^2+y^2=\frac{1}{4}</math>, because it is not translated and the radius is <math>\frac{1}{2}</math>.
  
−
[[File:2020 AMC 12B Problem 10 Solution.png|center]]
+
We have <math>A=\left(-\frac{1}{2}, \frac{1}{2}\right)</math> and <math>M=\left(0, -\frac{1}{2}\right)</math>. The slope of the line passing through these two points is <math>\frac{\frac{1}{2}+\frac{1}{2}}{-\frac{1}{2}-0}=\frac{1}{-\frac{1}{2}}=-2</math>, and the <math>y</math>-intercept is simply <math>M</math>. This gives us the line passing through both points as <math>y=-2x-\frac{1}{2}</math>.  
  
−
Since <math>AN=\frac12</math> and <math>NM=1,</math> we get <math>AM=\frac{\sqrt5}{2}</math> by the Pythagorean Theorem.  
+
We substitute this into the equation for the circle to get <math>x^2+\left(-2x-\frac{1}{2}\right)^2=\frac{1}{4}</math>, or <math>x^2+4x^2+2x+\frac{1}{4}=\frac{1}{4}</math>. Simplifying gives <math>x(5x+2)=0</math>. The roots of this quadratic are <math>x=0</math> and <math>x=-\frac{2}{5}</math>, but if <math>x=0</math> we get point <math>M</math>, so we only want <math>x=-\frac{2}{5}</math>.
  
−
Let <math>AP=x.</math> It follows that <math>PM=\frac{\sqrt5}{2}-x.</math> Applying the Pythagorean Theorem to right <math>\triangle ANP</math> gives <math>NP^2=\left(\frac12\right)^2-x^2,</math> and applying the Pythagorean Theorem to right <math>\triangle MNP</math> gives <math>NP^2=1^2-\left(\frac{\sqrt5}{2}-x\right)^2.</math> Equating the expressions for <math>NP^2</math> produces
+
We plug this back into the linear equation to find <math>y=\frac{3}{10}</math>, and so <math>P=\left(-\frac{2}{5}, \frac{3}{10}\right)</math>. Finally, we use distance formula on <math>A</math> and <math>P</math> to get <math>AP=\sqrt{\left(-\frac{5}{10}+\frac{4}{10}\right)^2+\left(\frac{5}{10}-\frac{3}{10}\right)^2}=\sqrt{\frac{1}{100}+\frac{4}{100}}=\boxed{\textbf{(B) } \frac{\sqrt5}{10}}</math>.
−
<cmath>\begin{align*}
 
−
\left(\frac12\right)^2-x^2&=1^2-\left(\frac{\sqrt5}{2}-x\right)^2 \\
 
−
\frac14-x^2&=1-\frac54+\sqrt5x-x^2 \\
 
−
\frac12&=\sqrt5x.
 
−
\end{align*}</cmath>
 
−
Finally, dividing both sides by <math>\sqrt5</math> and then rationalizing the denominator, we obtain <math>x=\frac{1}{2\sqrt5}=\boxed{\textbf{(B) } \frac{\sqrt5}{10}}.</math>
 
  
−
~MRENTHUSIASM
+
~Argonauts16
  
 
==Video Solution==
 
==Video Solution==

Revision as of 15:39, 30 September 2021

Problem

In unit square $ABCD,$ the inscribed circle $\omega$ intersects $\overline{CD}$ at $M,$ and $\overline{AM}$ intersects $\omega$ at a point $P$ different from $M.$ What is $AP?$

$\textbf{(A) } \frac{\sqrt5}{12} \qquad \textbf{(B) } \frac{\sqrt5}{10} \qquad \textbf{(C) } \frac{\sqrt5}{9} \qquad \textbf{(D) } \frac{\sqrt5}{8} \qquad \textbf{(E) } \frac{2\sqrt5}{15}$

Diagram

[asy] /* Made by MRENTHUSIASM */ size(180); pair A, B, C, D, M, O, P; O = origin; A = (-1/2,-1/2); B = (-1/2,1/2); C = (1/2,1/2); D = (1/2,-1/2); M = midpoint(C--D); path p; p = Circle(O,1/2); P = intersectionpoints(A--M,p)[0]; dot("$\omega$",O,1.5*S,linewidth(4)); dot("$A$",A,1.5*SW,linewidth(4)); dot("$B$",B,1.5*NW,linewidth(4)); dot("$C$",C,1.5*NE,linewidth(4)); dot("$D$",D,1.5*SE,linewidth(4)); dot("$M$",M,1.5*E,linewidth(4)); dot("$P$",P,1.5*N,linewidth(4)); draw(A--B--C--D--cycle^^A--M^^p); [/asy] ~MRENTHUSIASM

Solution 1 (Inscribed Angle Theorem and Pythagorean Theorem)

Let $N$ be the midpoint of $\overline{AB},$ from which $\angle ANM=90^\circ.$ Note that $\angle NPM=90^\circ$ by the Inscribed Angle Theorem.

We have the following diagram:

File:2020 AMC 12B Problem 10 Solution.png

Since $AN=\frac12$ and $NM=1,$ we get $AM=\frac{\sqrt5}{2}$ by the Pythagorean Theorem.

Let $AP=x.$ It follows that $PM=\frac{\sqrt5}{2}-x.$ Applying the Pythagorean Theorem to right $\triangle ANP$ gives $NP^2=\left(\frac12\right)^2-x^2,$ and applying the Pythagorean Theorem to right $\triangle MNP$ gives $NP^2=1^2-\left(\frac{\sqrt5}{2}-x\right)^2.$ Equating the expressions for $NP^2$ produces \begin{align*} \left(\frac12\right)^2-x^2&=1^2-\left(\frac{\sqrt5}{2}-x\right)^2 \\ \frac14-x^2&=1-\frac54+\sqrt5x-x^2 \\ \frac12&=\sqrt5x. \end{align*} Finally, dividing both sides by $\sqrt5$ and then rationalizing the denominator, we obtain $x=\frac{1}{2\sqrt5}=\boxed{\textbf{(B) } \frac{\sqrt5}{10}}.$

~MRENTHUSIASM

Solution 2 (Similar Triangles)

Call the midpoint of $\overline{AB}$ point $N$. Draw in $\overline{NM}$ and $\overline{NP}$. Note that $\angle{NPM}=90^{\circ}$ due to Thales's Theorem.

[asy] draw((0,0)--(0,1)--(1,1)--(1,0)--cycle); draw(circle((0.5,0.5),0.5)); draw((0,0)--(0,0.5)--(1,0.5)--cycle); label("A",(0,0),SW); label("B",(0,1),NW); label("C",(1,1),NE); label("D",(1,0),SE); label("M",(1,0.5),E); label("P",(0.2,0.1),S); label("N",(0,0.5),W); draw((0,0.5)--(0.2,0.1)); markscalefactor=0.007; draw(rightanglemark((0,0.5),(0.2,0.1),(1,0.5))); [/asy] Using the Pythagorean theorem, $AM=\frac{\sqrt{5}}{2}$. Now we just need to find $AP$ using similar triangles. \[\triangle APN\sim\triangle ANM\Rightarrow\frac{AP}{AN}=\frac{AN}{AM}\Rightarrow\frac{AP}{\frac{1}{2}}=\frac{\frac{1}{2}}{\frac{\sqrt{5}}{2}}\Rightarrow AP=\boxed{\textbf{(B) } \frac{\sqrt5}{10}}.\] ~QIDb602

Solution 3 (Power of a Point)

Let circle $\omega$ intersect $\overline{AB}$ at point $N$. By Power of a Point, we have $AN^2=AP\cdot AM$. We know $AN=\frac{1}{2}$ because $N$ is the midpoint of $\overline{AB}$, and we can easily find $AM$ by the Pythagorean Theorem, which gives us $AM=\sqrt{1^2+\left(\frac{1}{2}\right)^2}=\frac{\sqrt{5}}{2}$. Our equation is now $\frac{1}{4}=AP\cdot \frac{\sqrt{5}}{2}$, or $AP=\frac{2}{4\sqrt{5}}=\frac{1}{2\sqrt{5}}=\frac{\sqrt{5}}{2\cdot 5}$, thus our answer is $\boxed{\textbf{(B) } \frac{\sqrt5}{10}}.$

~Argonauts16

Solution 4 (Trigonometry)

Let $O$ be the center of the circle and the point of tangency between $\omega$ and $\overline{AD}$ be represented by $K$. We know that $\overline{AK} = \overline{KD} = \overline{DM} = \frac{1}{2}$. Consider the right triangle $\bigtriangleup ADM$. Let $\measuredangle AMD = \theta$.

Since $\omega$ is tangent to $\overline{DC}$ at $M$, $\measuredangle PMO = 90 - \theta$. Now, consider $\bigtriangleup POM$. This triangle is iscoceles because $\overline{PO}$ and $\overline{OM}$ are both radii of $\omega$. Therefore, $\measuredangle POM = 180 - 2(90 - \theta) = 2\theta$.

We can now use Law of Cosines on $\angle{POM}$ to find the length of ${PM}$ and subtract it from the length of ${AM}$ to find ${AP}$. Since $\cos{\theta} = \frac{1}{\sqrt{5}}$ and $\sin{\theta} = \frac{2}{\sqrt{5}}$, the double angle formula tells us that $\cos{2\theta} = -\frac{3}{5}$. We have \[PM^2 = \frac{1}{2} - \frac{1}{2}\cos{2\theta} \implies PM = \frac{2\sqrt{5}}{5}.\] By Pythagorean theorem, we find that $AM = \frac{\sqrt{5}}{2} \implies AP=\boxed{\textbf{(B) } \frac{\sqrt5}{10}}$.

~awesome1st

Solution 5 (Trigonometry)

Take $O$ as the center and draw segment $ON$ perpendicular to $AM$, $ON\cap AM=N$, link $OM$. Then we have $OM\parallel AD$. So $\angle DAM=\angle OMA$. Since $AD=2AM=2OM=1$, we have $\cos\angle DAM=\cos\angle OMP=\frac{2}{\sqrt{5}}$. As a result, $NM=OM\cos\angle OMP=\frac{1}{2}\cdot \frac{2}{\sqrt{5}}=\frac{1}{\sqrt{5}}.$ Thus $PM=2NM=\frac{2}{\sqrt{5}}=\frac{2\sqrt{5}}{5}$. Since $AM=\frac{\sqrt{5}}{2}$, we have $AP=AM-PM=\boxed{\textbf{(B) } \frac{\sqrt5}{10}}$.

~FANYUCHEN20020715

Solution 6 (Coordinate Geometry)

Place circle $\omega$ in the Cartesian plane such that the center lies on the origin. Then we can easily find the equation for $\omega$ as $x^2+y^2=\frac{1}{4}$, because it is not translated and the radius is $\frac{1}{2}$.

We have $A=\left(-\frac{1}{2}, \frac{1}{2}\right)$ and $M=\left(0, -\frac{1}{2}\right)$. The slope of the line passing through these two points is $\frac{\frac{1}{2}+\frac{1}{2}}{-\frac{1}{2}-0}=\frac{1}{-\frac{1}{2}}=-2$, and the $y$-intercept is simply $M$. This gives us the line passing through both points as $y=-2x-\frac{1}{2}$.

We substitute this into the equation for the circle to get $x^2+\left(-2x-\frac{1}{2}\right)^2=\frac{1}{4}$, or $x^2+4x^2+2x+\frac{1}{4}=\frac{1}{4}$. Simplifying gives $x(5x+2)=0$. The roots of this quadratic are $x=0$ and $x=-\frac{2}{5}$, but if $x=0$ we get point $M$, so we only want $x=-\frac{2}{5}$.

We plug this back into the linear equation to find $y=\frac{3}{10}$, and so $P=\left(-\frac{2}{5}, \frac{3}{10}\right)$. Finally, we use distance formula on $A$ and $P$ to get $AP=\sqrt{\left(-\frac{5}{10}+\frac{4}{10}\right)^2+\left(\frac{5}{10}-\frac{3}{10}\right)^2}=\sqrt{\frac{1}{100}+\frac{4}{100}}=\boxed{\textbf{(B) } \frac{\sqrt5}{10}}$.

~Argonauts16

Video Solution

https://youtu.be/6ujfjGLzVoE

~IceMatrix

See Also

2020 AMC 12B (Problems • Answer Key • Resources)
Preceded by
Problem 9 		Followed by
Problem 11
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

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