Difference between revisions of "2017 AIME I Problems/Problem 11"
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==Solution 1== | ==Solution 1== | ||
− | + | Assume that <math>5 \in \{a_1, a_2, a_3\}</math>, <math>m \neq 5</math>, and WLOG, <math>\max{(a_1, a_2, a_3)} = 5</math>. Then we know that the other two medians in <math>\{a_1, a_2, a_3\}</math> and the smallest number of rows 1, 2, and 3 are all less than 5. But there are only 4 numbers less than 5 in <math>1, 2, 3, \dots, 9</math>, a Contradiction. Thus, if <math>5 \in \{a_1, a_2, a_3\}</math>, then <math>m = 5</math>. | |
WLOG, assume <math>5</math> is in the upper left corner. One of the two other values in the top row needs to be below <math>5</math>, and the other needs to be above <math>5</math>. This can be done in <math>4\cdot4\cdot2=32</math> ways. | WLOG, assume <math>5</math> is in the upper left corner. One of the two other values in the top row needs to be below <math>5</math>, and the other needs to be above <math>5</math>. This can be done in <math>4\cdot4\cdot2=32</math> ways. | ||
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Finally, accounting for when <math>5</math> is in every other space, our answer is <math>32\cdot720\cdot9</math>, which is <math>207360</math>. But we only need the last <math>3</math> digits, so <math>\boxed{360}</math> is our answer. | Finally, accounting for when <math>5</math> is in every other space, our answer is <math>32\cdot720\cdot9</math>, which is <math>207360</math>. But we only need the last <math>3</math> digits, so <math>\boxed{360}</math> is our answer. | ||
− | ~Solution by SuperSaiyanOver9000, mathics42 | + | ~Solution by SuperSaiyanOver9000, mathics42, edited by zhaohm |
+ | ===Alternative Proof=== | ||
+ | '''Claim:''' As long as one median is equal to <math>5,</math> the overall median is equal to <math>5.</math> | ||
+ | |||
+ | '''Proof:''' Notice that to obtain a median of <math>5,</math> we need one number greater and one number lesser than <math>5</math> along with <math>5.</math> | ||
+ | |||
+ | Thus, we remain with 3 numbers greater than <math>5</math> and 3 less than <math>5,</math> meaning that the overall median, no matter the distribution, of the remaining two rows must be <math>5.</math> | ||
+ | |||
+ | mathboy282 | ||
==Solution 2== | ==Solution 2== | ||
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WLOG let <math>m=4</math> | WLOG let <math>m=4</math> | ||
− | There is a <math>\frac{15}{28}</math> chance that exactly one of 1, 2, 3 is in the same row. | + | 1. There is a <math>\frac{15}{28}</math> chance that exactly one of 1, 2, 3 is in the same row with 4. |
+ | |||
+ | There are 3 ways to select which of the smaller numbers will get in the row, and then 5 | ||
+ | |||
+ | ways to select the number larger than 4. | ||
+ | |||
+ | <math>\frac{\dbinom{3}{1}\cdot\dbinom{5}{1}}{\dbinom{8}{2}} = \frac{15}{28}</math> | ||
+ | |||
+ | 2. There is a <math>\frac{2}{5}</math> chance that the other two smaller numbers end up in the same row. | ||
+ | |||
+ | There are 2 ways to select the row that the two smaller number are in, and then <math>\dbinom{3}{2}</math> ways | ||
+ | |||
+ | to place the smaller numbers in the row. | ||
− | + | <math>\frac{\dbinom{2}{1}\cdot\dbinom{3}{2}}{\dbinom{6}{2}} = \frac{2}{5}</math> | |
<math>9!(1-2*\frac{15}{28}*\frac{2}{5})=362880*\frac{4}{7}=207\boxed{360}</math>. | <math>9!(1-2*\frac{15}{28}*\frac{2}{5})=362880*\frac{4}{7}=207\boxed{360}</math>. | ||
− | |||
==Solution 3== | ==Solution 3== | ||
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Hence, our answer is <math>3!(4\cdot4\cdot3!\cdot{10}\cdot{3!}\cdot{3!})=3!(16\cdot60\cdot36)=3!(34560)\implies{207\boxed{360}}</math> | Hence, our answer is <math>3!(4\cdot4\cdot3!\cdot{10}\cdot{3!}\cdot{3!})=3!(16\cdot60\cdot36)=3!(34560)\implies{207\boxed{360}}</math> | ||
+ | |||
+ | ==Solution 4== | ||
+ | We see that if one of the medians is 5, then there are two remaining numbers greater than 5 and two less than 5, so it follows that <math>m=5</math>. There are 3 ways to choose which row to have 5 in, <math>4\cdot 4=16</math> ways to choose the other two numbers in that row, <math>3!=6</math> ways to arrange the numbers in that row, and <math>6!=720</math> ways for the remaining numbers, for our answer is <math>3\cdot 16\cdot 6\cdot 720=207\boxed{360}</math>. | ||
+ | -Stormersyle | ||
+ | |||
+ | ==Solution 5== | ||
+ | We take the grid, and we do a bunch of stuff with it. First, we sort each row, smallest on the left, largest on the right, then we arrange these 3 rows such that the middle #s are increasing, from top to bottom. Thus, we get that the cell in the very center of the grid must be 5. (We need to multiply by 1296 at the end) | ||
+ | |||
+ | Let S mean a number that is < 5, L mean a number > 5. | ||
+ | |||
+ | [[Image:Bobby.jpg|200px]] | ||
+ | |||
+ | In the 2 blank corners, one can be S, and the other one has to be L. | ||
+ | |||
+ | If the top right corner is S, there are 16 ways, otherwise, there are 144 ways. (This is left as an exercise to the reader). | ||
+ | |||
+ | Thus, there are 160 * 1296 total configurations, which gets us an answer of <math>207\boxed{360}</math> | ||
+ | |||
+ | -AlexLikeMath | ||
+ | |||
+ | ==Solution 6== | ||
+ | We note that if <math>5</math> is a median of one of the rows, then <math>m=5</math>. First, focus on the row with <math>5</math> in it. There are <math>4^2</math> ways to choose the other numbers in that row and then <math>3!</math> ways to order it. Now, clearly, there are <math>6!</math> ways to put the other <math>6</math> numbers into the remaining slots so <math>Q=6!\cdot3!\cdot4^2\cdot3=207360</math>. Hence, our answer is <math>\boxed{360}</math>. | ||
+ | |||
+ | ~pleaseletmewin | ||
+ | |||
+ | ==Solution 7== | ||
+ | Let <math>A = \{1,2,3,4\}</math> and <math>B = \{6,7,8,9\}</math>. WLOG, if we let 5 be on the first row, then <math>m=5</math> if and only if <math>a_1=5</math>. Thus, the two other numbers that are on row 1 <math>p, q</math> satisfies one of the two following conditions: | ||
+ | |||
+ | <cmath>(1) \quad p\in A \quad \textrm{and} \quad q\in B</cmath> | ||
+ | or | ||
+ | <cmath>(2) \quad p\in B \quad \textrm{and} \quad q\in A</cmath> | ||
+ | |||
+ | The probability of this happening can be calculated through <math> \frac{4^2}{{8 \choose 2}} = \frac{4}{7}</math>. Therefore, <cmath>Q = 9! \cdot \frac{4}{7} = 207360 \equiv \boxed{360} \pmod {1000} .</cmath> | ||
+ | |||
+ | ~Kscv | ||
==See Also== | ==See Also== | ||
{{AIME box|year=2017|n=I|num-b=10|num-a=12}} | {{AIME box|year=2017|n=I|num-b=10|num-a=12}} | ||
{{MAA Notice}} | {{MAA Notice}} |
Latest revision as of 12:40, 24 January 2024
Contents
Problem 11
Consider arrangements of the numbers
in a
array. For each such arrangement, let
,
, and
be the medians of the numbers in rows
,
, and
respectively, and let
be the median of
. Let
be the number of arrangements for which
. Find the remainder when
is divided by
.
Solution 1
Assume that ,
, and WLOG,
. Then we know that the other two medians in
and the smallest number of rows 1, 2, and 3 are all less than 5. But there are only 4 numbers less than 5 in
, a Contradiction. Thus, if
, then
.
WLOG, assume is in the upper left corner. One of the two other values in the top row needs to be below
, and the other needs to be above
. This can be done in
ways.
The other
can be arranged in
ways.
Finally, accounting for when
is in every other space, our answer is
, which is
. But we only need the last
digits, so
is our answer.
~Solution by SuperSaiyanOver9000, mathics42, edited by zhaohm
Alternative Proof
Claim: As long as one median is equal to the overall median is equal to
Proof: Notice that to obtain a median of we need one number greater and one number lesser than
along with
Thus, we remain with 3 numbers greater than and 3 less than
meaning that the overall median, no matter the distribution, of the remaining two rows must be
mathboy282
Solution 2
(Complementary Counting with probability)
Notice that m can only equal 4, 5, or 6, and 4 and 6 are symmetric.
WLOG let
1. There is a chance that exactly one of 1, 2, 3 is in the same row with 4.
There are 3 ways to select which of the smaller numbers will get in the row, and then 5
ways to select the number larger than 4.
2. There is a chance that the other two smaller numbers end up in the same row.
There are 2 ways to select the row that the two smaller number are in, and then ways
to place the smaller numbers in the row.
.
Solution 3
We will make sure to multiply by in the end to account for all the possible permutation of the rows.
WLOG, let be present in the Row #
.
Notice that MUST be placed with a number lower than it and a number higher than it.
This happens in ways. You can permutate Row #
in
ways.
Now, take a look at Row and Row
.
Because there are numbers to choose from now, you can assign #'s to Row's #2&3 in
ways. There are
ways to permute the numbers in the individual Rows.
Hence, our answer is
Solution 4
We see that if one of the medians is 5, then there are two remaining numbers greater than 5 and two less than 5, so it follows that . There are 3 ways to choose which row to have 5 in,
ways to choose the other two numbers in that row,
ways to arrange the numbers in that row, and
ways for the remaining numbers, for our answer is
.
-Stormersyle
Solution 5
We take the grid, and we do a bunch of stuff with it. First, we sort each row, smallest on the left, largest on the right, then we arrange these 3 rows such that the middle #s are increasing, from top to bottom. Thus, we get that the cell in the very center of the grid must be 5. (We need to multiply by 1296 at the end)
Let S mean a number that is < 5, L mean a number > 5.
In the 2 blank corners, one can be S, and the other one has to be L.
If the top right corner is S, there are 16 ways, otherwise, there are 144 ways. (This is left as an exercise to the reader).
Thus, there are 160 * 1296 total configurations, which gets us an answer of
-AlexLikeMath
Solution 6
We note that if is a median of one of the rows, then
. First, focus on the row with
in it. There are
ways to choose the other numbers in that row and then
ways to order it. Now, clearly, there are
ways to put the other
numbers into the remaining slots so
. Hence, our answer is
.
~pleaseletmewin
Solution 7
Let and
. WLOG, if we let 5 be on the first row, then
if and only if
. Thus, the two other numbers that are on row 1
satisfies one of the two following conditions:
or
The probability of this happening can be calculated through . Therefore,
~Kscv
See Also
2017 AIME I (Problems • Answer Key • Resources) | ||
Preceded by Problem 10 |
Followed by Problem 12 | |
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.