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Difference between revisions of "2021 AIME I Problems/Problem 10"

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(I think the old one was #7 not #10; please let me know if I'm wrong.)
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==Problem==
 
==Problem==
−
Consider the sequence <math>(a_k)_{k\ge 1}</math> of positive rational numbers defined by <math>a_1 = \frac{2020}{2021}</math> and for <math>k\ge 1</math>, if <math>a_k = \frac{m}{n}</math> for relatively prime positive integers <math>m</math> and <math>n</math>, then
+
Find the number of pairs <math>(m,n)</math> of positive integers with <math>1\le m<n\le 30</math> such that there exists a real number <math>x</math> satisfying <cmath>\sin(mx)+\sin(nx)=2.</cmath>
  
−
<cmath>a_{k+1} = \frac{m + 18}{n+19}.</cmath>Determine the sum of all positive integers <math>j</math> such that the rational number <math>a_j</math> can be written in the form <math>\frac{t}{t+1}</math> for some positive integer <math>t</math>.
+
==Solution 1==
 +
The maximum value of <math>\sin \theta</math> is <math>1</math>, which is achieved at <math>\theta = \frac{\pi}{2}+2k\pi</math> for some integer <math>k</math>. This is left as an exercise to the reader.
 +
 
 +
This implies that <math>\sin(mx) = \sin(nx) = 1</math>, and that <math>mx = \frac{\pi}{2}+2a\pi</math> and <math>nx = \frac{\pi}{2}+2b\pi</math>, for integers <math>a, b</math>.
 +
 
 +
Taking their ratio, we have <cmath>\frac{mx}{nx} = \frac{\frac{\pi}{2}+2a\pi}{\frac{\pi}{2}+2b\pi} \implies \frac{m}{n} = \frac{4a + 1}{4b + 1} \implies \frac{m}{4a + 1} = \frac{n}{4b + 1} = k.</cmath>
 +
It remains to find all <math>m, n</math> that satisfy this equation.
 +
 
 +
If <math>k = 1</math>, then <math>m \equiv n \equiv 1 \pmod 4</math>. This corresponds to choosing two elements from the set <math>\{1, 5, 9, 13, 17, 21, 25, 29\}</math>. There are <math>\binom 82</math> ways to do so.
 +
 
 +
If <math>k < 1</math>, by multiplying <math>m</math> and <math>n</math> by the same constant <math>c = \frac{1}{k}</math>, we have that <math>mc \equiv nc \equiv 1 \pmod 4</math>. Then either <math>m \equiv n \equiv 1 \pmod 4</math>, or <math>m \equiv n \equiv 3 \pmod 4</math>. But the first case was already counted, so we don't need to consider that case. The other case corresponds to choosing two numbers from the set <math>\{3, 7, 11, 15, 19, 23, 27\}</math>. There are <math>\binom 72</math> ways here.
 +
 
 +
Finally, if <math>k > 1</math>, note that <math>k</math> must be an integer. This means that <math>m, n</math> belong to the set <math>\{k, 5k, 9k, \dots\}</math>, or <math>\{3k, 7k, 11k, \dots\}</math>. Taking casework on <math>k</math>, we get the sets <math>\{2, 10, 18, 26\}, \{6, 14, 22, 30\}, \{4, 20\}, \{12, 28\}</math>. Some sets have been omitted; this is because they were counted in the other cases already. This sums to <math>\binom 42 + \binom 42 + \binom 22 + \binom 22</math>.
 +
 
 +
In total, there are <math>\binom 82 + \binom 72 + \binom 42 + \binom 42 + \binom 22 + \binom 22 = \boxed{63}</math> pairs of <math>(m, n)</math>.
 +
 
 +
This solution was brought to you by ~Leonard_my_dude~
 +
 
 +
==Solution 2==
 +
In order for <math>\sin(mx) + \sin(nx) = 2</math>, <math>\sin(mx) = \sin(nx) = 1</math>.
 +
 
 +
This happens when
 +
<math>mx \equiv nx \equiv \frac{\pi}{2} (</math>mod <math>2\pi).</math>
 +
 
 +
This means that <math>mx = \frac{\pi}{2} + 2\pi\alpha</math> and <math>nx = \frac{\pi}{2} + 2\pi\beta</math> for any integers <math>\alpha</math> and <math>\beta</math>.
 +
 
 +
As in Solution 1, take the ratio of the two equations:
 +
<cmath>\frac{mx}{nx} = \frac{\frac{\pi}{2}+2\pi\alpha}{\frac{\pi}{2}+2\pi\beta} \implies \frac{m}{n} = \frac{\frac{1}{2}+2\alpha}{\frac{1}{2}+2\beta} \implies \frac{m}{n} = \frac{4\alpha+1}{4\beta+1}</cmath>
 +
 
 +
Now notice that the numerator and denominator of <math>\frac{4\alpha+1}{4\beta+1}</math> are both odd, which means that <math>m</math> and <math>n</math> have the same power of two (the powers of 2 cancel out).
 +
 
 +
Let the common power be <math>p</math>: then <math>m = 2^p\cdot a</math>, and <math>n = 2^p\cdot b</math> where <math>a</math> and <math>b</math> are integers between 1 and 30.
 +
 
 +
We can now rewrite the equation:
 +
<cmath>\frac{2^p\cdot a}{2^p\cdot b} = \frac{4\alpha+1}{4\beta+1} \implies \frac{a}{b} = \frac{4\alpha+1}{4\beta+1}</cmath>
  
−
==Solution 1==
+
Now it is easy to tell that <math>a \equiv 1 (</math>mod <math>4)</math> and <math>b \equiv 1 (</math>mod <math>4)</math>. However, there is another case: that  
−
We know that <math>a_{1}=\tfrac{t}{t+1}</math> when <math>t=2020</math> so <math>1</math> is a possible value of <math>j</math>. Note also that <math>a_{2}=\tfrac{2038}{2040}=\tfrac{1019}{1020}=\tfrac{t}{t+1}</math> for <math>t=1019</math>. Then <math>a_{2+q}=\tfrac{1019+18q}{1020+19q}</math> unless <math>1019+18q</math> and <math>1020+19q</math> are not relatively prime which happens when <math>q+1</math> divides <math>18q+1019</math> or <math>q+1</math> divides <math>1001</math>, so the least value of <math>q</math> is <math>6</math> and <math>j=2+6=8</math>. We know <math>a_{8}=\tfrac{1019+108}{1020+114}=\tfrac{1127}{1134}=\tfrac{161}{162}</math>. Now <math>a_{8+q}=\tfrac{161+18q}{162+19q}</math> unless <math>18q+161</math> and <math>19q+162</math> are not relatively prime which happens the first time <math>q+1</math> divides <math>18q+161</math> or <math>q+1</math> divides <math>143</math> or <math>q=10</math>, and <math>j=8+10=18</math>. We have <math>a_{18}=\tfrac{161+180}{162+190}=\tfrac{341}{352}=\tfrac{31}{32}</math>. Now <math>a_{18+q}=\tfrac{31+18q}{32+19q}</math> unless <math>18q+31</math> and <math>19q+32</math> are not relatively prime. This happens the first time <math>q+1</math> divides <math>18q+31</math> implying <math>q+1</math> divides <math>13</math>, which is prime so <math>q=12</math> and <math>j=18+12=30</math>. We have <math>a_{30}=\tfrac{31+216}{32+228}=\tfrac{247}{260}=\tfrac{19}{20}</math>. We have <math>a_{30+q}=\tfrac{18q+19}{19q+20}</math>, which is always reduced by EA, so the sum of all <math>j</math> is <math>1+2+8+18+30=\boxed{059}</math>.
+
 
 +
<math>a \equiv 3 (</math>mod <math>4)</math> and <math>b \equiv 3 (</math>mod <math>4)</math>. This is because multiplying both <math>4\alpha+1</math> and <math>4\beta+1</math> by <math>-1</math> will not change the fraction, but each congruence will be changed to <math>-1 (</math>mod <math>4) \equiv 3 (</math>mod <math>4)</math>.
 +
 
 +
From the first set of congruences, we find that <math>a</math> and <math>b</math> can be two of
 +
<math>\{1, 5, 9, \ldots, 29\}</math>.
 +
 
 +
From the second set of congruences, we find that <math>a</math> and <math>b</math> can be two of
 +
<math>\{3, 7, 11, \ldots, 27\}</math>.
 +
 
 +
Now all we have to do is multiply by <math>2^p</math> to get back to <math>m</math> and <math>n</math>.
 +
Let’s organize the solutions in order of increasing values of <math>p</math>, keeping in mind that <math>m</math> and <math>n</math> are bounded between 1 and 30.
 +
 
 +
For <math>p = 0</math> we get <math>\{1, 5, 9, \ldots, 29\}, \{3, 7, 11, \ldots, 27\}</math>.
 +
 
 +
For <math>p = 1</math> we get <math>\{2, 10, 18, 26\}, \{6, 14, 22, 30\}</math>
 +
 
 +
For <math>p = 2</math> we get <math>\{4, 20\}, \{12, 28\}</math>
 +
 
 +
If we increase the value of <math>p</math> more, there will be less than two integers in our sets, so we are done there.
 +
 
 +
There are 8 numbers in the first set, 7 in the second, 4 in the third, 4 in the fourth, 2 in the fifth, and 2 in the sixth.
 +
 +
In each of these sets we can choose 2 numbers to be <math>m</math> and <math>n</math> and then assign them in increasing order. Thus there are:
 +
 
 +
<cmath>\dbinom{8}{2}+\dbinom{7}{2}+\dbinom{4}{2}+\dbinom{4}{2}+\dbinom{2}{2}+\dbinom{2}{2} = 28+21+6+6+1+1 = \boxed{63}</cmath> possible pairs <math>(m,n)</math> that satisfy the conditions.
 +
 
 +
-KingRavi
 +
 
 +
==Solution 3==
 +
We know that the range of sine is between <math>-1</math> and <math>1</math>, inclusive.
  
−
==Solution 2 (Euclidean Algorithm and Generalization)==
+
Thus, the only way for the sum to be <math>2</math> is for <math>\sin(mx)=\sin(nx)=1</math>.
−
Let <math>a_{j_1}, a_{j_2}, a_{j_3}, \ldots, a_{j_u}</math> be all terms in the form <math>\frac{t}{t+1},</math> where <math>j_1<j_2<j_3<\cdots<j_u,</math> and <math>t</math> is some positive integer.
 
  
−
We wish to find <math>\sum_{i=1}^{u}{j_i}.</math> Suppose <math>a_{j_i}=\frac{m}{m+1}</math> for some positive integer <math>m.</math>
+
Note that <math>\sin(90+360k)=1</math>.
  
−
<i><b>To find <math>\boldsymbol{a_{j_{i+1}},}</math> we look for the smallest positive integer <math>\boldsymbol{k'}</math> for which <cmath>\boldsymbol{a_{j_{i+1}}=a_{j_i+k'}=\frac{m+18k'}{m+1+19k'}}</cmath> is reducible:</b></i>
+
Assuming <math>mx</math> and <math>nx</math> are both positive, <math>m</math> and <math>n</math> could be <math>1,5,9,13,17,21,25,29</math>. There are <math>8</math> ways, so <math>\dbinom{8}{2}</math>.
  
−
If <math>\frac{m+18k'}{m+1+19k'}</math> is reducible, then there exists a common factor <math>d>1</math> for <math>m+18k'</math> and <math>m+1+19k'.</math> By the [[Euclidean algorithm|Euclidean Algorithm]], we have
+
If both are negative, <math>m</math> and <math>n</math> could be <math>3,7,11,15,19,23,27</math>. There are <math>7</math> ways, so <math>\dbinom{7}{2}</math>.
−
<cmath>\begin{align*}
 
−
d\mid m+18k' \text{ and } d\mid m+1+19k' &\implies d\mid m+18k' \text{ and } d\mid k'+1 \\
 
−
&\implies d\mid m-18 \text{ and } d\mid k'+1.
 
−
\end{align*}</cmath>
 
−
Since <math>m-18</math> and <math>k'+1</math> are not relatively prime, and <math>m</math> is fixed, the smallest value of <math>k'</math> such that <math>\frac{m+18k'}{m+1+19k'}</math> is reducible occurs when <math>k'+1</math> is the smallest prime factor of <math>m-18.</math>
 
  
−
<i><b>We will prove that for such value of <math>\boldsymbol{k',}</math> the number <math>\boldsymbol{a_{j_{i+1}}}</math> can be written in the form <math>\boldsymbol{\frac{t}{t+1}:}</math></b></i> <cmath>a_{j_{i+1}}=a_{j_i+k'}=\frac{m+18k'}{m+1+19k'}=\frac{(m-18)+18(k'+1)}{(m-18)+19(k'+1)}=\frac{\frac{m-18}{k'+1}+18}{\frac{m-18}{k'+1}+19}, \hspace{10mm} (*)</cmath> where <math>t=\frac{m-18}{k'+1}+18</math> must be a positive integer.
+
However, the pair <math>(1,5)</math> could also be <math>(2, 10)</math> and so on. The same goes for some other pairs.
  
−
We start with <math>m=2020</math> and <math>a_{j_1}=a_1=\frac{2020}{2021},</math> then find <math>a_{j_2}, a_{j_3}, \ldots, a_{j_u}</math> by filling out the table below recursively:
+
In total there are <math>14</math> of these extra pairs.
−
<cmath>\begin{array}{c|c|c|c|c|c}
 
−
& & & & & \\ [-2ex]
 
−
\boldsymbol{i} & \boldsymbol{m} & \boldsymbol{m-18} & \boldsymbol{k'+1} & \boldsymbol{k'} & \boldsymbol{a_{j_{i+1}} \left(\textbf{by } (*)\right)} \\ [0.5ex]
 
−
\hline 
 
−
& & & & & \\ [-1.5ex]
 
−
1 & 2020 & 2002 & 2 & 1 & \hspace{4.25mm} a_2 = \frac{1019}{1020} \\ [1ex]   
 
−
2 & 1019 & 1001 & 7 & 6 & \hspace{2.75mm} a_8 = \frac{161}{162} \\ [1ex]   
 
−
3 & 161 & 143 & 11 & 10 & a_{18} = \frac{31}{32} \\ [1ex]
 
−
4 & 31 & 13 & 13 & 12 & a_{30} = \frac{19}{20} \\ [1ex]
 
−
5 & 19 & 1 & \text{N/A} & \text{N/A} & \text{N/A} \\ [1ex]
 
−
\end{array}</cmath>
 
−
As <math>\left(j_1,j_2,j_3,j_4,j_5\right)=(1,2,8,18,30),</math> the answer is <math>\sum_{i=1}^{5}{j_i}=\boxed{059}.</math>
 
  
−
<b><u>Remark</u></b>
+
The answer is <math>28+21+14 = \boxed{063}</math>.
  
−
Alternatively, from <math>(*)</math> we can set <cmath>\frac{m+18k'}{m+1+19k'}=\frac{t}{t+1}.</cmath>
+
==Remark==
−
We cross-multiply, rearrange, and apply Simon's Favorite Factoring Trick to get <cmath>\left(k'+1\right)(t-18)=m-18.</cmath>
+
The graphs of <math>r\leq\sin(m\theta)+\sin(n\theta)</math> and <math>r=2</math> are shown here in Desmos: https://www.desmos.com/calculator/busxadywja
−
Since <math>k'+1\geq2,</math> to find the smallest <math>k',</math> we need <math>k'+1</math> to be the smallest prime factor of <math>m-18.</math> Now we continue with the last two paragraphs of the solution above.
 
  
−
~MRENTHUSIASM
+
Move the sliders around for <math>1\leq m \leq 29</math> and <math>2\leq m+1\leq n\leq30</math> to observe the geometric representation generated by each pair <math>(m,n).</math>
  
 +
~MRENTHUSIASM (inspired by TheAMCHub)
  
 
==Video Solution==
 
==Video Solution==
−
https://youtu.be/oiUcYn1uYMM ~Math Problem Solving Skills
+
https://youtu.be/O84aJ5OTZ2E
  
 +
~mathproblemsolvingskills
  
−
==Video Solution by Punxsutawney Phil==
+
==Video Solution==
−
https://youtube.com/watch?v=LIjTty3rVso
+
https://www.youtube.com/watch?v=LUkQ7R1DqKo
 +
 
 +
~Mathematical Dexterity
  
 
==See Also==
 
==See Also==
 
{{AIME box|year=2021|n=I|num-b=9|num-a=11}}
 
{{AIME box|year=2021|n=I|num-b=9|num-a=11}}
−
 
−
[[Category:Intermediate Number Theory Problems]]
 
 
{{MAA Notice}}
 
{{MAA Notice}}

Revision as of 11:47, 13 March 2023

Problem

Find the number of pairs $(m,n)$ of positive integers with $1\le m<n\le 30$ such that there exists a real number $x$ satisfying \[\sin(mx)+\sin(nx)=2.\]

Solution 1

The maximum value of $\sin \theta$ is $1$, which is achieved at $\theta = \frac{\pi}{2}+2k\pi$ for some integer $k$. This is left as an exercise to the reader.

This implies that $\sin(mx) = \sin(nx) = 1$, and that $mx = \frac{\pi}{2}+2a\pi$ and $nx = \frac{\pi}{2}+2b\pi$, for integers $a, b$.

Taking their ratio, we have \[\frac{mx}{nx} = \frac{\frac{\pi}{2}+2a\pi}{\frac{\pi}{2}+2b\pi} \implies \frac{m}{n} = \frac{4a + 1}{4b + 1} \implies \frac{m}{4a + 1} = \frac{n}{4b + 1} = k.\] It remains to find all $m, n$ that satisfy this equation.

If $k = 1$, then $m \equiv n \equiv 1 \pmod 4$. This corresponds to choosing two elements from the set $\{1, 5, 9, 13, 17, 21, 25, 29\}$. There are $\binom 82$ ways to do so.

If $k < 1$, by multiplying $m$ and $n$ by the same constant $c = \frac{1}{k}$, we have that $mc \equiv nc \equiv 1 \pmod 4$. Then either $m \equiv n \equiv 1 \pmod 4$, or $m \equiv n \equiv 3 \pmod 4$. But the first case was already counted, so we don't need to consider that case. The other case corresponds to choosing two numbers from the set $\{3, 7, 11, 15, 19, 23, 27\}$. There are $\binom 72$ ways here.

Finally, if $k > 1$, note that $k$ must be an integer. This means that $m, n$ belong to the set $\{k, 5k, 9k, \dots\}$, or $\{3k, 7k, 11k, \dots\}$. Taking casework on $k$, we get the sets $\{2, 10, 18, 26\}, \{6, 14, 22, 30\}, \{4, 20\}, \{12, 28\}$. Some sets have been omitted; this is because they were counted in the other cases already. This sums to $\binom 42 + \binom 42 + \binom 22 + \binom 22$.

In total, there are $\binom 82 + \binom 72 + \binom 42 + \binom 42 + \binom 22 + \binom 22 = \boxed{63}$ pairs of $(m, n)$.

This solution was brought to you by ~Leonard_my_dude~

Solution 2

In order for $\sin(mx) + \sin(nx) = 2$, $\sin(mx) = \sin(nx) = 1$.

This happens when $mx \equiv nx \equiv \frac{\pi}{2} ($mod $2\pi).$

This means that $mx = \frac{\pi}{2} + 2\pi\alpha$ and $nx = \frac{\pi}{2} + 2\pi\beta$ for any integers $\alpha$ and $\beta$.

As in Solution 1, take the ratio of the two equations: \[\frac{mx}{nx} = \frac{\frac{\pi}{2}+2\pi\alpha}{\frac{\pi}{2}+2\pi\beta} \implies \frac{m}{n} = \frac{\frac{1}{2}+2\alpha}{\frac{1}{2}+2\beta} \implies \frac{m}{n} = \frac{4\alpha+1}{4\beta+1}\]

Now notice that the numerator and denominator of $\frac{4\alpha+1}{4\beta+1}$ are both odd, which means that $m$ and $n$ have the same power of two (the powers of 2 cancel out).

Let the common power be $p$: then $m = 2^p\cdot a$, and $n = 2^p\cdot b$ where $a$ and $b$ are integers between 1 and 30.

We can now rewrite the equation: \[\frac{2^p\cdot a}{2^p\cdot b} = \frac{4\alpha+1}{4\beta+1} \implies \frac{a}{b} = \frac{4\alpha+1}{4\beta+1}\]

Now it is easy to tell that $a \equiv 1 ($mod $4)$ and $b \equiv 1 ($mod $4)$. However, there is another case: that

$a \equiv 3 ($mod $4)$ and $b \equiv 3 ($mod $4)$. This is because multiplying both $4\alpha+1$ and $4\beta+1$ by $-1$ will not change the fraction, but each congruence will be changed to $-1 ($mod $4) \equiv 3 ($mod $4)$.

From the first set of congruences, we find that $a$ and $b$ can be two of $\{1, 5, 9, \ldots, 29\}$.

From the second set of congruences, we find that $a$ and $b$ can be two of $\{3, 7, 11, \ldots, 27\}$.

Now all we have to do is multiply by $2^p$ to get back to $m$ and $n$. Let’s organize the solutions in order of increasing values of $p$, keeping in mind that $m$ and $n$ are bounded between 1 and 30.

For $p = 0$ we get $\{1, 5, 9, \ldots, 29\}, \{3, 7, 11, \ldots, 27\}$.

For $p = 1$ we get $\{2, 10, 18, 26\}, \{6, 14, 22, 30\}$

For $p = 2$ we get $\{4, 20\}, \{12, 28\}$

If we increase the value of $p$ more, there will be less than two integers in our sets, so we are done there.

There are 8 numbers in the first set, 7 in the second, 4 in the third, 4 in the fourth, 2 in the fifth, and 2 in the sixth.

In each of these sets we can choose 2 numbers to be $m$ and $n$ and then assign them in increasing order. Thus there are:

\[\dbinom{8}{2}+\dbinom{7}{2}+\dbinom{4}{2}+\dbinom{4}{2}+\dbinom{2}{2}+\dbinom{2}{2} = 28+21+6+6+1+1 = \boxed{63}\] possible pairs $(m,n)$ that satisfy the conditions.

-KingRavi

Solution 3

We know that the range of sine is between $-1$ and $1$, inclusive.

Thus, the only way for the sum to be $2$ is for $\sin(mx)=\sin(nx)=1$.

Note that $\sin(90+360k)=1$.

Assuming $mx$ and $nx$ are both positive, $m$ and $n$ could be $1,5,9,13,17,21,25,29$. There are $8$ ways, so $\dbinom{8}{2}$.

If both are negative, $m$ and $n$ could be $3,7,11,15,19,23,27$. There are $7$ ways, so $\dbinom{7}{2}$.

However, the pair $(1,5)$ could also be $(2, 10)$ and so on. The same goes for some other pairs.

In total there are $14$ of these extra pairs.

The answer is $28+21+14 = \boxed{063}$.

Remark

The graphs of $r\leq\sin(m\theta)+\sin(n\theta)$ and $r=2$ are shown here in Desmos: https://www.desmos.com/calculator/busxadywja

Move the sliders around for $1\leq m \leq 29$ and $2\leq m+1\leq n\leq30$ to observe the geometric representation generated by each pair $(m,n).$

~MRENTHUSIASM (inspired by TheAMCHub)

Video Solution

https://youtu.be/O84aJ5OTZ2E

~mathproblemsolvingskills

Video Solution

https://www.youtube.com/watch?v=LUkQ7R1DqKo

~Mathematical Dexterity

See Also

2021 AIME I (Problems • Answer Key • Resources)
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Problem 11
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All AIME Problems and Solutions

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