Difference between revisions of "2024 AIME I Problems/Problem 12"
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== Note 1== | == 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). | 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). | ||
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+ | ==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 the graph of <math>f(x)</math> oscillates between <math>y=0</math> and <math>y=1</math>, and the graph of <math>g(x)</math> oscillates between <math>x=0</math> and <math>x=1</math>. 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>. | ||
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+ | 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. | ||
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+ | 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>. | ||
==See also== | ==See also== |
Revision as of 21:24, 3 February 2024
Contents
[hide]Problem
Define and . Find the number of intersections of the graphs of
Graph
https://www.desmos.com/calculator/wml09giaun
Solution 1
If we graph , we see it forms a sawtooth graph that oscillates between and (for values of between and , which is true because the arguments are between and ). Thus by precariously drawing the graph of the two functions in the square bounded by , , , and , and hand-counting each of the intersections, we get
Note
While this solution might seem unreliable (it probably is), the only parts where counting the intersection might be tricky is near . Make sure to count them as two points and not one, or you'll get .
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 for simplicity. Denote as the first equation and as the graph of the second. We notice that the graph of oscillates between and , and the graph of oscillates between and . The intersections are thus all in the square , , , and .
Every wave going up and down crosses every wave. Now, we need to find the number of times each wave touches 0 and 1. We notice that occurs at , and occurs at . A sinusoid passes through each point twice during each period, but it only passes through the extrema once. has 1 period between 0 and 1, giving 8 solutions for and 9 solutions for , or 16 up and down waves. has 1.5 periods, giving 12 solutions for and 13 solutions for , or 24 up and down waves. This amounts to intersections.
However, we have to be very careful when counting around . At this point, has an infinite downwards slope and is slanted, giving us an extra intersection; thus, we need to add 1 to our answer to get .
See also
2024 AIME I (Problems • Answer Key • Resources) | ||
Preceded by Problem 11 |
Followed by Problem 13 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.