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Difference between revisions of "2024 AIME I Problems/Problem 12"

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Define <math>f(x)=|| x|-\tfrac{1}{2}|</math> and <math>g(x)=|| x|-\tfrac{1}{4}|</math>. Find the number of intersections of the graphs of <cmath>y=4 g(f(\sin (2 \pi x))) \quad\text{ and }\quad x=4 g(f(\cos (3 \pi y))).</cmath>
 
Define <math>f(x)=|| x|-\tfrac{1}{2}|</math> and <math>g(x)=|| x|-\tfrac{1}{4}|</math>. Find the number of intersections of the graphs of <cmath>y=4 g(f(\sin (2 \pi x))) \quad\text{ and }\quad x=4 g(f(\cos (3 \pi y))).</cmath>
  
−
==Solution 1 (BASH, DO NOT ATTEMPT IF INSUFFICIENT TIME)==
+
==Graph==
−
If we graph <math>4g(f(x))</math>, we see it forms a sawtooth graph that oscillates between <math>0</math> and <math>1</math> (for values of <math>x</math> between <math>-1</math> and <math>1</math>, which is true because the arguments are between <math>-1</math> and <math>1</math>). Thus by precariously drawing the graph of the two functions in the square bounded by <math>(0,0)</math>, <math>(0,1)</math>, <math>(1,1)</math>, and <math>(1,0)</math>, and hand-counting each of the intersections, we get <math>\boxed{384}</math>
+
https://www.desmos.com/calculator/wml09giaun
−
===Note===
+
 
−
While this solution might seem unreliable (it probably is), the only parts where counting the intersection might be tricky is near <math>(1,1)</math>. Make sure to count them as two points and not one, or you'll get <math>383</math>.
+
==Solution 1==
 +
If we graph <math>4g(f(x))</math>, we see it forms a sawtooth graph that oscillates between <math>0</math> and <math>1</math> (for values of <math>x</math> between <math>-1</math> and <math>1</math>, which is true because the arguments are between <math>-1</math> and <math>1</math>). Thus by precariously drawing the graph of the two functions in the square bounded by <math>(0,0)</math>, <math>(0,1)</math>, <math>(1,1)</math>, and <math>(1,0)</math>, and hand-counting each of the intersections, we get <math>\boxed{385}</math>
 +
 
 +
===Remark===
 +
While this solution might seem unreliable (it probably is), the only parts where counting the intersection might be tricky is near <math>(1,1)</math>. Make sure to count them as two points and not one, or you'll get <math>384</math>.
 +
 
 +
== Note 1==
 +
The answer should be 385 since there are 16 intersections in each of 24 smaller boxes of dimensions 1/6 x 1/4 and then another one at the corner (1,1).
 +
 
 +
==Solution 2==
 +
We will denote <math>h(x)=4g(f(x))</math> for simplicity. Denote <math>p(x)</math> as the first equation and <math>q(y)</math> as the graph of the second. We notice that both <math>f(x)</math> and <math>g(x)</math> oscillate between 0 and 1. The intersections are thus all in the square <math>(0,0)</math>, <math>(0,1)</math>, <math>(1,1)</math>, and <math>(1,0)</math>. Every <math>p(x)</math> wave going up and down crosses every <math>q(y)</math> wave. Now, we need to find the number of times each wave touches 0 and 1.
 +
 
 +
 
 +
We notice that <math>h(x)=0</math> occurs at <math>x=-\frac{3}{4}, -\frac{1}{4}, \frac{1}{4}, \frac{3}{4}</math>, and <math>h(x)=1</math> occurs at <math>x=-1, -\frac{1}{2}, 0,\frac{1}{2},1</math>. A sinusoid passes through each point twice during each period, but it only passes through the extrema once. <math>p(x)</math> has 1 period between 0 and 1, giving 8 solutions for <math>p(x)=0</math> and 9 solutions for <math>p(x)=1</math>, or 16 up and down waves. <math>q(y)</math> has 1.5 periods, giving 12 solutions for <math>q(y)=0</math> and 13 solutions for <math>q(y)=1</math>, or 24 up and down waves. This amounts to <math>16\cdot24=384</math> intersections.
 +
 
 +
 
 +
However, we have to be very careful when counting around <math>(1, 1)</math>. At this point, <math>q(y)</math> has an infinite downwards slope and <math>p(x)</math> is slanted, giving us an extra intersection; thus, we need to add 1 to our answer to get <math>\boxed{385}</math>.
 +
 
 +
~Xyco
 +
 
 +
==Solution 3 (Rigorous analysis of why there are two solutions near (1,1))==
 +
 
 +
We can easily see that only <math>x, y \in \left[0,1 \right]</math> may satisfy both functions.
 +
We call function <math>y = 4g \left( f \left( \sin \left( 2 \pi x \right) \right) \right)</math> as Function 1 and function <math>x = 4g \left( f \left( \cos \left( 3 \pi y \right) \right) \right)</math> as Function 2.
 +
 
 +
For Function 1, in each interval <math>\left[ \frac{i}{4} , \frac{i+1}{4} \right]</math> with <math>i \in \left\{ 0, 1, \cdots, 3 \right\}</math>, Function 1's value oscillates between 0 and 1. It attains 1 at <math>x = \frac{i}{4}</math>, <math>\frac{i+1}{4}</math> and another point between these two.
 +
Between two consecutive points whose functional values are 1, the function first decreases from 1 to 0 and then increases from 0 to 1.
 +
So the graph of this function in this interval consists of 4 monotonic pieces.
 +
 
 +
For Function 2, in each interval <math>\left[ \frac{i}{6} , \frac{i+1}{6} \right]</math> with <math>i \in \left\{ 0, 1, \cdots, 5 \right\}</math>, Function 2's value oscillates between 0 and 1. It attains 1 at <math>y = \frac{i}{6}</math>, <math>\frac{i+1}{6}</math> and another point between these two.
 +
Between two consecutive points whose functional values are 1, the function first decreases from 1 to 0 and then increases from 0 to 1.
 +
So the graph of this function in this interval consists of 4 monotonic curves.
 +
 
 +
Consider any region <math>\left[ \frac{i}{4} , \frac{i+1}{4} \right] \times \left[ \frac{j}{6} , \frac{j+1}{6} \right]</math> with <math>i \in \left\{ 0, 1, \cdots, 3 \right\}</math> and <math>j \in \left\{0, 1, \cdots , 5 \right\}</math> but <math>\left( i, j \right) \neq \left( 3, 5 \right)</math>.
 +
Both functions have four monotonic pieces.
 +
Because Function 1's each monotonic piece can take any value in  <math>\left[ \frac{j}{6} , \frac{j+1}{6} \right]</math> and Function 2' each monotonic piece can take any value in  <math>\left[ \frac{i}{4} , \frac{i+1}{4} \right]</math>, Function 1's each monotonic piece intersects with Function 2's each monotonic piece.
 +
Therefore, in the interval <math>\left[ \frac{i}{4} , \frac{i+1}{4} \right] \times \left[ \frac{j}{6} , \frac{j+1}{6} \right]</math>, the number of intersecting points is <math>4 \cdot 4 = 16</math>.
 +
 
 +
Next, we prove that if an intersecting point is on a line <math>x = \frac{i}{4}</math> for <math>i \in \left\{ 0, 1, \cdots, 4 \right\}</math>, then this point must be <math>\left( 1, 1 \right)</math>.
 +
 
 +
For <math>x = \frac{i}{4}</math>, Function 1 attains value 1.
 +
For Function 2, if <math>y = 1</math>, then <math>x = 1</math>.
 +
Therefore, the intersecting point is <math>\left( 1, 1 \right)</math>.
 +
 
 +
Similarly, we can prove that if an intersecting point is on a line <math>y = \frac{i}{6}</math> for <math>i \in \left\{ 0, 1, \cdots, 6 \right\}</math>, then this point must be <math>\left( 1, 1 \right)</math>.
 +
 
 +
Therefore, in each region <math>\left[ \frac{i}{4} , \frac{i+1}{4} \right] \times \left[ \frac{j}{6} , \frac{j+1}{6} \right]</math> with <math>i \in \left\{ 0, 1, \cdots, 3 \right\}</math> and <math>j \in \left\{0, 1, \cdots , 5 \right\}</math> but <math>\left( i, j \right) \neq \left( 3, 5 \right)</math>, all 16 intersecting points are interior.
 +
That is, no two regions share any common intersecting point.
 +
 
 +
Next, we study region  <math>\left[ \frac{3}{4} , 1 \right] \times \left[ \frac{5}{6} , 1 \right]</math>.
 +
Consider any pair of monotonic pieces, where one is from Function 1 and one is from Function 2, except the pair of two monotonic pieces from two functions that attain <math>\left( 1 , 1 \right)</math>.
 +
Two pieces in each pair intersects at an interior point on the region.
 +
So the number of intersecting points is <math>4 \cdot 4 - 1 = 15</math>.
 +
 
 +
Finally, we compute the number intersection points of two functions' monotonic pieces that both attain <math>\left( 1, 1 \right)</math>.
 +
 
 +
One trivial intersection point is <math>\left( 1, 1 \right)</math>.
 +
Now, we study whether they intersect at another point.
 +
 
 +
Define <math>x = 1 - x'</math> and <math>y = 1 - y'</math>.
 +
Thus, for positive and sufficiently small <math>x'</math> and <math>y'</math>, Function 1 is reduced to
 +
<cmath>
 +
\[
 +
y' = 4 \sin 2 \pi x' \hspace{1cm} (1)
 +
\]
 +
</cmath>
 +
and Function 2 is reduced to
 +
<cmath>
 +
\[
 +
x' = 4 \left( 1 - \cos 3 \pi y' \right) . \hspace{1cm} (2)
 +
\]
 +
</cmath>
 +
 
 +
Now, we study whether there is a non-zero solution.
 +
Because we consider sufficiently small <math>x'</math> and <math>y'</math>, to get an intuition and quick estimate, we do approximations of the above equations.
 +
 
 +
Equation (1) is approximated as
 +
<cmath>
 +
\[
 +
y' = 4 \cdot 2 \pi x'
 +
\]
 +
</cmath>
 +
and Equation (2) is approximated as
 +
<cmath>
 +
\[
 +
x' = 2 \left( 3 \pi y' \right)^2
 +
\]
 +
</cmath>
 +
 
 +
To solve these equations, we get <math>x' = \frac{1}{8^2 \cdot 18 \pi^4}</math> and <math>y' = \frac{1}{8 \cdot 18 \pi^3}</math>.
 +
Therefore, two functions' two monotonic pieces that attain <math>\left( 1, 1 \right)</math> have two intersecting points.
 +
 
 +
Putting all analysis above, the total number of intersecting points is <math>16 \cdot 4 \cdot 6 + 1 = \boxed{\textbf{(385) }}</math>.
 +
 
 +
~Steven Chen (Professor Chen Education Palace, www.professorchenedu.com)
 +
 
 +
==Video Solution==
 +
https://youtu.be/PA3aLjPinhI?t=7722
 +
 
 +
~Math Gold Medalist
 +
 
 +
 
 +
==Video Solution (Rigorous analysis of why there are two solutions near (1,1))==
 +
 
 +
https://youtu.be/XjXwWfFzSrM
 +
 
 +
~Steven Chen (Professor Chen Education Palace, www.professorchenedu.com)
 +
 
 +
==Drawing of two solutions near <math>\left( 1, 1 \right)</math>==
 +
 
 +
https://artofproblemsolving.com/wiki/index.php/File:2024_AIME_I_Problem_12,_two_solutions_near_(1,1).png
 +
 
 +
~Steven Chen (Professor Chen Education Palace, www.professorchenedu.com)
 +
 
  
 
==See also==
 
==See also==

Latest revision as of 06:09, 24 November 2024

Problem

Define $f(x)=|| x|-\tfrac{1}{2}|$ and $g(x)=|| x|-\tfrac{1}{4}|$. Find the number of intersections of the graphs of \[y=4 g(f(\sin (2 \pi x))) \quad\text{ and }\quad x=4 g(f(\cos (3 \pi y))).\]

Graph

https://www.desmos.com/calculator/wml09giaun

Solution 1

If we graph $4g(f(x))$, we see it forms a sawtooth graph that oscillates between $0$ and $1$ (for values of $x$ between $-1$ and $1$, which is true because the arguments are between $-1$ and $1$). Thus by precariously drawing the graph of the two functions in the square bounded by $(0,0)$, $(0,1)$, $(1,1)$, and $(1,0)$, and hand-counting each of the intersections, we get $\boxed{385}$

Remark

While this solution might seem unreliable (it probably is), the only parts where counting the intersection might be tricky is near $(1,1)$. Make sure to count them as two points and not one, or you'll get $384$.

Note 1

The answer should be 385 since there are 16 intersections in each of 24 smaller boxes of dimensions 1/6 x 1/4 and then another one at the corner (1,1).

Solution 2

We will denote $h(x)=4g(f(x))$ for simplicity. Denote $p(x)$ as the first equation and $q(y)$ as the graph of the second. We notice that both $f(x)$ and $g(x)$ oscillate between 0 and 1. The intersections are thus all in the square $(0,0)$, $(0,1)$, $(1,1)$, and $(1,0)$. Every $p(x)$ wave going up and down crosses every $q(y)$ wave. Now, we need to find the number of times each wave touches 0 and 1.


We notice that $h(x)=0$ occurs at $x=-\frac{3}{4}, -\frac{1}{4}, \frac{1}{4}, \frac{3}{4}$, and $h(x)=1$ occurs at $x=-1, -\frac{1}{2}, 0,\frac{1}{2},1$. A sinusoid passes through each point twice during each period, but it only passes through the extrema once. $p(x)$ has 1 period between 0 and 1, giving 8 solutions for $p(x)=0$ and 9 solutions for $p(x)=1$, or 16 up and down waves. $q(y)$ has 1.5 periods, giving 12 solutions for $q(y)=0$ and 13 solutions for $q(y)=1$, or 24 up and down waves. This amounts to $16\cdot24=384$ intersections.


However, we have to be very careful when counting around $(1, 1)$. At this point, $q(y)$ has an infinite downwards slope and $p(x)$ is slanted, giving us an extra intersection; thus, we need to add 1 to our answer to get $\boxed{385}$.

~Xyco

Solution 3 (Rigorous analysis of why there are two solutions near (1,1))

We can easily see that only $x, y \in \left[0,1 \right]$ may satisfy both functions. We call function $y = 4g \left( f \left( \sin \left( 2 \pi x \right) \right) \right)$ as Function 1 and function $x = 4g \left( f \left( \cos \left( 3 \pi y \right) \right) \right)$ as Function 2.

For Function 1, in each interval $\left[ \frac{i}{4} , \frac{i+1}{4} \right]$ with $i \in \left\{ 0, 1, \cdots, 3 \right\}$, Function 1's value oscillates between 0 and 1. It attains 1 at $x = \frac{i}{4}$, $\frac{i+1}{4}$ and another point between these two. Between two consecutive points whose functional values are 1, the function first decreases from 1 to 0 and then increases from 0 to 1. So the graph of this function in this interval consists of 4 monotonic pieces.

For Function 2, in each interval $\left[ \frac{i}{6} , \frac{i+1}{6} \right]$ with $i \in \left\{ 0, 1, \cdots, 5 \right\}$, Function 2's value oscillates between 0 and 1. It attains 1 at $y = \frac{i}{6}$, $\frac{i+1}{6}$ and another point between these two. Between two consecutive points whose functional values are 1, the function first decreases from 1 to 0 and then increases from 0 to 1. So the graph of this function in this interval consists of 4 monotonic curves.

Consider any region $\left[ \frac{i}{4} , \frac{i+1}{4} \right] \times \left[ \frac{j}{6} , \frac{j+1}{6} \right]$ with $i \in \left\{ 0, 1, \cdots, 3 \right\}$ and $j \in \left\{0, 1, \cdots , 5 \right\}$ but $\left( i, j \right) \neq \left( 3, 5 \right)$. Both functions have four monotonic pieces. Because Function 1's each monotonic piece can take any value in $\left[ \frac{j}{6} , \frac{j+1}{6} \right]$ and Function 2' each monotonic piece can take any value in $\left[ \frac{i}{4} , \frac{i+1}{4} \right]$, Function 1's each monotonic piece intersects with Function 2's each monotonic piece. Therefore, in the interval $\left[ \frac{i}{4} , \frac{i+1}{4} \right] \times \left[ \frac{j}{6} , \frac{j+1}{6} \right]$, the number of intersecting points is $4 \cdot 4 = 16$.

Next, we prove that if an intersecting point is on a line $x = \frac{i}{4}$ for $i \in \left\{ 0, 1, \cdots, 4 \right\}$, then this point must be $\left( 1, 1 \right)$.

For $x = \frac{i}{4}$, Function 1 attains value 1. For Function 2, if $y = 1$, then $x = 1$. Therefore, the intersecting point is $\left( 1, 1 \right)$.

Similarly, we can prove that if an intersecting point is on a line $y = \frac{i}{6}$ for $i \in \left\{ 0, 1, \cdots, 6 \right\}$, then this point must be $\left( 1, 1 \right)$.

Therefore, in each region $\left[ \frac{i}{4} , \frac{i+1}{4} \right] \times \left[ \frac{j}{6} , \frac{j+1}{6} \right]$ with $i \in \left\{ 0, 1, \cdots, 3 \right\}$ and $j \in \left\{0, 1, \cdots , 5 \right\}$ but $\left( i, j \right) \neq \left( 3, 5 \right)$, all 16 intersecting points are interior. That is, no two regions share any common intersecting point.

Next, we study region $\left[ \frac{3}{4} , 1 \right] \times \left[ \frac{5}{6} , 1 \right]$. Consider any pair of monotonic pieces, where one is from Function 1 and one is from Function 2, except the pair of two monotonic pieces from two functions that attain $\left( 1 , 1 \right)$. Two pieces in each pair intersects at an interior point on the region. So the number of intersecting points is $4 \cdot 4 - 1 = 15$.

Finally, we compute the number intersection points of two functions' monotonic pieces that both attain $\left( 1, 1 \right)$.

One trivial intersection point is $\left( 1, 1 \right)$. Now, we study whether they intersect at another point.

Define $x = 1 - x'$ and $y = 1 - y'$. Thus, for positive and sufficiently small $x'$ and $y'$, Function 1 is reduced to \[ y' = 4 \sin 2 \pi x' \hspace{1cm} (1) \] and Function 2 is reduced to \[ x' = 4 \left( 1 - \cos 3 \pi y' \right) . \hspace{1cm} (2) \]

Now, we study whether there is a non-zero solution. Because we consider sufficiently small $x'$ and $y'$, to get an intuition and quick estimate, we do approximations of the above equations.

Equation (1) is approximated as \[ y' = 4 \cdot 2 \pi x' \] and Equation (2) is approximated as \[ x' = 2 \left( 3 \pi y' \right)^2 \]

To solve these equations, we get $x' = \frac{1}{8^2 \cdot 18 \pi^4}$ and $y' = \frac{1}{8 \cdot 18 \pi^3}$. Therefore, two functions' two monotonic pieces that attain $\left( 1, 1 \right)$ have two intersecting points.

Putting all analysis above, the total number of intersecting points is $16 \cdot 4 \cdot 6 + 1 = \boxed{\textbf{(385) }}$.

~Steven Chen (Professor Chen Education Palace, www.professorchenedu.com)

Video Solution

https://youtu.be/PA3aLjPinhI?t=7722

~Math Gold Medalist


Video Solution (Rigorous analysis of why there are two solutions near (1,1))

https://youtu.be/XjXwWfFzSrM

~Steven Chen (Professor Chen Education Palace, www.professorchenedu.com)

Drawing of two solutions near $\left( 1, 1 \right)$

https://artofproblemsolving.com/wiki/index.php/File:2024_AIME_I_Problem_12,_two_solutions_near_(1,1).png

~Steven Chen (Professor Chen Education Palace, www.professorchenedu.com)


See also

2024 AIME I (Problems • Answer Key • Resources)
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Problem 13
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