Difference between revisions of "2019 AIME II Problems/Problem 10"
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at which point we realize a pattern emerging. Specifically, the intervals repeat every <math>3</math> after <math>n=1</math>. We can use these repeating intervals to determine the desired value of <math>\theta</math> since the upper and lower bounds will converge to such a value (since it is unique, as indicated in the problem). Let's keep track of the lower bound. | at which point we realize a pattern emerging. Specifically, the intervals repeat every <math>3</math> after <math>n=1</math>. We can use these repeating intervals to determine the desired value of <math>\theta</math> since the upper and lower bounds will converge to such a value (since it is unique, as indicated in the problem). Let's keep track of the lower bound. | ||
− | Initially, the lower bound is <math>0</math> (at <math>n=0</math>), then increases to <math>\frac{90}{2}=45</math> at <math>n=1</math>. This then becomes <math>45+\frac{45}{2}</math> at <math>n=2</math>, <math>45+\frac{45}{2}</math> at <math>n=3</math>, <math>45+\frac{45}{2}+\frac{45}{2^3}</math> at <math>n=4</math>,<math>45+\frac{45}{2}+\frac{45}{2^3}+\frac{45}{2^4}</math> at <math>n=5</math>. Due to the observed pattern of the intervals, the lower bound follows a partial geometric series. Hence, as <math>n</math> approaches infinity, the lower bound converges to | + | Initially, the lower bound is <math>0</math> (at <math>n=0</math>), then increases to <math>\frac{90}{2}=45</math> at <math>n=1</math>. This then becomes <math>45+\frac{45}{2}</math> at <math>n=2</math>, <math>45+\frac{45}{2}</math> at <math>n=3</math>, <math>45+\frac{45}{2}+\frac{45}{2^3}</math> at <math>n=4</math>, <math>45+\frac{45}{2}+\frac{45}{2^3}+\frac{45}{2^4}</math> at <math>n=5</math>, etc. Due to the observed pattern of the intervals, the lower bound follows a partial geometric series. Hence, as <math>n</math> approaches infinity, the lower bound converges to |
<cmath>\sum_{k=0}^{\infty}\left(45+\frac{45}{2}\right)\cdot \left(\frac{1}{8}\right)^k=\frac{45+\frac{45}{2}}{1-\frac{1}{8}}=\frac{\frac{135}{2}}{\frac{7}{8}}=\frac{540}{7}\implies p+q=540+7=\boxed{547}</cmath>-ktong | <cmath>\sum_{k=0}^{\infty}\left(45+\frac{45}{2}\right)\cdot \left(\frac{1}{8}\right)^k=\frac{45+\frac{45}{2}}{1-\frac{1}{8}}=\frac{\frac{135}{2}}{\frac{7}{8}}=\frac{540}{7}\implies p+q=540+7=\boxed{547}</cmath>-ktong | ||
+ | |||
+ | <b>Extra note:</b> If you are still unsure, you can check the upper bound to see if the top converges. This value comes out to be: | ||
+ | |||
+ | <math>90-\frac{90}{2^3}-\frac{90}{2^6}-\cdots=90\left(1-\sum_{k=1}^{\infty}\frac{1}{2^{3k}}\right)=90\left(1-\frac{1}{7}\right)=\frac{540}{7}</math>. | ||
+ | |||
+ | Thus, the limit does in fact converge to the same value on both sides. ~eevee9406 | ||
==Solution 3== | ==Solution 3== | ||
− | Since <math>tan\left(\theta\right) > 0</math>, <math>0 < \theta < 90</math>. Since <math>tan\left(2\theta\right) < 0</math>, <math>\theta</math> has to be in the second half of the interval (0, 90) ie (45, 90). Since <math>tan\left(4\theta\right) < 0</math>, <math>\theta</math> has to be in the second half of that interval ie (67.5, 90). And since <math>tan\left(8\theta\right) > 0</math>, <math>\theta</math> has to be in the first half of (67.5, 90). Inductively, the pattern repeats: <math>\theta</math> is in | + | Since <math>\tan\left(\theta\right) > 0</math>, <math>0 < \theta < 90</math>. Since <math>\tan\left(2\theta\right) < 0</math>, <math>\theta</math> has to be in the second half of the interval (0, 90) ie (45, 90). Since <math>\tan\left(4\theta\right) < 0</math>, <math>\theta</math> has to be in the second half of that interval ie (67.5, 90). And since <math>\tan\left(8\theta\right) > 0</math>, <math>\theta</math> has to be in the first half of (67.5, 90). Inductively, the pattern repeats: <math>\theta</math> is in the second half of the second half of the first half of the second half of the second half... of the interval (0, 90). Consider the binary representation of numbers in the interval (0, 1). Numbers in the first half of the interval start with 0.0... and numbers in the second half start with 0.1... . Similarly, numbers in the second half of the second half start with 0.11... etc. So if we want a number in the first half of the second half of the second half... of the interval, we want its binary representation to be <math>0.11011011011..._2 = \frac{6}{7}_{10}</math>. So we want the number which is 6/7 of the way through the interval (0, 90) so <math>\theta = \frac{6}{7}\cdot 90 = \frac{540}{7}</math> and <math>p+q = 540 + 7 = \boxed{547}</math> |
+ | |||
+ | ~minor edit by [[Mathkiddie]] | ||
==Solution 4== | ==Solution 4== | ||
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==Solution 5== | ==Solution 5== | ||
− | Since 7 is the only number n such that f(x) = 2^ | + | Since <math>7</math> is the only number n such that f(x) = <math>2^{\lfloor x\rfloor}</math> <math>\text{(mod 7)}</math> has a period of 3, we find that <math>\theta</math> is a multiple of <math>{\frac{180}{7}}^\circ</math>. Note that the tangents of <math>{\frac{180}{7}}^\circ</math>, <math>{\frac{360}{7}}^\circ</math>, <math>{\frac{540}{7}}^\circ</math> are positive while those of <math>{\frac{720}{7}}^\circ</math>, <math>{\frac{900}{7}}^\circ</math>, and <math>{\frac{1080}{7}}^\circ</math> are negative. With a bit of trial and error, we find <math>\theta</math> is <math>{\frac{540}{7}}^\circ</math> and <math>540+7</math> is <math>\boxed{547}</math>. |
~ [[User:Afly|Afly]] ([[User talk:Afly|talk]]) | ~ [[User:Afly|Afly]] ([[User talk:Afly|talk]]) |
Latest revision as of 18:21, 8 August 2024
Problem
There is a unique angle between
and
such that for nonnegative integers
, the value of
is positive when
is a multiple of
, and negative otherwise. The degree measure of
is
, where
and
are relatively prime integers. Find
.
Solution 1
Note that if is positive, then
is in the first or third quadrant, so
.
Furthermore, the only way can be positive for all
that are multiples of
is when:
(This is because if it isn't the same value, the terminal angle will gradually shift from the first quadrant into different quadrants, making the condition for positive tan untrue. This must also be true in order for
to be unique.)
This is the case if , so
. Therefore, recalling that
the possible
are:
does not work since
is positive.
does not work because
is positive.
Thus, , and a quick check verifies that it does work. Our desired answer is
.
Solution 2
As in the previous solution, we note that is positive when
is in the first or third quadrant. In order for
to be positive for all
divisible by
, we must have
,
,
, etc to lie in the first or third quadrants. We already know that
. We can keep track of the range of
for each
by considering the portion in the desired quadrants, which gives
at which point we realize a pattern emerging. Specifically, the intervals repeat every
after
. We can use these repeating intervals to determine the desired value of
since the upper and lower bounds will converge to such a value (since it is unique, as indicated in the problem). Let's keep track of the lower bound.
Initially, the lower bound is (at
), then increases to
at
. This then becomes
at
,
at
,
at
,
at
, etc. Due to the observed pattern of the intervals, the lower bound follows a partial geometric series. Hence, as
approaches infinity, the lower bound converges to
-ktong
Extra note: If you are still unsure, you can check the upper bound to see if the top converges. This value comes out to be:
.
Thus, the limit does in fact converge to the same value on both sides. ~eevee9406
Solution 3
Since ,
. Since
,
has to be in the second half of the interval (0, 90) ie (45, 90). Since
,
has to be in the second half of that interval ie (67.5, 90). And since
,
has to be in the first half of (67.5, 90). Inductively, the pattern repeats:
is in the second half of the second half of the first half of the second half of the second half... of the interval (0, 90). Consider the binary representation of numbers in the interval (0, 1). Numbers in the first half of the interval start with 0.0... and numbers in the second half start with 0.1... . Similarly, numbers in the second half of the second half start with 0.11... etc. So if we want a number in the first half of the second half of the second half... of the interval, we want its binary representation to be
. So we want the number which is 6/7 of the way through the interval (0, 90) so
and
~minor edit by Mathkiddie
Solution 4
With some simple arithmetic and guess and check, we can set the lower bound and upper bounds for the "first round of powers of two", which are
and
. Going on to the "second round of
powers of two, we set the new lower and upper bounds as
and
using some guess and check and bashing. Now, it is obvious that the bounds for the "zeroth round of
powers of two" are
and
, and notice that
and
and
.
This is obviously a geometric series, so setting
as
, we obtain
=
=
=
which simplifies to
. We can now finally subtract
from
and then we get
as the unique angle, so
is our answer.
-fidgetboss_4000
Solution 5
Since is the only number n such that f(x) =
has a period of 3, we find that
is a multiple of
. Note that the tangents of
,
,
are positive while those of
,
, and
are negative. With a bit of trial and error, we find
is
and
is
.
See Also
2019 AIME II (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 | ||
All AIME Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.