Difference between revisions of "2021 AMC 12A Problems/Problem 18"

(Solution 2: Revised the first paragraph--combined the results of JHawk0224 and awesomediabrine)
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<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}\qquad</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}\qquad</math>
  
==Solution 1 (but where do you get <math>10=11+f(\frac{25}{11})</math> ==
+
==Solution 1==
 
Looking through the solutions we can see that <math>f(\frac{25}{11})</math> can be expressed as <math>f(\frac{25}{11} \cdot 11) = f(11) + f(\frac{25}{11})</math> so using the prime numbers to piece together what we have we can get <math>10=11+f(\frac{25}{11})</math>, so <math>f(\frac{25}{11})=-1</math> or <math>\boxed{E}</math>.
 
Looking through the solutions we can see that <math>f(\frac{25}{11})</math> can be expressed as <math>f(\frac{25}{11} \cdot 11) = f(11) + f(\frac{25}{11})</math> so using the prime numbers to piece together what we have we can get <math>10=11+f(\frac{25}{11})</math>, so <math>f(\frac{25}{11})=-1</math> or <math>\boxed{E}</math>.
  
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==Solution 2==
 
==Solution 2==
We know that <math>f(p) = f(p \cdot 1) = f(p) + f(1)</math>. By Transitive, we have<cmath>f(p) &= f(p) + f(1).</cmath>
+
We know that <math>f(p) = f(p \cdot 1) = f(p) + f(1)</math>. By transitive, we have <cmath>f(p) = f(p) + f(1).</cmath>
 
Subtracting <math>f(p)</math> from both sides gives <math>0 = f(1).</math>
 
Subtracting <math>f(p)</math> from both sides gives <math>0 = f(1).</math>
 
Also
 
Also
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Thus, our answer is <math>\boxed{\textbf{(E)} \frac{25}{11}}</math>
 
Thus, our answer is <math>\boxed{\textbf{(E)} \frac{25}{11}}</math>
~JHawk0224 -awesomediabrine
+
 
 +
~JHawk0224 ~awesomediabrine
  
 
==Solution 3 (Deeper)==
 
==Solution 3 (Deeper)==
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~yofro
 
~yofro
 +
 +
==Solution 4 (Extremely Comprehensive, Similar to Solution 3)==
 +
===Results===
 +
We have the following important results:
 +
 +
<ol style="margin-left: 1.5em;">
 +
  <li><math>f\left(\prod_{k=1}^{n}a_k\right)=\sum_{k=1}^{n}f(a_k)</math> for all positive rational numbers <math>a_k</math> and positive integers <math>n</math></li><p>
 +
  <li><math>f\left(a^n\right)=nf(a)</math> for all positive rational numbers <math>a</math> and positive integers <math>n</math></li><p>
 +
  <li><math>f(1)=0</math></li><p>
 +
  <li><math>f\left({\frac 1a}\right)=-f(a)</math> for all positive rational numbers <math>a</math></li><p>
 +
</ol>
 +
 +
~MRENTHUSIASM
 +
 +
===Proofs===
 +
<ol style="margin-left: 1.5em;">
 +
  <li>Result 1: We can show Result 1 by induction.</li><p>
 +
  <li>Result 2: Since positive powers are just repeated multiplication of the base, we will use Result 1 to prove Result 2: <cmath>f\left(a^n\right)=f\left(\prod_{k=1}^{n}a\right)=\sum_{k=1}^{n}f(a)=nf(a).</cmath></li><p>
 +
  <li>Result 3: For all positive rational numbers <math>a,</math> we have <cmath>f(a)=f(a\cdot1)=f(a)+f(1).</cmath> Therefore, we get <math>f(1)=0,</math> from which Result 3 is true.</li><p>
 +
  <li>Result 4: We have <cmath>f(a)+f\left(\frac1a\right)=f\left(a\cdot\frac1a\right)=f(1)=0.</cmath> Therefore, we get <math>f\left({\frac 1a}\right)=-f(a),</math> from which Result 4 is true.</li><p>
 +
</ol>
 +
 +
~MRENTHUSIASM
 +
 +
===Solution===
 +
For all positive integers <math>x</math> and <math>y,</math> suppose <math>\prod_{k=1}^{m}p_k^{e_k}</math> and <math>\prod_{k=1}^{n}q_k^{d_k}</math> are their respective prime factorizations, we have
 +
<cmath>\begin{align*}
 +
f\left(\frac xy\right)&=f(x)+f\left(\frac 1y\right) & \text{by Result 1} \\
 +
&=f(x)-f(y) & \text{by Result 4} \\
 +
&=f\left(\prod_{k=1}^{m}p_k^{e_k}\right)-f\left(\prod_{k=1}^{n}q_k^{d_k}\right) \\
 +
&=\left[\sum_{k=1}^{m}f\left(p_k^{e_k}\right)\right]-\left[\sum_{k=1}^{n}f\left(q_k^{d_k}\right)\right] & \text{by Result 1} \\
 +
&=\left[\sum_{k=1}^{m}e_k f\left(p_k\right)\right]-\left[\sum_{k=1}^{n}d_k f\left(q_k\right)\right] &\hspace{10mm} \text{by Result 2} \\
 +
&=\left[\sum_{k=1}^{m}e_k p_k \right]-\left[\sum_{k=1}^{n}d_k q_k \right].
 +
\end{align*}</cmath>
 +
 +
We apply function <math>f</math> to each fraction in the choices:
 +
 +
<cmath>\begin{alignat*}{10}
 +
&\textbf{(A)} \qquad && f\left(\frac{17}{32}\right) \quad && = \quad && f\left(\frac{17^1}{2^5}\right) \quad && = \quad && [1(17)]-[5(2)] \quad && = \quad && 7 \\
 +
&\textbf{(B)} \qquad && f\left(\frac{11}{16}\right) \quad && = \quad && f\left(\frac{11^1}{2^4}\right) \quad && = \quad && [1(11)]-[4(2)] \quad && = \quad && 3 \\
 +
&\textbf{(C)} \qquad && f\left(\frac{7}{9}\right) \quad && = \quad && f\left(\frac{7^1}{3^2}\right)  \quad && = \quad && [1(7)]-[2(3)]  \quad && = \quad && 1 \\
 +
&\textbf{(D)} \qquad && f\left(\frac{7}{6}\right) \quad && = \quad && f\left(\frac{7^1}{2^1\cdot3^1}\right) \quad && = \quad && [1(7)]-[1(2)+1(3)] \quad && = \quad && 2 \\
 +
&\textbf{(E)} \qquad && f\left(\frac{25}{11}\right) \quad && = \quad && f\left(\frac{5^2}{11^1}\right) \quad && = \quad && [2(5)]-[1(11)] \quad && = \quad && -1
 +
\end{alignat*}</cmath>
 +
Therefore, the answer is <math>\boxed{\textbf{(E) }\frac{25}{11}}.</math>
 +
 +
~MRENTHUSIASM
 +
 +
==Solution 5==
 +
The problem gives us that f(p)=p.  If we let a=p and b=1, we get f(p)=f(p)+f(1), which implies f(1)=0.  Notice that the answer choices are all fractions, which means we will have to multiply an integer by a fraction to be able to solve it. Therefore, let's try plugging in fractions and try to solve them.  Note that if we plug in a=p and b=1/p, we get f(1)=f(p)+f(1/p).  We can solve for f(1/p) as -f(p)!  This gives us the information we need to solve the problem.  Testing out the answer choices gives us the answer of E.
  
 
==Video Solution by Hawk Math==
 
==Video Solution by Hawk Math==
 
https://www.youtube.com/watch?v=dvlTA8Ncp58
 
https://www.youtube.com/watch?v=dvlTA8Ncp58
 +
 +
==Video Solution by North America Math Contest Go Go Go Through Induction==
 +
https://www.youtube.com/watch?v=ffX0fTgJN0w&list=PLexHyfQ8DMuKqltG3cHT7Di4jhVl6L4YJ&index=12
  
 
==Video Solution by Punxsutawney Phil==
 
==Video Solution by Punxsutawney Phil==
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~ pi_is_3.14
 
~ pi_is_3.14
 +
 +
==Video Solution by TheBeautyofMath==
 +
https://youtu.be/IUJ_A9KiLEE
 +
 +
~IceMatrix
  
 
==See also==
 
==See also==
 
{{AMC10 box|year=2021|ab=A|num-b=17|num-a=19}}
 
{{AMC10 box|year=2021|ab=A|num-b=17|num-a=19}}
 
{{AMC12 box|year=2021|ab=A|num-b=17|num-a=19}}
 
{{AMC12 box|year=2021|ab=A|num-b=17|num-a=19}}
 +
[[Category:Intermediate Algebra Problems]]
 
{{MAA Notice}}
 
{{MAA Notice}}

Revision as of 12:25, 15 April 2021

The following problem is from both the 2021 AMC 10A #18 and 2021 AMC 12A #18, so both problems redirect to this page.

Problem

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 $f(x) < 0$?

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

Solution 1

Looking through the solutions we can see that $f(\frac{25}{11})$ can be expressed as $f(\frac{25}{11} \cdot 11) = f(11) + f(\frac{25}{11})$ so using the prime numbers to piece together what we have we can get $10=11+f(\frac{25}{11})$, so $f(\frac{25}{11})=-1$ or $\boxed{E}$.

-Lemonie

$f(\frac{25}{11} \cdot 11) = f(25) = f(5) + f(5) = 10$

- awesomediabrine

Solution 2

We know that $f(p) = f(p \cdot 1) = f(p) + f(1)$. By transitive, we have \[f(p) = f(p) + f(1).\] Subtracting $f(p)$ from both sides gives $0 = f(1).$ Also \[f(2)+f\left(\frac{1}{2}\right)=f(1)=0 \implies 2+f\left(\frac{1}{2}\right)=0 \implies f\left(\frac{1}{2}\right) = -2\] \[f(3)+f\left(\frac{1}{3}\right)=f(1)=0 \implies 3+f\left(\frac{1}{3}\right)=0 \implies f\left(\frac{1}{3}\right) = -3\] \[f(11)+f\left(\frac{1}{11}\right)=f(1)=0 \implies 11+f\left(\frac{1}{11}\right)=0 \implies f\left(\frac{1}{11}\right) = -11\] In $\textbf{(A)}$ we have $f\left(\frac{17}{32}\right)=17+5f\left(\frac{1}{2}\right)=17-5(2)=7$.

In $\textbf{(B)}$ we have $f\left(\frac{11}{16}\right)=11+4f\left(\frac{1}{2}\right)=11-4(2)=3$.

In $\textbf{(C)}$ we have $f\left(\frac{7}{9}\right)=7+2f\left(\frac{1}{3}\right)=7-2(3)=1$.

In $\textbf{(D)}$ we have $f\left(\frac{7}{6}\right)=7+f\left(\frac{1}{2}\right)+f\left(\frac{1}{3}\right)=7-2-3=2$.

In $\textbf{(E)}$ we have $f\left(\frac{25}{11}\right)=10+f\left(\frac{1}{11}\right)=10-11=-1$.

Thus, our answer is $\boxed{\textbf{(E)} \frac{25}{11}}$

~JHawk0224 ~awesomediabrine

Solution 3 (Deeper)

Consider the rational $\frac{a}{b}$, for $a,b$ integers. We have $f(a)=f\left(\frac{a}{b}\cdot b\right)=f\left(\frac{a}{b}\right)+f(b)$. So $f\left(\frac{a}{b}\right)=f(a)-f(b)$. Let $p$ be a prime. Notice that $f(p^k)=kf(p)$. And $f(p)=p$. So if $a=p_1^{a_1}p_2^{a_2}\cdots p_k^{a_k}$, $f(a)=a_1p_1+a_2p_2+....+a_kp_k$. We simply need this to be greater than what we have for $f(b)$. Notice that for answer choices $A,B,C,$ and $D$, the numerator $(a)$ has less prime factors than the denominator, and so they are less likely to work. We check $E$ first, and it works, therefore the answer is $\boxed{\textbf{(E)}}$.

~yofro

Solution 4 (Extremely Comprehensive, Similar to Solution 3)

Results

We have the following important results:

  1. $f\left(\prod_{k=1}^{n}a_k\right)=\sum_{k=1}^{n}f(a_k)$ for all positive rational numbers $a_k$ and positive integers $n$
  2. $f\left(a^n\right)=nf(a)$ for all positive rational numbers $a$ and positive integers $n$
  3. $f(1)=0$
  4. $f\left({\frac 1a}\right)=-f(a)$ for all positive rational numbers $a$

~MRENTHUSIASM

Proofs

  1. Result 1: We can show Result 1 by induction.
  2. Result 2: Since positive powers are just repeated multiplication of the base, we will use Result 1 to prove Result 2: \[f\left(a^n\right)=f\left(\prod_{k=1}^{n}a\right)=\sum_{k=1}^{n}f(a)=nf(a).\]
  3. Result 3: For all positive rational numbers $a,$ we have \[f(a)=f(a\cdot1)=f(a)+f(1).\] Therefore, we get $f(1)=0,$ from which Result 3 is true.
  4. Result 4: We have \[f(a)+f\left(\frac1a\right)=f\left(a\cdot\frac1a\right)=f(1)=0.\] Therefore, we get $f\left({\frac 1a}\right)=-f(a),$ from which Result 4 is true.

~MRENTHUSIASM

Solution

For all positive integers $x$ and $y,$ suppose $\prod_{k=1}^{m}p_k^{e_k}$ and $\prod_{k=1}^{n}q_k^{d_k}$ are their respective prime factorizations, we have \begin{align*} f\left(\frac xy\right)&=f(x)+f\left(\frac 1y\right) & \text{by Result 1} \\ &=f(x)-f(y) & \text{by Result 4} \\ &=f\left(\prod_{k=1}^{m}p_k^{e_k}\right)-f\left(\prod_{k=1}^{n}q_k^{d_k}\right) \\ &=\left[\sum_{k=1}^{m}f\left(p_k^{e_k}\right)\right]-\left[\sum_{k=1}^{n}f\left(q_k^{d_k}\right)\right] & \text{by Result 1} \\ &=\left[\sum_{k=1}^{m}e_k f\left(p_k\right)\right]-\left[\sum_{k=1}^{n}d_k f\left(q_k\right)\right] &\hspace{10mm} \text{by Result 2} \\ &=\left[\sum_{k=1}^{m}e_k p_k \right]-\left[\sum_{k=1}^{n}d_k q_k \right]. \end{align*}

We apply function $f$ to each fraction in the choices:

\begin{alignat*}{10} &\textbf{(A)} \qquad && f\left(\frac{17}{32}\right) \quad && = \quad && f\left(\frac{17^1}{2^5}\right) \quad && = \quad && [1(17)]-[5(2)] \quad && = \quad && 7 \\  &\textbf{(B)} \qquad && f\left(\frac{11}{16}\right) \quad && = \quad && f\left(\frac{11^1}{2^4}\right) \quad && = \quad && [1(11)]-[4(2)] \quad && = \quad && 3 \\  &\textbf{(C)} \qquad && f\left(\frac{7}{9}\right) \quad && = \quad && f\left(\frac{7^1}{3^2}\right)  \quad && = \quad && [1(7)]-[2(3)]  \quad && = \quad && 1 \\  &\textbf{(D)} \qquad && f\left(\frac{7}{6}\right) \quad && = \quad && f\left(\frac{7^1}{2^1\cdot3^1}\right) \quad && = \quad && [1(7)]-[1(2)+1(3)] \quad && = \quad && 2 \\ &\textbf{(E)} \qquad && f\left(\frac{25}{11}\right) \quad && = \quad && f\left(\frac{5^2}{11^1}\right) \quad && = \quad && [2(5)]-[1(11)] \quad && = \quad && -1 \end{alignat*} Therefore, the answer is $\boxed{\textbf{(E) }\frac{25}{11}}.$

~MRENTHUSIASM

Solution 5

The problem gives us that f(p)=p. If we let a=p and b=1, we get f(p)=f(p)+f(1), which implies f(1)=0. Notice that the answer choices are all fractions, which means we will have to multiply an integer by a fraction to be able to solve it. Therefore, let's try plugging in fractions and try to solve them. Note that if we plug in a=p and b=1/p, we get f(1)=f(p)+f(1/p). We can solve for f(1/p) as -f(p)! This gives us the information we need to solve the problem. Testing out the answer choices gives us the answer of E.

Video Solution by Hawk Math

https://www.youtube.com/watch?v=dvlTA8Ncp58

Video Solution by North America Math Contest Go Go Go Through Induction

https://www.youtube.com/watch?v=ffX0fTgJN0w&list=PLexHyfQ8DMuKqltG3cHT7Di4jhVl6L4YJ&index=12

Video Solution by Punxsutawney Phil

https://youtu.be/8gGcj95rlWY

Video Solution by OmegaLearn (Using Functions and manipulations)

https://youtu.be/aGv99CLzguE

~ pi_is_3.14

Video Solution by TheBeautyofMath

https://youtu.be/IUJ_A9KiLEE

~IceMatrix

See also

2021 AMC 10A (ProblemsAnswer KeyResources)
Preceded by
Problem 17
Followed by
Problem 19
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
2021 AMC 12A (ProblemsAnswer KeyResources)
Preceded by
Problem 17
Followed by
Problem 19
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