Difference between revisions of "2014 AMC 10B Problems/Problem 24"
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==Solution 2== | ==Solution 2== | ||
− | Like Solution 1, we note that the numbers from <math>1, 2, 3, 4, and 5</math> are always going to be able to be made. Also, by selecting all numbers but one of <math>{1, 2, 3, 4, 5}</math>, we can obtain the numbers from <math>10, 11, 12, 13, 14,</math> and <math>15</math> will always be made from all the numbers. So we must check only <math>6</math> through <math>9</math>. But if one circle can make a 6, we can select all of the other numbers and get a 9. Similarly, we can do the same for 7 and 8. So we must check only for 6 and 7. | + | Like Solution 1, we note that the numbers from <math>1, 2, 3, 4</math>, and <math>5</math> are always going to be able to be made. Also, by selecting all numbers but one of <math>{1, 2, 3, 4, 5}</math>, we can obtain the numbers from <math>10, 11, 12, 13, 14,</math> and <math>15</math> will always be made from all the numbers. So we must check only <math>6</math> through <math>9</math>. But if one circle can make a 6, we can select all of the other numbers and get a 9. Similarly, we can do the same for 7 and 8. So we must check only for 6 and 7. |
We can make <math>6</math> by having <math>4, 2</math>, or <math>3, 2, 1</math>, or <math>5, 1</math>. We can start with the group of three. To seperate <math>3, 2, 1</math> from eachother, they must be grouped two together and one seperate, like this. | We can make <math>6</math> by having <math>4, 2</math>, or <math>3, 2, 1</math>, or <math>5, 1</math>. We can start with the group of three. To seperate <math>3, 2, 1</math> from eachother, they must be grouped two together and one seperate, like this. |
Revision as of 17:13, 15 August 2016
- The following problem is from both the 2014 AMC 12B #18 and 2014 AMC 10B #24, so both problems redirect to this page.
Contents
Problem
The numbers are to be arranged in a circle. An arrangement is
if it is not true that for every
from
to
one can find a subset of the numbers that appear consecutively on the circle that sum to
. Arrangements that differ only by a rotation or a reflection are considered the same. How many different bad arrangements are there?
Solution 1
We see that there are total ways to arrange the numbers. However, we can always rotate these numbers so that, for example, the number
is always at the top of the circle. Thus, there are only
ways under rotation, which is not difficult to list out. We systematically list out all
cases.
Now, we must examine if they satisfy the conditions. We can see that by choosing one number at a time, we can always obtain subsets with sums and
. By choosing the full circle, we can obtain
. By choosing everything except for
and
, we can obtain subsets with sums of
and
.
This means that we now only need to check for and
. However, once we have found a set summing to
, we can choose everything else and obtain a set summing to
, and similarly for
and
. Thus, we only need to check each case for whether or not we can obtain
or
.
We find that there are only arrangements that satisfy these conditions. However, each of these is a reflection of another. We divide by
for these reflections to obtain a final answer of
.
Solution 2
Like Solution 1, we note that the numbers from , and
are always going to be able to be made. Also, by selecting all numbers but one of
, we can obtain the numbers from
and
will always be made from all the numbers. So we must check only
through
. But if one circle can make a 6, we can select all of the other numbers and get a 9. Similarly, we can do the same for 7 and 8. So we must check only for 6 and 7.
We can make by having
, or
, or
. We can start with the group of three. To seperate
from eachother, they must be grouped two together and one seperate, like this.
Now, we note that is next to both blank spots, so we can't have a number from one of the pairs. So since we can't have
, because it is part of the
pair, and we can't have
there, because it's part of the
pair, we must have
inserted into the
spot. We can insert
and
in
and
interchangably, since reflections are considered the same.
We have and
left to insert. We can't place the
next to the
or the
next to the
, so we must place
next to the
and
next to the
.
This is the only solution to make "bad."
Next we move on to , which can be made by
, or
, or
. We do this the same way as before. We start with the three group. Since we can't have 4 or 2 in the top slot, we must have one there, and 4 and 2 are next to eachother on the bottom. When we have
and
left to insert, we place them such that we don't have the two pairs adjacent.
This is the only solution to make "bad."
We've covered all needed cases, and the two examples we found are distinct, therefore the answer is .
See Also
2014 AMC 10B (Problems • Answer Key • Resources) | ||
Preceded by Problem 23 |
Followed by Problem 25 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | ||
All AMC 10 Problems and Solutions |
2014 AMC 12B (Problems • Answer Key • Resources) | |
Preceded by Problem 17 |
Followed by Problem 19 |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | |
All AMC 12 Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.