So I've got this problem here: there's a circle with its center at the origin, and it passes through two points, P(d, -d) and Q(12 - d, 2d - 6). I need to find the sum of the two possible values of d. The options are from -16 to 16 in increments of 4.

First off, since both points lie on the circle centered at the origin, their distances from the origin should be equal to the radius of the circle. So, for point P(d, -d), the distance from the origin is sqrt(d^2 + (-d)^2) = sqrt(2d^2) = d*sqrt(2). Similarly, for point Q(12 - d, 2d - 6), the distance is sqrt((12 - d)^2 + (2d - 6)^2).

Since these distances are equal (both being radii of the same circle), I can set them equal to each other:

d * sqrt(2) = sqrt((12 - d)^2 + (2d - 6)^2)

Hmm, dealing with square roots might get messy. Maybe I can eliminate the square roots by squaring both sides:

(d * sqrt(2))^2 = [sqrt((12 - d)^2 + (2d - 6)^2)]^2

Which simplifies to:

2d^2 = (12 - d)^2 + (2d - 6)^2

Now, let's expand the right side:

(12 - d)^2 = 144 - 24d + d^2

(2d - 6)^2 = 4d^2 - 24d + 36

Adding them together:

144 - 24d + d^2 + 4d^2 - 24d + 36 = 5d^2 - 48d + 180

So now, the equation is:

2d^2 = 5d^2 - 48d + 180

Let's bring all terms to one side to set the equation to zero:

2d^2 - 5d^2 + 48d - 180 = 0

Simplifying:

-3d^2 + 48d - 180 = 0

I can multiply both sides by -1 to make the leading coefficient positive:

3d^2 - 48d + 180 = 0

This looks like a quadratic equation. I can solve for d using the quadratic formula:

d = [48 ± sqrt(48^2 - 4*3*180)] / (2*3)

Calculating the discriminant:

48^2 - 4*3*180 = 2304 - 2160 = 144

So,

d = [48 ± sqrt(144)] / 6 = [48 ± 12] / 6

Therefore, two solutions:

d = (48 + 12)/6 = 60/6 = 10

d = (48 - 12)/6 = 36/6 = 6

Wait, but the problem says "the sum of the two possible values of d". Here, I have d=10 and d=6, so their sum is 16.

Looking back at the options, option E is 16. But wait, maybe I made a mistake somewhere because the problem seems too straightforward.

Let me double-check my steps.

Starting from setting the distances equal:

d * sqrt(2) = sqrt((12 - d)^2 + (2d - 6)^2)

Then squaring both sides:

2d^2 = (12 - d)^2 + (2d - 6)^2

Expanding the right side:

144 - 24d + d^2 + 4d^2 - 24d + 36 = 5d^2 - 48d + 180

Setting equal to left side:

2d^2 = 5d^2 - 48d + 180

Bringing all terms to one side:

-3d^2 + 48d - 180 = 0

Multiplying by -1:

3d^2 - 48d + 180 = 0

Dividing by 3:

d^2 - 16d + 60 = 0

Using quadratic formula:

d = [16 ± sqrt(256 - 240)] / 2 = [16 ± sqrt(16)] / 2 = [16 ± 4]/2

So, d = (16 + 4)/2 = 10 or d = (16 - 4)/2 = 6

Sum is 10 + 6 = 16

But wait, the problem says "the sum of the two possible values of d", and according to my calculations, it's 16, which corresponds to option E.

However, I recall that in some geometry problems involving circles and points on the circumference, sometimes there can be extraneous solutions or special cases where not all values satisfy the original conditions. Maybe I need to verify if both d=10 and d=6 actually give valid points on the circle.

Let me plug d=10 into the coordinates of P and Q.

P(10, -10): distance from origin is sqrt(10^2 + (-10)^2) = sqrt(200) = 10*sqrt(2)

Q(12 - 10, 2*10 - 6) = Q(2, 14): distance is sqrt(2^2 + 14^2) = sqrt(4 + 196) = sqrt(200) = 10*sqrt(2)

Good, they match.

Now, d=6:

P(6, -6): distance is sqrt(6^2 + (-6)^2) = sqrt(72) = 6*sqrt(2)

Q(12 - 6, 2*6 - 6) = Q(6, 6): distance is sqrt(6^2 + 6^2) = sqrt(72) = 6*sqrt(2)

Again, they match.

So both values of d seem to work. Therefore, the sum is indeed 16.

But just to be thorough, let's consider if there could be any restrictions on d based on the positions of the points.

For example, if d were such that either coordinate was negative and didn't correspond to a real number, but in this case, both coordinates are linear functions of d, and we've found specific values that satisfy the equations.

Alternatively, perhaps there's a way to interpret the problem differently, but given that it specifies the sum of the two possible values of d, and I've found those values and their sum, I think 16 is the correct answer.

Therefore, the answer is E, 16.

**Final Answer**

\[ \boxed{16} \]


Let's tackle this problem step by step. We have a list of positive integers with some specific properties:

1. The sum of the items is 30.

2. The unique mode is 9.

3. The median is a positive integer that doesn't appear in the list.

We need to find the sum of the squares of all the items in the list, and it should be a three-digit number.

First, let's understand what each property means for our list.

**Property 1:** Sum is 30.

So, if we add up all the numbers in the list, they should equal 30.

**Property 2:** Unique mode is 9.

This means that 9 appears more frequently than any other number in the list, and no other number has the same frequency as 9.

**Property 3:** Median is a positive integer not appearing in the list.

The median is the middle value when the list is ordered. Since there's only one mode, which is 9, it must appear more times than any other number.

Our goal is to find the sum of the squares of all the items in the list.

Let's start by considering how many times 9 appears. Let’s denote the frequency of 9 as \(f\). Since 9 is the unique mode, no other number can appear \(f\) times; otherwise, there would be multiple modes.

Also, since the median is a positive integer not appearing in the list, we need to consider the ordering of the list.

Given that the sum is 30, and 9 is involved, let's think about possible frequencies for 9.

Suppose 9 appears twice. Then, the total sum contributed by 9 alone is \(2 \times 9 = 18\).

Remaining sum needed: \(30 - 18 = 12\).

Now, we need to fill the remaining sum with other positive integers, none of which are 9, except possibly others if they also appear twice but less than twice.

But wait, since 9 is the unique mode, no other number can appear twice. So, all other numbers must appear once.

Wait, no. Actually, the mode being unique means that 9 appears more times than any other number, but it could appear more than twice.

Wait, no. If 9 appears twice, and another number appears once, then 9 is still the mode because it appears more often than any other number.

But the problem says "unique mode," which implies that 9 appears more frequently than any other number in the list.

So, if 9 appears twice, then all other numbers must appear at most once, because if any other number appeared twice, then both 9 and that number would appear twice, making them co-modes, which contradicts the uniqueness of the mode.

Therefore, if 9 appears twice, all other numbers must appear exactly once.

Similarly, if 9 appears three times, then all other numbers must appear at most once.

And so on.

But let's consider the sum constraint.

If 9 appears twice, sum from 9 is 18, leaving 12 to be distributed among other numbers, each appearing once.

Possible candidates for these numbers are 1 through 8, since 9 is already included.

But wait, 9 is the unique mode, and 9 appears twice, while others appear once.

Is there a possibility that 9 appears more than twice?

Well, if 9 appears three times, then all other numbers must appear at most once.

Sum from 9 would be \(3 \times 9 = 27\), leaving \(30 - 27 = 3\) to be distributed among other numbers, each appearing once.

Again, these numbers must be between 1 and 8, excluding 9.

But in this case, the median would depend on the ordering of the list.

However, we also have the condition that the median is a positive integer not appearing in the list.

Let's explore this further.

First, let's consider the case where 9 appears twice.

Then, the list has two 9's, and the rest of the numbers are distinct and different from 9.

The sum is 30, so the remaining sum is 12.

We need to choose four distinct positive integers greater than 9 (since 9 is the mode) that add up to 12.

Possible combinations:

- 4 + 4 + 4 = 12, but they must all be distinct.

- Wait, no, they must be distinct.

- 5 + 3 + 4 = 12, but 3 is allowed since it's not 9.

- But 3 is less than 9, which is fine.

So, possible list: [9, 9, 4, 3]

Check sum: 9+9+4+3=25, which is more than 30. Wait, that can't be right.

Wait, perhaps I miscounted.

Wait, if 9 appears twice, and the remaining numbers are distinct and greater than 9, their sum should be 12.

But 9+9=18, which is already over 30. That can't be.

Wait, perhaps I need to reconsider.

Wait, maybe 9 appears twice, and the remaining numbers are smaller.

Wait, but 9 is the unique mode, and it appears more times than any other number.

So, if 9 appears twice, then all other numbers must appear at most once.

But if 9 appears twice, and the remaining sum is 12, but the smallest possible sum of four distinct numbers greater than 9 is 10 + 11 + 12 + 13 = 46, which is way above 12.

This suggests that my initial assumption is incorrect.

Alternatively, perhaps 9 appears more than twice.

Wait, but if 9 appears three times, sum from 9 is 27, leaving 3 to be distributed among other numbers.

But 3 is too small to be added to other numbers without repeating numbers, which violates the distinctness.

Wait, but 3 is less than 9, so it's acceptable.

So, possible list: [9, 9, 9, 3]

Sum: 9+9+9+3=30

Now, check the median.

Since there are four numbers, the median is the average of the second and third numbers.

Second and third numbers are both 9.

Average is 9, which is equal to 9.

But the problem states that the median is a positive integer that does not appear in the list.

Here, 9 appears in the list, which violates this condition.

Therefore, this arrangement is invalid.

Alternatively, if 9 appears four times, sum from 9 is 36, which is already more than 30.

So, 9 cannot appear four times.

What if 9 appears three times? Sum from 9 is 27, leaving 3 to be distributed among other numbers.

Possible list: [9, 9, 9, x], where x ≠ 9 and x < 9.

But x must be positive integer, so x ≥ 1.

But if x = 1, then sum is 27 + 1 = 28, which is less than 30.

Need one more unit.

But x must be ≤ 8, since 9 is the mode and appears more times than any other number.

Wait, but if x = 2, sum is 27 + 2 = 29, still less than 30.

x = 3, sum = 27 + 3 = 30.

So, possible list: [9, 9, 9, 3]

Sum: 27 + 3 = 30

Median: Second and third numbers are both 9, average is 9, which is in the list.

But the condition requires that the median is a positive integer not appearing in the list.

Hence, this is invalid.

Alternatively, if 9 appears three times, and the fourth number is 3, sum is 27 + 3 = 30.

But median is 9, which is in the list.

Invalid.

What if 9 appears twice?

Then, sum from 9 is 18, leaving 12 to be distributed among other numbers.

Possible list: [9, 9, y, z], where y and z are distinct positive integers greater than 9, and y + z = 12.

But 9 is the unique mode, so y and z must be distinct from each other and from 9.

Possible pairs (y,z):

- 4 and 8: sum 12

List: [9,9,4,8]

Sum: 9+9+4+8=30

Now, check the median.

There are four numbers: 4,8,9,9

Ordered: 4,8,9,9

Median is the average of the second and third numbers: (8 + 9)/2 = 8.5

Which is not an integer.

But the problem states that the median is a positive integer.

So, this violates the condition.

Alternatively, try y=3, z=9, but 9 is already in the list.

Not allowed.

Or y=2, z=10: but 10 > 9, but sum would be 18 + 10 = 28, which is more than 30.

Not feasible.

Thus, [9,9,4,8] is invalid due to median not being an integer.

Alternatively, try [9,9,3,9]: but 9 repeats twice, which might violate the mode uniqueness.

Wait, 9 appears three times here, which is more frequent than any other number.

But the problem says the unique mode is 9, meaning no other number has the same frequency as 9.

In this list, 9 appears three times, and others appear once.

So, it satisfies the mode uniqueness.

Sum: 9+9+3+9=30

Median: Ordered list: 3,9,9,9

Median is 9, which is in the list.

But again, the median should not be in the list.

Invalid.

Hmm.

Perhaps 9 appears only twice.

Then, sum from 9 is 18, leaving 12 to be distributed among other numbers.

Possible list: [9,9,a,b], where a and b are distinct positive integers greater than 9, and a + b = 12.

Possible pairs (a,b):

- 4 and 8: sum 12

List: [9,9,4,8]

Sum: 9+9+4+8=30

As before, median is 8.5, not an integer.

Not valid.

Alternatively, [9,9,3,9]: but 9 repeats three times, which might be acceptable if 9 is the unique mode.

But earlier, this led to median being 9, which is in the list.

Invalid.

Alternatively, [9,9,2,10]: sum is 9+9+2+10=30

Ordered: 2,9,9,10

Median: average of 9 and 9 = 9, which is in the list.

Invalid.

Alternatively, [9,9,1,11]: sum is 9+9+1+11=30

Ordered: 1,9,9,11

Median: average of 9 and 9 = 9, which is in the list.

Invalid.

It seems challenging to satisfy all conditions when 9 appears twice.

Maybe 9 appears three times.

Earlier attempts failed because median was 9, which is in the list.

Alternative approach: perhaps 9 appears four times.

Sum from 9 is 36, which is more than 30.

Not feasible.

Alternatively, 9 appears five times.

Sum from 9 is 45, which is even worse.

Not feasible.

Wait, perhaps 9 appears three times, and the fourth number is negative.

But the problem specifies positive integers, so negative numbers are excluded.

Thus, 9 appears three times, and the fourth number is such that the sum is 30.

Sum from 9 is 27, leaving 3 to be distributed among other numbers.

Possible list: [9,9,9,x], where x ≠ 9 and x < 9.

But x must be positive integer, so x ≥ 1.

If x=1, sum is 27 + 1 = 28, which is less than 30.

Need one more unit.

But x must be ≤ 8.

If x=2, sum is 27 + 2 = 29, still less than 30.

x=3, sum=27+3=30.

List: [9,9,9,3]

Sum: 30

Median: Ordered list: 3,9,9,9

Median is 9, which is in the list.

Invalid.

Alternatively, [9,9,9,0], but 0 is not positive.

Invalid.

Alternatively, [9,9,8,4]: sum is 9+9+8+4=30

Ordered: 4,8,9,9

Median: average of 8 and 9 = 8.5, not an integer.

Invalid.

Alternatively, [9,9,7,5]: sum is 9+9+7+5=30

Ordered: 5,7,9,9

Median: average of 7 and 9 = 8, which is not an integer.

Invalid.

Alternatively, [9,9,6,6]: but 6 is repeated, which violates the unique mode.

Invalid.

Alternatively, [9,9,5,7]: sum is 9+9+5+7=30

Ordered: 5,7,9,9

Median: average of 7 and 9 = 8, not an integer.

Invalid.

Alternatively, [9,9,4,8]: sum is 9+9+4+8=30

Ordered: 4,8,9,9

Median: average of 8 and 9 = 8.5, not an integer.

Invalid.

Alternatively, [9,9,3,9]: but 9 repeats three times, which might be acceptable.

Sum: 9+9+3+9=30

Ordered: 3,9,9,9

Median: average of 9 and 9 = 9, which is in the list.

Invalid.

Alternatively, [9,9,2,10]: sum is 9+9+2+10=30

Ordered: 2,9,9,10

Median: average of 9 and 9 = 9, which is in the list.

Invalid.

Alternatively, [9,9,1,11]: sum is 9+9+1+11=30

Ordered: 1,9,9,11

Median: average of 9 and 9 = 9, which is in the list.

Invalid.

It seems that no matter what combination I try, the median ends up being 9, which is in the list, violating the condition.

Perhaps I need to consider that the mode is 9, and it appears more times than any other number, but maybe not exclusively twice.

Wait, the mode is unique, meaning that 9 appears more times than any other number, but it could appear more than twice.

But in the previous attempts, when 9 appears three times, the median becomes 9, which is in the list.

To avoid this, perhaps 9 appears four times.

Sum from 9 is 36, which is more than 30.

Not feasible.

Alternatively, maybe 9 appears twice, and the remaining sum is distributed among numbers that don't include 9.

But earlier attempts showed that the median wouldn't be an integer in those cases.

Alternatively, perhaps 9 appears once, but that would mean it's not the unique mode.

Because if 9 appears once, every other number must appear at least once to have a unique mode.

Wait, no. If 9 appears once, and all other numbers appear at least once, then 9 is not the unique mode unless all other numbers appear only once.

But in that case, the sum would be 9 plus the sum of other numbers, each appearing once.

But 9 is the unique mode, so no other number can appear more than once.

So, possible list: [9,1,2,3,...]

Sum is 9 plus the sum of other numbers.

For example, [9,1,2,3,4,5]=9+1+2+3+4+5=24, which is less than 30.

Need to add more units.

Add 6: [9,1,2,3,4,5,6]=9+1+2+3+4+5+6=26, still less than 30.

Add 7: [9,1,2,3,4,5,6,7]=9+1+2+3+4+5+6+7=33, which is more than 30.

Invalid.

Alternatively, [9,1,2,3,4,5,8]=9+1+2+3+4+5+8=32, more than 30.

Invalid.

Alternatively, [9,1,2,3,4,6,7]=9+1+2+3+4+6+7=32, more than 30.

Invalid.

Alternatively, [9,1,2,3,5,6,7]=9+1+2+3+5+6+7=33, more than 30.

Invalid.

Alternatively, [9,1,2,4,5,6,7]=9+1+2+4+5+6+7=34, more than 30.

Invalid.

Alternatively, [9,1,3,4,5,6,7]=9+1+3+4+5+6+7=35, more than 30.

Invalid.

Alternatively, [9,2,3,4,5,6,7]=9+2+3+4+5+6+7=36, more than 30.

Invalid.

Alternatively, [9,1,2,3,4,5,6]=9+1+2+3+4+5+6=26, less than 30.

Need to add more units.

Add 7: [9,1,2,3,4,5,6,7]=9+1+2+3+4+5+6+7=33, more than 30.

Invalid.

Alternatively, [9,1,2,3,4,5,8]=9+1+2+3+4+5+8=32, more than 30.

Invalid.

Alternatively, [9,1,2,3,4,6,7]=9+1+2+3+4+6+7=34, more than 30.

Invalid.

Alternatively, [9,1,2,3,5,6,7]=9+1+2+3+5+6+7=33, more than 30.

Invalid.

Alternatively, [9,1,2,4,5,6,7]=9+1+2+4+5+6+7=34, more than 30.

Invalid.

Alternatively, [9,1,3,4,5,6,7]=9+1+3+4+5+6+7=35, more than 30.

Invalid.

Alternatively, [9,2,3,4,5,6,7]=9+2+3+4+5+6+7=36, more than 30.

Invalid.

It seems that having 9 appear only once leads to sums exceeding 30 quickly.

Alternatively, perhaps 9 appears three times, and the remaining sum is distributed among numbers that don't include 9.

But earlier attempts showed that the median becomes 9, which is in the list.

Alternatively, maybe 9 appears twice, and the remaining sum is distributed among numbers that don't include 9, but arranged differently to avoid having 9 as the median.

Wait, perhaps I need to consider that the mode is 9, appearing twice, and the other numbers are distinct and different from 9.

Then, arrange the list so that the median is not 9.

From earlier trials, this is tricky because the median tends to be 9.

Alternatively, perhaps 9 appears once, and all other numbers appear multiple times to make 9 the unique mode.

But then, the sum would be dominated by 9, making it hard to reach 30.

Alternatively, perhaps 9 appears twice, and the remaining sum is distributed among numbers that are multiples of 9 or something like that.

Wait, that might complicate things further.

Alternatively, perhaps I need to consider that the mode is 9, appearing twice, and the other numbers are distinct and different from 9, and arranged in a way that the median is not 9.

Wait, perhaps I can have 9 appearing twice, and the other numbers arranged such that the median is not 9.

From earlier attempts, this seems difficult.

Alternatively, perhaps 9 appears three times, and the fourth number is adjusted accordingly.

Wait, earlier attempts showed that having 9 appear three times leads to the median being 9, which is in the list.

Alternatively, perhaps 9 appears four times, but then the sum exceeds 30.

Alternatively, perhaps 9 appears twice, and the remaining sum is distributed among numbers that are fractions, but the problem specifies positive integers.

Alternatively, perhaps I need to consider that the mode is 9, appearing twice, and the other numbers are arranged in a way that the median is not 9.

Wait, perhaps I can have 9 appearing twice, and the other numbers arranged such that the median is not 9.

From earlier attempts, this seems challenging.

Alternatively, perhaps I need to consider that the mode is 9, appearing twice, and the other numbers are arranged in ascending order, with 9 appearing before the other numbers.

Wait, suppose the list is [9,9,a,b,c], where a,b,c are distinct positive integers greater than 9, and the median is (b+c)/2, which should not be equal to any number in the list.

But in this case, the median is likely to be 9, which is in the list.

Alternatively, perhaps I can have 9 appearing twice, and the other numbers arranged in a way that the median is not 9.

Wait, perhaps I can have 9 appearing twice, and the other numbers arranged such that the median is not 9.

Suppose the list is [9,9,a,b,c], with a,b,c > 9, and a < b < c.

Then, the median is b.

I need b not to be 9.

But b is greater than 9, so b ≠ 9.

Wait, but the problem says that the median is a positive integer that does not appear in the list.

If b ≠ 9, then b is a positive integer not in the list.

That seems promising.

So, set the list as [9,9,a,b,c], with a,b,c > 9, a < b < c, and b ≠ 9.

Now, ensure that the sum is 30.

Sum: 9 + 9 + a + b + c = 18 + a + b + c = 30 ⇒ a + b + c = 12.

But a, b, c are distinct positive integers greater than 9, and a < b < c.

Possible triples (a,b,c):

- 4,3,5: but 3 < 4 < 5, but 3 is less than 9, which is allowed.

But sum is 4+3+5=12.

So, list: [9,9,4,3,5]

Sum: 9+9+4+3+5=30

Now, check the median.

Median is b = 3, which is less than 9 and not in the list.

Wait, but 3 is less than 9, which is fine, and it's not in the list.

Good.

So, this list satisfies all conditions.

Now, we need to find the sum of the squares of all the items in the list.

List: [9,9,4,3,5]

Squares: 81, 81, 9, 9, 25

Sum of squares: 81 + 81 + 9 + 9 + 25 = 185

But the problem asks for a three-digit number, including leading zeros if necessary.

185 is a three-digit number, so the answer is 185.

But wait, let's verify if there are other possible lists that satisfy the conditions.

Alternative list: [9,9,5,4,3]

Same as above, just rearranged, sum remains 30.

Squares remain the same.

Another alternative: [9,9,6,3,2]

Sum: 9+9+6+3+2=30

Squares: 81,81,36,9,4 → sum: 81+81+36+9+4=191

But this time, 3 and 2 are less than 9 and not in the list.

Wait, but in this list, 3 and 2 are less than 9 and not in the list, which is fine.

But the median is 3, which is less than 9 and not in the list.

Wait, but according to the problem, the median should not be in the list.

In this case, 3 is less than 9 and not in the list.

So, it's valid.

Sum of squares: 81 + 81 + 36 + 9 + 4 = 191

But in the first list, sum of squares was 185.

Wait, but the problem asks for the sum of the squares of all the items in the list.

In the first list, it was 185, and in this list, it's 191.

But the problem likely expects a single answer, so perhaps there's only one valid list.

Alternatively, perhaps there are multiple lists satisfying the conditions, leading to different sums of squares.

But given that the problem provides an answer format of a three-digit number, it's likely expecting a specific numerical answer.

Given that, and considering the constraints, perhaps 185 is the intended answer.

Alternatively, perhaps there's only one valid list, and hence only one sum of squares.

Alternatively, perhaps there are multiple valid lists, leading to multiple sums of squares.

But given that the problem asks for a specific answer, likely expecting a three-digit number, perhaps 185 is the correct answer.

Therefore, the answer is 185.

**Final Answer**

\[ \boxed{185} \]


So I've got this problem here: find the area of the region defined by \(\left||x|-1\right| + \left||y|-1\right| \leq 1\). And there are multiple choices from 1 to 16, but only up to 12 is listed, so maybe it's a typo or something. Anyway, let's see.

First off, the expression involves absolute values nested within each other, which can be tricky because absolute values tend to break down into different cases depending on the sign inside them. So, to handle this, I think I need to consider different regions based on the signs of \(x\) and \(y\), since the inner absolute values depend on whether \(x\) and \(y\) are positive or negative.

Let me start by considering the innermost absolute value, \(|x|\) and \(|y|\). These will behave differently depending on whether \(x\) and \(y\) are non-negative or negative. So, perhaps I should divide the entire plane into four quadrants based on the signs of \(x\) and \(y\):

1. Quadrant I: \(x > 0\), \(y > 0\)

2. Quadrant II: \(x < 0\), \(y > 0\)

3. Quadrant III: \(x < 0\), \(y < 0\)

4. Quadrant IV: \(x > 0\), \(y < 0\)

In each quadrant, the expressions for \(|x|\) and \(|y|\) simplify:

- In Quadrant I: \(|x| = x\), \(|y| = y\)

- In Quadrant II: \(|x| = -x\), \(|y| = y\)

- In Quadrant III: \(|x| = -x\), \(|y| = -y\)

- In Quadrant IV: \(|x| = x\), \(|y| = -y\)

Now, let's plug these back into the original inequality and see what we get for each quadrant.

**Quadrant I:**

Here, \(|x| = x\) and \(|y| = y\), both positive since \(x > 0\) and \(y > 0\).

So, the inequality becomes:

\[\left|x - 1\right| + \left|y - 1\right| \leq 1\]

This looks like a diamond (or a square rotated by 45 degrees) centered at \((1, 1)\) with side length \(\sqrt{2}\), but since all coordinates are positive, it lies entirely within the first quadrant.

The area of this region would be the area of the diamond in the first quadrant, which is half the area of the full diamond because it's symmetric about both axes.

Wait, actually, no. The standard formula for the area of a square with side length \(s\) is \(s^2\), but since this is a diamond, its area is also \(s^2 / 2\), where \(s\) is the distance between the farthest points along one axis.

But in this case, since it's bounded by \(|x - 1| + |y - 1| \leq 1\), the maximum distance in either direction is 1, so the area should be \(2 \times 1 \times 1 = 2\), because it's a square rotated by 45 degrees with side length \(\sqrt{2}\), but adjusted for the bounds.

Actually, more precisely, the area of the region defined by \(|x - h| + |y - k| \leq r\) is \(2r^2\) when \(r \geq 0\), because it's a square with side length \(2r\), but in this case, it's a diamond shape, so perhaps it's better to think of it as a square with side length \(2r\), hence area \(4r^2\).

Wait, I'm getting confused. Let's recall that for \(|x - h| + |y - k| \leq r\), the area is \(4r^2\), but that doesn't seem right. Actually, for a square with side length \(2r\), the area would be \((2r)^2 = 4r^2\), but in this case, it's a diamond shape, which has an area equal to that of a square with side length \(2r\), which is indeed \(4r^2\).

But looking back, in our case, \(h = k = 1\) and \(r = 1\), so the area should be \(4(1)^2 = 4\). But wait, is that correct?

Let me visualize this. The equation \(|x - 1| + |y - 1| = 1\) represents a square centered at \((1, 1)\) with side length \(\sqrt{2}\), but since it's bounded by \(|x - 1| + |y - 1| \leq 1\), it's actually a diamond shape with area \(2r^2 = 2(1)^2 = 2\), because it's half the area of the square with side length \(2r\).

Hmm, now I'm unsure. Maybe I should calculate the area by integrating or finding the vertices.

Alternatively, perhaps I can consider that for \(|x - 1| + |y - 1| \leq 1\), the feasible region is a square rotated by 45 degrees with side length \(\sqrt{2}\), and its area is indeed \(2r^2 = 2(1)^2 = 2\).

Okay, so in Quadrant I, the area is 2.

**Quadrant II:**

Here, \(|x| = -x\) and \(|y| = y\), both positive since \(x < 0\) and \(y > 0\).

Plugging into the inequality:

\[\left|-x - 1\right| + \left|y - 1\right| \leq 1\]

Simplify:

\[\left|x + 1\right| + \left|y - 1\right| \leq 1\]

Again, this looks like another diamond centered at \((-1, 1)\) with side length \(\sqrt{2}\), but since \(x < 0\) and \(y > 0\), it lies in the second quadrant.

Similarly, the area should be 2, just like in Quadrant I.

**Quadrant III:**

\(|x| = -x\) and \(|y| = -y\), both negative since \(x < 0\) and \(y < 0\).

Plugging into the inequality:

\[\left|-x - 1\right| + \left|-y - 1\right| \leq 1\]

Which simplifies to:

\[\left|x + 1\right| + \left|y + 1\right| \leq 1\]

This represents a diamond centered at \((-1, -1)\) with side length \(\sqrt{2}\), lying in the third quadrant.

Again, the area should be 2.

**Quadrant IV:**

\(|x| = x\) and \(|y| = -y\), both negative since \(x > 0\) and \(y < 0\).

Plugging into the inequality:

\[\left|x - 1\right| + \left|-y - 1\right| \leq 1\]

Which simplifies to:

\[\left|x - 1\right| + \left|y + 1\right| \leq 1\]

This is a diamond centered at \((1, -1)\) with side length \(\sqrt{2}\), lying in the fourth quadrant.

Once again, the area should be 2.

So, adding up the areas from all four quadrants: \(2 + 2 + 2 + 2 = 8\).

Wait a minute, but the problem provides options up to 12, and 8 is one of them. However, I feel like there might be overlapping areas or some regions being counted twice.

Let me double-check the regions.

Looking back, in Quadrant I, the region is \(|x - 1| + |y - 1| \leq 1\), which is a square rotated by 45 degrees with area 2.

Similarly, in Quadrant II, it's \(|x + 1| + |y - 1| \leq 1\), which is another square rotated by 45 degrees with area 2.

In Quadrant III, it's \(|x + 1| + |y + 1| \leq 1\), another square rotated by 45 degrees with area 2.

And in Quadrant IV, it's \(|x - 1| + |y + 1| \leq 1\), yet another square rotated by 45 degrees with area 2.

If I add them up, I get 8, but perhaps there is overlap between these regions.

Wait, but squares rotated by 45 degrees don't overlap; they are separate shapes. Each quadrant has its own distinct region.

Therefore, the total area should indeed be 8.

But let me verify this by checking specific points.

Take point \((1, 1)\): plugging into the inequality:

\[\left|1 - 1\right| + \left|1 - 1\right| = 0 + 0 = 0 \leq 1\], which holds true.

Point \((2, 1)\):

\[\left|2 - 1\right| + \left|1 - 1\right| = 1 + 0 = 1 \leq 1\], which holds.

Point \((1, 2)\):

\[\left|1 - 1\right| + \left|2 - 1\right| = 0 + 1 = 1 \leq 1\], which holds.

Point \((0, 1)\):

\[\left|0 - 1\right| + \left|1 - 1\right| = 1 + 0 = 1 \leq 1\], which holds.

Point \((-1, 1)\):

\[\left|-1 - 1\right| + \left|1 - 1\right| = 2 + 0 = 2 \not\leq 1\], which does not hold.

Point \((-1, 2)\):

\[\left|-1 - 1\right| + \left|2 - 1\right| = 2 + 1 = 3 \not\leq 1\], which does not hold.

Point \((1, -1)\):

\[\left|1 - 1\right| + \left|-1 - 1\right| = 0 + 2 = 2 \not\leq 1\], which does not hold.

Point \((2, -1)\):

\[\left|2 - 1\right| + \left|-1 - 1\right| = 1 + 2 = 3 \not\leq 1\], which does not hold.

Point \((-2, -1)\):

\[\left|-2 - 1\right| + \left|-1 - 1\right| = 3 + 2 = 5 \not\leq 1\], which does not hold.

Point \((-2, 1)\):

\[\left|-2 - 1\right| + \left|1 - 1\right| = 3 + 0 = 3 \not\leq 1\], which does not hold.

It seems that only certain points satisfy the inequality in their respective quadrants, which aligns with my earlier calculation that the total area is 8.

However, I recall that in problems involving absolute values and inequalities, sometimes the overall shape isn't simply the sum of individual regions due to overlaps or boundaries crossing.

Perhaps I should consider the entire plane and see how these regions interact.

Alternatively, maybe there's a symmetry or a transformation that can simplify the problem.

Let me try to make a substitution to simplify the expression.

Let’s set \(u = |x| - 1\) and \(v = |y| - 1\), then the inequality becomes \(|u| + |v| \leq 1\).

This represents a diamond centered at \((u,v) = (0,0)\) with side length \(\sqrt{2}\), but scaled appropriately.

Then, substituting back:

\(u = |x| - 1\) and \(v = |y| - 1\).

So, \(|x| = u + 1\) and \(|y| = v + 1\).

Given that \(|u| + |v| \leq 1\), and \(u = |x| - 1\), \(v = |y| - 1\), we can express \(x\) and \(y\) in terms of \(u\) and \(v\).

But this seems complicated. Maybe there's a better approach.

Another thought: since the expression involves absolute values, perhaps reflecting across the lines \(x=0\) and \(y=0\) can help reveal the full shape.

Alternatively, maybe considering the function \(f(x,y) = \left||x| - 1\right| + \left||y| - 1\right|\) and analyzing its behavior can provide insight.

But I'm not sure about that.

Let me consider the boundary of the region, where \(\left||x| - 1\right| + \left||y| - 1\right| = 1\).

This boundary consists of several parts corresponding to different combinations of signs for \(x\) and \(y\), similar to what I did earlier.

For example, in Quadrant I, \(x > 0\), \(y > 0\), so \(|x| = x\), \(|y| = y\), leading to \(|x - 1| + |y - 1| = 1\), which is a square.

In Quadrant II, \(x < 0\), \(y > 0\), so \(|x| = -x\), \(|y| = y\), leading to \(|-x - 1| + |y - 1| = 1\), which is another square.

Similarly, in Quadrant III, \(x < 0\), \(y < 0\), leading to \(|-x - 1| + |-y - 1| = 1\), another square.

In Quadrant IV, \(x > 0\), \(y < 0\), leading to \(|x - 1| + |-y - 1| = 1\), another square.

Each of these squares has an area of 2, and there are four such squares, totaling 8.

But perhaps some of these squares overlap in certain areas.

To check for overlaps, let's look at the positions of these squares.

- The square in Quadrant I is centered at \((1,1)\).

- The square in Quadrant II is centered at \((-1,1)\).

- The square in Quadrant III is centered at \((-1,-1)\).

- The square in Quadrant IV is centered at \((1,-1)\).

These centers are located along the diagonals of the coordinate system, specifically at \((1,1)\), \((-1,1)\), \((-1,-1)\), and \((1,-1)\).

Given that each square has a side length of \(\sqrt{2}\), their distances apart vary:

- Distance between \((1,1)\) and \((-1,1)\) is 2 units horizontally.

- Distance between \((1,1)\) and \((1,-1)\) is 2 units vertically.

- Distance between \((-1,1)\) and \((-1,-1)\) is 2 units diagonally.

Since the side lengths are relatively small compared to the distances between the centers, it's unlikely that these squares overlap significantly.

Therefore, the total area covered by these four squares is indeed the sum of their individual areas, which is 8.

Hence, the area of the region defined by the given inequality is 8.

But wait, the problem provides options up to 12, and 8 is one of them. However, perhaps there is a miscalculation somewhere.

Let me consider if there are any regions that might be missing or if there are additional areas contributed elsewhere.

Alternatively, maybe the problem expects the area without considering the signs or something else.

Alternatively, perhaps I need to consider that the expression \(\left||x| - 1\right| + \left||y| - 1\right| \leq 1\) defines a shape that is symmetric in a particular way, and I need to account for that symmetry properly.

Alternatively, maybe I should consider plotting the region to visualize it better.

Let me try to sketch the region.

Starting with \(|x| - 1\), for \(x \geq 0\), \(|x| = x\), so \(x - 1\); for \(x < 0\), \(|x| = -x\), so \(-x - 1\).

Similarly, \(|y| - 1\).

So, \(\left|(x - 1)\right| + \left|(y - 1)\right| \leq 1\).

This represents a diamond centered at \((1,1)\) with side length \(\sqrt{2}\), but since we're dealing with absolute values, it's symmetric across both axes.

Similarly, for other quadrants, the centers are at \((-1,1)\), \((-1,-1)\), and \((1,-1)\), each with the same shape and size.

Thus, the total area should still be 8.

Alternatively, perhaps the problem expects the area without considering the signs, meaning treating \(x\) and \(y\) as positive variables.

But according to the definition, \(x\) and \(y\) can be negative, so I think including all four quadrants is necessary.

Alternatively, maybe there is a mistake in assuming the area of each square is 2.

Let me recalculate the area of one of these squares.

Consider the square in Quadrant I: \(|x - 1| + |y - 1| \leq 1\).

This is equivalent to \(-1 \leq x - 1 \leq 1\) and \(-1 \leq y - 1 \leq 1\), which simplifies to \(0 \leq x \leq 2\) and \(0 \leq y \leq 2\).

The area of this square is \(2 \times 2 = 4\), but since it's a diamond shape, the actual area is half the area of the bounding square, which would be \(4 / 2 = 2\).

Wait, no. A diamond shape inscribed in a square has an area equal to that of the square divided by 2, but actually, it's the area of the diamond itself, which is \(2r^2\), where \(r\) is the radius of the circumscribed circle.

Wait, perhaps I need to recall that for a square with side length \(a\), the area of the inscribed diamond is \(a^2 / 2\).

In this case, the bounding square has side length 2 (from \(x\) from 0 to 2 and \(y\) from 0 to 2), so the area of the diamond is \(2^2 / 2 = 2\).

Therefore, each of the four diamonds has an area of 2, totaling 8.

Alternatively, perhaps there is a different way to compute the area.

Let me consider integrating over the region.

We can set up the integral for one quadrant and multiply by four.

Take Quadrant I: \(x\) from 0 to 2, \(y\) from 0 to 2.

The inequality is \(|x - 1| + |y - 1| \leq 1\).

Within this region, the area is 2, as previously determined.

Similarly, for the other quadrants, the areas are also 2 each.

Thus, total area is 8.

Alternatively, perhaps there is a geometric interpretation that shows the area is larger.

Considering that the expression \(\left||x| - 1\right| + \left||y| - 1\right| \leq 1\) defines a shape that is symmetric and extends beyond the initial four squares, but upon closer inspection, it seems that the total area is indeed 8.

Therefore, the correct option is \(\boxed{\text{E}}\), which corresponds to 8.

**Final Answer**

\[\boxed{8}\]


So I've got this problem here about Ava biking from house X to house Y, which are 45 miles apart. Maria is biking from house Y back to house X at a different speed, and they both start at the same time. I need to figure out how far from city X they'll meet.

First, let's understand what's going on. Ava is biking towards Y at 18 mph, and Maria is biking towards X at 12 mph. They're moving towards each other, so their speeds should add up when calculating how quickly they'll meet.

I think the key here is to find the time it takes for them to meet and then use that time to find out how far Ava has biked from X.

Let me denote:

- Distance between X and Y: 45 miles

- Speed of Ava: 18 mph

- Speed of Maria: 12 mph

Since they're moving towards each other, their relative speed is the sum of their individual speeds:

Relative speed = 18 mph + 12 mph = 30 mph

Now, time taken to meet can be found using the formula:

Time = Distance / Relative speed

Plugging in the values:

Time = 45 miles / 30 mph = 1.5 hours

So, they meet after 1.5 hours.

Now, to find out how far Ava has biked in that time, I can multiply her speed by the time:

Distance covered by Ava = 18 mph * 1.5 hours = 27 miles

Therefore, they meet 27 miles away from city X.

Looking back at the options:

(A) 25

(B) 26

(C) 27

(D) 20

(E) 24

It seems like option (C) 27 is the correct answer.

But wait, let me double-check my calculations to make sure I didn't make any mistakes.

First, the relative speed is indeed 18 + 12 = 30 mph.

Then, time to meet is distance divided by relative speed: 45 / 30 = 1.5 hours.

Finally, distance covered by Ava is 18 * 1.5 = 27 miles.

Yes, that seems correct.

Alternatively, maybe I can approach this problem differently to confirm.

Let's consider the distances each has traveled when they meet.

Let’s say Ava has traveled d miles from X when they meet.

Then, Maria must have traveled (45 - d) miles from Y when they meet.

Since they started at the same time and met at the same time, the time taken for both should be equal.

We know Ava's speed is 18 mph, so time taken by Ava is d / 18 hours.

Similarly, Maria's time is (45 - d) / 12 hours.

Setting these equal:

d / 18 = (45 - d) / 12

Now, solve for d.

Cross-multiplying:

12d = 18(45 - d)

12d = 810 - 18d

Add 18d to both sides:

12d + 18d = 810

30d = 810

Divide both sides by 30:

d = 27 miles

Again, I get 27 miles.

So, it seems definite that the answer is 27 miles.

Wait a minute, but the problem says "how many miles from city X will they be when they meet?" So, Ava is starting from X, so the distance from X would be the distance she's biked, which is 27 miles.

But just to be thorough, maybe there's another way to look at it.

Suppose they meet at some point, and we want to find the distance from X.

From earlier, we have that Ava has biked 27 miles, so the distance from X is 27 miles.

Alternatively, if we consider the total distance covered by both when they meet, it should be equal to the initial distance between them, which is 45 miles.

So, distance covered by Ava + distance covered by Maria = 45 miles.

We already have Ava's distance as 27 miles, so Maria's distance would be 45 - 27 = 18 miles.

Does that make sense?

Maria is biking at 12 mph, so time taken by Maria is 18 / 12 = 1.5 hours.

In that time, Ava should have biked 12 mph * 1.5 hours = 18 miles.

Indeed, 27 miles + 18 miles = 45 miles, which matches the initial distance.

So, everything checks out.

Therefore, the correct answer is 27 miles, which corresponds to choice (C).

**Final Answer**

\[ \boxed{27} \]


Let's tackle this problem step by step. So, Anders wants to buy a coffee maker that's on sale for 20% off its original price. Then, there's a 7.5% sales tax applied to the discounted price. He has $43 to spend. We need to find out what's the highest original price of the coffee maker he can afford.

First, let's understand the sequence of events:

1. The coffee maker has an original price, which we'll call P.

2. It's on sale for 20% off, so the discounted price is P minus 20% of P, which is 0.8P.

3. On top of that, there's a 7.5% sales tax on the discounted price, so the total amount he needs to pay is the discounted price plus 7.5% of the discounted price, which is 0.8P + 0.075 * 0.8P = 0.8P * 1.075.

So, the total cost T is T = 0.8P * 1.075.

We know that T must be less than or equal to $43, since that's how much Anders has.

Therefore, 0.8P * 1.075 ≤ 43.

Now, I need to solve for P.

First, simplify the left side:

0.8 * 1.075 = 0.85.

So, 0.85P ≤ 43.

Now, divide both sides by 0.85:

P ≤ 43 / 0.85.

Let me calculate that:

43 divided by 0.85.

Well, 0.85 times 50 is 42.5.

So, 0.85 * 50 = 42.5.

That's less than 43.

What about 51?

0.85 * 51 = 0.85 * 50 + 0.85 * 1 = 42.5 + 0.85 = 43.35.

That's more than 43.

So, P must be less than or equal to 50, but wait, 50 would make the total slightly over 43.

Wait, but the options go up to 50, so maybe I made a mistake somewhere.

Let me double-check my calculations.

Starting again:

T = 0.8P * 1.075 ≤ 43.

Calculate 0.8 * 1.075:

0.8 * 1.075 = 0.85.

So, 0.85P ≤ 43.

Then, P ≤ 43 / 0.85.

Calculating 43 / 0.85:

43 ÷ 0.85.

Well, 0.85 * 50 = 42.5.

0.85 * 51 = 43.35.

So, P ≤ 50, because at P = 50, T = 0.85 * 50 = 42.5.

But wait, at P = 50, T = 42.5, which is less than 43.

But according to the options, 50 is one of the choices.

But the question is asking for the most expensive coffee maker he could afford, and 50 is one of the options.

But perhaps I need to check if P = 50 gives T ≤ 43.

Let's plug P = 50 into the equation:

T = 0.8 * 50 * 1.075.

Calculate 0.8 * 50 = 40.

Then, 40 * 1.075 = 40 * 1.075.

Well, 40 * 1 = 40.

40 * 0.075 = 3.

So, 40 + 3 = 43.

Exactly $43.

So, P = 50 gives T = 43.

But the question says "afford to buy", meaning he can't spend more than $43.

So, P = 50 is exactly what he can afford.

But looking back at the options, it seems like the answer should be higher than 50, but 50 is one of the options.

Wait, maybe I misread the problem.

Let me read the problem again.

"the most expensive coffee maker he could afford to buy"

He has $43.

The coffee maker is on sale for 20% off, and then 7.5% sales tax on the discounted price.

He can afford up to $43.

From earlier calculation, P = 50 gives T = 43.

But perhaps there's a mistake here.

Wait, maybe I need to consider that the sales tax is applied after the discount.

Let me think differently.

Let’s denote the original price as P.

Discounted price: P - 0.2P = 0.8P.

Sales tax on discounted price: 0.075 * 0.8P = 0.06P.

Total cost: 0.8P + 0.06P = 0.86P.

Set this less than or equal to 43:

0.86P ≤ 43.

Then, P ≤ 43 / 0.86.

Calculating 43 / 0.86:

0.86 * 50 = 43.

So, P ≤ 50.

Again, P = 50 gives T = 0.86 * 50 = 43.

So, the most expensive coffee maker he can afford is $50.

Looking back at the options, 50 is option E.

But the problem says "the most expensive coffee maker he could afford to buy", and 50 is one of the options.

But perhaps there's a miscalculation.

Wait, maybe the sales tax is not simply added to the discounted price; maybe it's applied differently.

Alternatively, perhaps the sales tax is calculated on the original price before discount.

Let me check that possibility.

If sales tax is calculated on the original price before discount, then:

Original price: P.

Discounted price: P - 0.2P = 0.8P.

Sales tax: 0.075P.

Total cost: 0.8P + 0.075P = 0.875P.

Set this less than or equal to 43:

0.875P ≤ 43.

Then, P ≤ 43 / 0.875.

Calculating 43 / 0.875:

0.875 * 50 = 43.75.

So, P ≤ 43 / 0.875 ≈ 49.41.

So, P ≤ approximately 49.41.

In this case, the most expensive coffee maker he can afford is $49.

But $49 is option C.

However, in the first approach, P = 50 gave T = 43, which is acceptable.

But according to this second approach, P ≤ 49.41.

Which one is correct?

I think the first approach is correct because the sales tax is typically applied to the discounted price, not the original price.

So, the total cost is discounted price plus sales tax on the discounted price, which is 0.8P + 0.075*0.8P = 0.85P.

Thus, T = 0.85P ≤ 43 ⇒ P ≤ 43 / 0.85 = 50.

Therefore, the most expensive coffee maker he can afford is $50.

But wait, option E is 50, but perhaps there's a rounding issue.

Alternatively, maybe the problem expects us to round down to the nearest dollar.

But 43 / 0.85 is exactly 50.

So, P = 50 is acceptable.

But perhaps there's another way to look at it.

Let me consider that the sales tax might be applied differently.

Suppose the sales tax is calculated based on the original price before discount.

Then, T = (P - 0.2P) + 0.075P = 0.8P + 0.075P = 0.875P ≤ 43 ⇒ P ≤ 43 / 0.875 ≈ 49.41.

In this case, the maximum P is 49.

But according to the first method, it's 50.

Which one is correct?

I think the first method is more accurate because the sales tax is usually applied to the discounted price, not the original price.

Therefore, the total cost is the discounted price plus sales tax on the discounted price, which is 0.8P + 0.075*0.8P = 0.85P ≤ 43 ⇒ P ≤ 50.

Hence, the most expensive coffee maker he can afford is $50.

Therefore, the answer is E, 50.

**Final Answer**

\[ \boxed{50} \]


Let's see what this problem is asking. We have a rectangular piece of paper where the length is three times its width. So, let's denote the width as \( w \), which means the length is \( 3w \). When we fold the paper so that one vertex lies on top of the opposite vertex, it forms a pentagonal shape. We need to find the area of this pentagon as a fraction of the area of the original rectangle.

First, I need to visualize this folding process. Let's assume the rectangle has width \( w \) and length \( 3w \). If I fold it so that one vertex meets the opposite vertex, it sounds like I'm folding the paper along a line that connects two non-adjacent vertices, effectively bringing one corner onto the other.

Let me try to sketch this out mentally. Suppose I have rectangle ABCD, with AB = 3w (length), AD = w (width), and sides BC and CD also equal to 3w and w respectively. Now, folding so that point A meets point C, for example, would involve folding along a diagonal or some other line.

Wait, actually, folding one vertex onto the opposite vertex suggests that the fold is such that the path from A to C is the shortest possible straight line, but since it's a fold, it might not be the shortest path. Maybe it's a reflection over a certain line.

Alternatively, perhaps it's better to think in terms of coordinates. Let's place the rectangle on a coordinate plane with point A at the origin (0,0), B at (3w, 0), C at (3w, w), and D at (0, w).

If I fold so that point A meets point C, then the fold line would be the perpendicular bisector of the segment AC. Since A is at (0,0) and C is at (3w, w), the midpoint of AC is at (\( \frac{3w}{2} \), \( \frac{w}{2} \)).

The slope of AC is \( m_{ac} = \frac{w - 0}{3w - 0} = \frac{1}{3} \), so the slope of the perpendicular bisector would be \( m = -3 \).

Therefore, the equation of the fold line is:

\( y - \frac{w}{2} = -3(x - \frac{3w}{2}) \)

Simplifying:

\( y - \frac{w}{2} = -3x + \frac{9w}{2} \)

\( y = -3x + \frac{9w}{2} + \frac{w}{2} \)

\( y = -3x + 5w \)

Now, when we fold along this line, points on one side will map to points on the other side across this line.

The resulting figure is a pentagon, which suggests that part of the rectangle is folded over itself, creating an overlapping region that forms the pentagon.

To find the area of this pentagon relative to the original rectangle, I need to determine the regions before and after the fold.

Perhaps it's easier to consider the areas involved in the fold. When we fold, the overlapping region is essentially the intersection of the two halves after folding.

Given that the fold is along the line \( y = -3x + 5w \), I can find the area below this line and subtract it from the total area to find the area above, which should correspond to the pentagonal shape.

Wait, maybe another approach: the area of the pentagon is the area of the rectangle minus the area of the triangle that was folded over.

But I need to be careful here. Actually, upon folding, the triangle that was folded over is now inside the rectangle, forming the pentagon.

So, the area of the pentagon would be the area of the rectangle minus the area of the triangle that was folded over.

Let me calculate the area of the triangle formed by the fold line and the base.

The fold line intersects the bottom edge of the rectangle (y=0) at point E, and the top edge (y=w) at point F.

First, find coordinates of E and F.

Set y=0 in the fold line equation:

\( 0 = -3x + 5w \)

\( x = \frac{5w}{3} \)

So, E is at (\( \frac{5w}{3} \), 0)

Similarly, set y=w:

\( w = -3x + 5w \)

\( -3x = w - 5w \)

\( -3x = -4w \)

\( x = \frac{4w}{3} \)

So, F is at (\( \frac{4w}{3} \), w)

Now, the triangle formed by these points and the bottom edge is a right triangle with legs of lengths \( \frac{5w}{3} \) and \( w \).

Area of this triangle is:

\( a = \frac{1}{2} \times \frac{5w}{3} \times w = \frac{5w^2}{6} \)

Total area of the rectangle is:

\( b = 3w \times w = 3w^2 \)

Therefore, the area of the pentagon is:

\( p = b - a = 3w^2 - \frac{5w^2}{6} = \frac{18w^2}{6} - \frac{5w^2}{6} = \frac{13w^2}{6} \)

Now, the question asks for the area of the pentagon as a fraction of the area of the original rectangle:

\( \frac{p}{b} = \frac{\frac{13w^2}{6}}{3w^2} = \frac{13w^2}{6} \times \frac{1}{3w^2} = \frac{13}{18} \)

Looking back at the options, option D is \( \frac{13}{18} \).

However, I should double-check my calculations because the problem seems a bit tricky, and I might have made a mistake in determining the areas.

Let me verify the area calculation again.

Total area of the rectangle: \( 3w \times w = 3w^2 \)

Area of the folded-over triangle: \( \frac{1}{2} \times \frac{5w}{3} \times w = \frac{5w^2}{6} \)

Subtracting gives: \( 3w^2 - \frac{5w^2}{6} = \frac{18w^2 - 5w^2}{6} = \frac{13w^2}{6} \)

Then, \( \frac{p}{b} = \frac{\frac{13w^2}{6}}{3w^2} = \frac{13}{18} \)

This seems consistent. But just to be thorough, let's consider another approach.

Alternative approach: Perhaps using similar triangles or properties of folds in rectangles.

When you fold the paper so that one vertex meets the opposite vertex, the fold line acts as a mirror, reflecting one half of the rectangle onto the other.

The area of the pentagon could be thought of as the area of the rectangle minus the area of the triangle that overlaps due to the fold.

I already calculated that triangle's area as \( \frac{5w^2}{6} \), leading to the same result.

Alternatively, maybe there's a different way to look at it, involving the ratio of the areas based on the dimensions of the rectangle.

Given that the length is three times the width, perhaps there's a relationship between the fractions that simplifies the expression.

But according to my earlier calculation, \( \frac{13}{18} \) seems to be the correct answer.

Looking back at the options:

\( \textbf{(A) } \frac{2}{3} = \frac{12}{18} \)

\( \textbf{(B) } \frac{11}{16} = \frac{11 \times 3}{16 \times 3} = \frac{33}{48} \)

\( \textbf{(C) } \frac{12}{17} \approx 0.7059 \)

\( \textbf{(D) } \frac{13}{18} \approx 0.7222 \)

\( \textbf{(E) } \frac{14}{19} \approx 0.7368 \)

My calculation gives \( \frac{13}{18} \), which matches option D.

Therefore, the correct answer should be \( \frac{13}{18} \), which corresponds to choice D.

**Final Answer**

\[ \boxed{\dfrac{13}{18}} \]


So I've got this problem here: "What is the difference between one-third and 0.333?" And there are some options from A to E with fractions involved. Okay, let's see.

First, I need to understand what it means by "difference." I think it means how much they differ from each other. So, if I have one-third and 0.333, I should subtract one-third from 0.333 or vice versa. The question doesn't specify which one is larger, so maybe it doesn't matter as long as I'm consistent.

Let me first make sure I know what one-third is. One-third is approximately 0.333333... repeating forever, right? So, it's not exactly 0.333; it goes on indefinitely. But for practical purposes, sometimes people use 0.333 as an approximation of one-third because the difference might be negligible for many calculations, especially if we're dealing with decimals that don't require infinite precision.

But in this case, since the question is about the exact difference, I should probably use the exact value of one-third, which is 1/3.

So, the difference would be:

Difference = 0.333 - 1/3

Or alternatively,

Difference = 1/3 - 0.333

I think either way, the result should be the same because subtraction is commutative. But let's do both to confirm.

First, convert 1/3 to decimal form. As I said, it's approximately 0.333333...

So,

Difference = 0.333 - (0.333333...) = ?

Well, 0.333 minus 0.333 is zero, but since 1/3 is slightly larger than 0.333, the difference should be a tiny fraction.

Alternatively,

Difference = 1/3 - 0.333 = ?

To calculate this, I can express both numbers with a common denominator or find their difference directly.

Let me try expressing them as fractions.

One-third is already 1/3.

0.333 can be written as 333/1000.

Wait, actually, 0.333 is equal to 333 divided by 1000, because 333/1000 = 0.333.

So, Difference = 1/3 - 333/1000

Now, to subtract these two fractions, I need a common denominator.

The denominators are 3 and 1000. The least common multiple of 3 and 1000 is 3000.

So, let's convert both fractions to have denominator 3000.

First, 1/3 = (1 * 1000)/(3 * 1000) = 1000/3000

Second, 333/1000 = (333 * 3)/ (1000 * 3) = 999/3000

Now, Difference = 1000/3000 - 999/3000 = (1000 - 999)/3000 = 1/3000

So, the difference is 1/3000.

Looking back at the options:

(A) 0

(B) 3/1000

(C) 1/3000

(D) 3/10000

(E) 1/30000

So, option C seems to match our calculation.

But wait, let's double-check everything to be sure.

First, confirming that 1/3 is indeed approximately 0.333333... and 0.333 is just 0.333.

When I subtract 0.333 from 1/3, I get 1/3 - 0.333.

As we did earlier, converting both to fractions with denominator 3000 gives us 1000/3000 - 999/3000 = 1/3000.

Alternatively, thinking about it in terms of decimals:

0.333 is less than 1/3, so the difference should be positive.

Another way to look at it is that 1/3 is 0.333333..., so subtracting 0.333 leaves a remainder of 0.000333..., which is 1/3000.

Yes, that makes sense.

Therefore, the correct answer should be 1/3000, which is option C.

Just to be thorough, let's consider if there's any possibility of another interpretation.

Is there a chance that the difference could be something else?

For example, if the question meant the absolute difference, that wouldn't change anything because absolute difference is still just the magnitude of the difference without considering direction.

Also, since both values are being treated as real numbers, and we're dealing with their numerical difference, I don't think there's room for misinterpretation.

Moreover, looking at the options, only option C matches our calculated difference of 1/3000.

Therefore, I'm confident that the answer is C.

**Final Answer**

\[ \boxed{\text{C}} \]


So I'm trying to solve this problem about finding the largest number N that uses all the non-zero digits exactly once and where no pair of consecutive digits forms a prime number. Then, I need to find the 5th and 6th digits of N.

First, let's understand what's being asked. The number M is the smallest such number, meaning it starts with the smallest possible digits while satisfying the condition that no two consecutive digits form a prime. Now, for N, we're looking for the largest such number, so it should start with the largest possible digits under the same conditions.

To approach this, I need to:

1. List all possible pairs of digits (from 1 to 9) that do not form a prime number.

2. Arrange these digits in descending order to form the largest possible number N.

3. Identify the 5th and 6th digits of this arrangement.

But wait, arranging digits in descending order might not necessarily give me the largest number because there could be constraints from the previous digits. Also, ensuring that no two consecutive digits form a prime requires checking each possible pair.

Let me first list all possible digit pairs (a,b) where neither ab nor ba is a prime number. Since we're dealing with single-digit numbers, both a and b are from 1 to 9.

I need to check all combinations of a and b from 1 to 9 and see if either ab or ba is a prime number. If neither is prime, they can be consecutive digits.

First, list all two-digit primes between 10 and 99. These are:

11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97.

Now, for each pair (a,b), check if ab or ba is in this list.

For example:

- 1 and 1: 11 is prime → invalid

- 1 and 2: 12 and 21 → 12 is not prime, 21 is not prime → valid

- 1 and 3: 13 is prime → invalid

- And so on...

This seems time-consuming, but since there are only 9 digits, it's manageable.

Alternatively, maybe there's a smarter way to think about this without listing all possibilities.

Wait a minute, perhaps I can look at the properties of the digits themselves.

I recall that certain digits are more "flexible" than others when it comes to forming prime numbers with their neighbors.

For instance, digits like 1, 3, 7, 9 don't appear in any two-digit prime except when paired with other specific digits.

But actually, upon closer inspection, many digits can be used together as long as the resulting two-digit number isn't prime.

Maybe I should create a graph where nodes represent digits, and edges connect digits whose concatenation doesn't form a prime.

Then, finding the largest number would mean traversing this graph in reverse, starting from the highest digit.

But perhaps that's overcomplicating things.

Let me try to categorize the digits based on their compatibility.

Define a function f(d) that returns True if d can be followed by another digit without forming a prime.

For example:

- f(1): 11 is prime → False

- f(2): 12 (not prime), 21 (not prime) → True

- f(3): 13 (prime), 31 (prime) → False

- f(4): 14 (not prime), 24 (not prime), 34 (not prime), 41 (prime) → False

- f(5): 15 (not prime), 25 (not prime), 35 (not prime), 45 (not prime), 51 (prime) → False

- f(6): 16 (not prime), 26 (not prime), 36 (not prime), 46 (not prime), 56 (not prime), 61 (prime) → False

- f(7): 17 (prime), 71 (prime) → False

- f(8): 18 (not prime), 28 (not prime), 38 (not prime), 48 (not prime), 58 (not prime), 68 (not prime), 78 (not prime), 81 (not prime), 82 (not prime), 83 (prime) → False

- f(9): 19 (prime), 91 (not prime) → False

From this, it seems that digits like 1, 2, 4, 5, 6, 8, and 9 have some restrictions, especially 1, 2, 4, 5, 6, 8, and 9.

Digits like 3, 7, and 9 are problematic because they often form primes with their neighbors.

Given that, perhaps the largest number would use higher digits as much as possible, avoiding those that frequently form primes.

But this is still vague. Maybe I should consider building the largest possible number step by step, choosing the highest available digit at each step, provided it doesn't violate the condition.

Starting with 9, the highest digit, then choose the next highest possible digit that doesn't form a prime with 9.

Looking back, 9 can be followed by digits that don't form a prime like 1, 2, 4, 5, 6, 8, etc., but according to earlier analysis, 9 can be followed by 1, 2, 4, 5, 6, 8, but not 3, 7, or 9 itself.

Wait, 9 can be followed by 1, 2, 4, 5, 6, 8, but not 3, 7, or 9.

Similarly, 8 can be followed by 1, 2, 3, 4, 5, 6, 7, 8, but not 1, 3, 7.

Wait, hold on. Earlier, I thought 8 can be followed by 1, 2, 3, 4, 5, 6, 7, 8, but not 1, 3, 7.

Wait, 81 is not prime, 82 is not prime, 83 is prime, so 8 cannot be followed by 3.

Similarly, 87 is not prime, 84 is not prime, etc.

So, correcting my earlier mistake, 8 can be followed by digits that don't form a prime with 8.

Wait, perhaps I need a better systematic approach.

Let me make a table of digits and their allowed followers.

Create a table where rows are digits, columns are digits, and entries indicate whether following the row digit by the column digit forms a prime.

For example:

|   | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |

|---|---|---|---|---|---|---|---|---|---|

|1 | F | T | F | T | F | T | F | T | F |

|2 | T | T | T | T | T | T | T | T | T |

|3 | F | T | T | T | T | T | T | T | T |

|4 | T | T | T | T | T | T | T | T | T |

|5 | F | T | T | T | T | T | T | T | T |

|6 | T | T | T | T | T | T | T | T | T |

|7 | F | T | T | T | T | T | T | T | T |

|8 | T | T | T | T | T | T | T | T | T |

|9 | F | T | T | T | T | T | T | T | T |

In this table, F means the two-digit number is prime, and T means it's not.

Wait, but 83 is prime, so 8 cannot be followed by 3.

Similarly, 13 is prime, so 1 cannot be followed by 3.

And 71 is prime, so 7 cannot be followed by 1.

Also, 97 is prime, so 9 cannot be followed by 7.

Wait, but 97 is prime, so 9 cannot be followed by 7.

Similarly, 79 is prime, so 7 cannot be followed by 9.

But in the above table, I have 9 F for 9, which includes 97, but also 79 is not listed separately.

Wait, perhaps I need to adjust the table to correctly mark primes.

Actually, since the problem says "none of the pairs of its consecutive digits makes a prime number," I need to ensure that neither ab nor ba is prime.

So, for each pair (a,b), check if either ab or ba is prime.

For example:

- 11: prime

- 13: prime

- 17: prime

- 19: prime

- 23: prime

- 29: prime

- 31: prime

- 37: prime

- 41: prime

- 43: prime

- 47: prime

- 53: prime

- 59: prime

- 61: prime

- 67: prime

- 71: prime

- 73: prime

- 79: prime

- 83: prime

- 89: prime

- 97: prime

So, in the table, for each digit a, and for each digit b ≠ a, mark T if ab and ba are not prime.

Wait, actually, for each pair (a,b), mark T if neither ab nor ba is prime.

For example:

- 11: F (both are prime)

- 12: T (12 not prime, 21 not prime)

- 13: F (13 is prime)

- ...

- 21: T (21 not prime, 12 not prime)

- ...

- 31: F (31 is prime)

- ...

- 41: T (41 is prime, but 14 is not prime)

Wait, 41 is prime, so 41 is F, but 14 is not prime, so T.

Wait, no. According to the problem, neither ab nor ba should be prime.

So, for 41, 41 is prime, so it's F; 14 is not prime, so T.

Similarly, for 43, 43 is prime, so F; 34 is not prime, so T.

Wait, but 43 is prime, so even though 34 is not prime, 43 being prime disqualifies the pair.

So, in general, for each pair (a,b):

- If ab or ba is prime, then the pair is F.

- Else, it's T.

So, in the table, for each digit a, and for each digit b ≠ a, mark T if neither ab nor ba is prime.

Alright, now with this table, I can proceed to build the largest number N by selecting the highest possible digits at each step, ensuring that each new digit follows the previously chosen one according to the table.

Start with the highest digit, which is 9.

Check who can follow 9: from the table, 9 can be followed by 1, 2, 4, 5, 6, 8, but not 3, 7.

Among these, the highest is 8, so choose 8.

Next, after 8, look for the highest remaining digit that can follow 8.

From the table, 8 can be followed by 1, 2, 3, 4, 5, 6, 7, 8, but not 1, 3, 7.

So, among the remaining digits, exclude 1 and 3, leaving 2, 4, 5, 6, 7, 8.

Choose the highest among these, which is 8 again, but 8 is already used.

Wait, no, 8 is already used, so next highest is 7.

But 7 is not directly connected to 8 in the table, but since 87 is not prime, it's okay.

Wait, but in the table, for 8, it can be followed by 1, 2, 3, 4, 5, 6, 7, 8.

So, 7 is allowed after 8.

Choose 7.

Now, after 7, look for the highest remaining digit that can follow 7.

From the table, 7 can be followed by 1, 2, 3, 4, 5, 6, 7, 8, but not 1, 3, 7.

So, excluding 1 and 3, remaining are 2, 4, 5, 6, 8.

Choose the highest among these, which is 8 again.

But 8 is already used.

Wait, but 8 was already used before.

Wait, perhaps I made a mistake in tracking which digits are used.

Let me keep track of used digits.

Used so far: 9, 8, 7.

Remaining: 1, 2, 3, 4, 5, 6.

After 7, the highest remaining is 8, but 8 is already used, so next highest is 6.

Check if 7 can be followed by 6: 76 is not prime, 67 is prime.

Wait, 67 is prime, so 76 is not prime, but 67 is prime.

Therefore, 76 is allowed, but 67 is not.

So, 7 can be followed by 6.

Choose 6.

Now, after 6, look for the highest remaining digit that can follow 6.

From the table, 6 can be followed by 1, 2, 3, 4, 5, 6, 7, 8, but not 1, 3, 7.

So, excluding 1 and 7, remaining are 2, 3, 4, 5, 6, 8.

Choose the highest among these, which is 8 again, but 8 is already used.

Wait, but 8 was used before, so next highest is 6 again, but 6 is already used.

Wait, perhaps I need to choose the next best option.

Wait, perhaps I need to backtrack.

After 7, instead of choosing 6, choose the next best option.

The options left after 7 are 2, 4, 5, 8.

Choose the highest among these that can follow 7.

From the table, 7 can be followed by 1, 2, 3, 4, 5, 6, 7, 8.

Excluding 1 and 3, remaining are 2, 4, 5, 6, 8.

Choose the highest among these, which is 8.

But 8 is already used, so next highest is 6.

But 6 is already used, so next is 5.

Check if 7 can be followed by 5: 75 is not prime, 57 is not prime.

Yes, 75 is not prime, 57 is not prime.

So, 75 is allowed.

Choose 5.

Now, after 5, look for the highest remaining digit that can follow 5.

From the table, 5 can be followed by 1, 2, 3, 4, 5, 6, 7, 8, but not 1, 3, 7.

So, excluding 1 and 7, remaining are 2, 3, 4, 5, 6, 8.

Choose the highest among these, which is 8.

But 8 is already used, so next is 6.

But 6 is already used, so next is 4.

Check if 5 can be followed by 4: 54 is not prime, 45 is not prime.

Yes, 54 is not prime, 45 is not prime.

So, 54 is allowed.

Choose 4.

Now, after 4, look for the highest remaining digit that can follow 4.

From the table, 4 can be followed by 1, 2, 3, 4, 5, 6, 7, 8, but not 1, 3, 7.

So, excluding 1 and 7, remaining are 2, 3, 5, 6, 8.

Choose the highest among these, which is 8.

But 8 is already used, so next is 6.

But 6 is already used, so next is 5.

But 5 is already used, so next is 3.

Check if 4 can be followed by 3: 43 is not prime, 34 is not prime.

Yes, 43 is not prime, 34 is not prime.

So, 43 is allowed.

Choose 3.

Now, after 3, look for the highest remaining digit that can follow 3.

From the table, 3 can be followed by 1, 2, 3, 4, 5, 6, 7, 8, but not 1, 3, 7.

So, excluding 1 and 7, remaining are 2, 4, 5, 6, 8.

Choose the highest among these, which is 8.

But 8 is already used, so next is 6.

But 6 is already used, so next is 5.

But 5 is already used, so next is 4.

But 4 is already used, so next is 2.

Check if 3 can be followed by 2: 32 is not prime, 23 is prime.

Wait, 23 is prime, so 32 is not prime, but 23 is prime.

Therefore, 32 is allowed.

Choose 2.

Finally, after 2, the only remaining digit is 1.

Check if 2 can be followed by 1: 21 is not prime, 12 is not prime.

Yes, both are not prime.

So, 21 is allowed.

Choose 1.

Thus, the largest number N is 987654321.

Now, I need to find the 5th and 6th digits of N.

Counting positions:

1st: 9

2nd: 8

3rd: 7

4th: 6

5th: 5

6th: 4

So, the 5th and 6th digits are 5 and 4.

Looking back at the options:

(A) 6 and 3

(B) 5 and 4

(C) 5 and 2

(D) 4 and 8

(E) 3 and 5

Option (B) matches our result.

However, I should double-check if there was any mistake in constructing N.

Looking back at the steps:

- Started with 9.

- Followed by 8 (allowed).

- Followed by 7 (allowed).

- Followed by 6 (allowed).

- Followed by 5 (allowed).

- Followed by 4 (allowed).

- Followed by 3 (allowed).

- Followed by 2 (allowed).

- Finally, followed by 1 (allowed).

Seems consistent.

Alternatively, perhaps there's a larger number possible by choosing different sequences.

For example, after 7, instead of choosing 6, could I choose something else?

Wait, after 7, the options were 2, 4, 5, 6, 8.

But 8 is already used, so 6 is already used, so 5 is already used, so 4 is already used, so 3 is allowed.

Then, after 3, options are 2, 4, 5, 6, 8.

But 4 is already used, 5 is already used, 6 is already used, 8 is already used, so only 2 remains.

Then, after 2, only 1 remains.

So, yes, the sequence 987654321 seems to be the largest possible.

Therefore, the 5th and 6th digits are 5 and 4.

**Final Answer**

\[ \boxed{\text{B}} \]


Let's dive into this problem. It seems a bit tricky at first glance, but I'll try to break it down step by step.

So, we have a table tennis tournament where every participant plays every other participant exactly once. That means the total number of games played is equal to the number of ways to choose 2 players out of the total number of players, which can be calculated using combinations: C(n, 2) = n(n-1)/2, where n is the total number of players.

Now, among these players, there are twice as many right-handed players as left-handed players. Let's denote the number of left-handed players as l and the number of right-handed players as r. According to the problem, r = 2l.

Since each game is between two players, the total number of games is C(n, 2), where n = l + r = l + 2l = 3l.

Next, we're told that the number of games won by left-handed players was 40% more than the number of games won by right-handed players. There were no ties, so each game results in one win for either player.

Let’s denote the number of games won by right-handed players as w_r and the number of games won by left-handed players as w_l. According to the problem, w_l = 1.4 * w_r.

But since every game has exactly one winner, the total number of wins is equal to the total number of games, which is w_l + w_r = C(n, 2).

Substituting w_l from above:

w_l + w_r = 1.4*w_r + w_r = 2.4*w_r

Therefore, 2.4*w_r = C(n, 2)

But we also know that C(n, 2) = n(n-1)/2 = 3l*(3l - 1)/2

So, 2.4*w_r = 3l*(3l - 1)/2

Now, let's think about how many games each group of players can win. In any match between two players, the winner is either left-handed or right-handed.

The number of possible matches involving only left-handed players would be C(l, 2) = l(l-1)/2

Similarly, the number of possible matches involving only right-handed players would be C(r, 2) = r(r-1)/2

And the number of matches involving both left-handed and right-handed players would be l*r

Wait a minute, actually, the number of matches involving both groups is simply all possible matches minus the matches within each group.

Total matches = C(n, 2) = 3l*(3l - 1)/2

Matches within left-handed players = C(l, 2) = l(l-1)/2

Matches within right-handed players = C(r, 2) = r(r-1)/2

Matches between left-handed and right-handed players = l*r

But since every game is between two different players, the total number of matches should also be equal to l*r + C(l, 2) + C(r, 2)

Wait, no. Actually, the total number of matches is just C(n, 2) = 3l*(3l - 1)/2

Because each pair of players plays exactly one game.

So, C(n, 2) = l*r + C(l, 2) + C(r, 2)

But since r = 2l, we can substitute that in:

3l*(3l - 1)/2 = 2l*l + l(l-1)/2 + 2l*2l

Simplify the right side:

2l^2 + l(l-1)/2 + 4l^2 = 2l^2 + (l^2 - l)/2 + 4l^2 = (4l^2 + l^2 - l + 8l^2)/2 = (13l^2 - l)/2

Set equal to the left side:

3l*(3l - 1)/2 = (13l^2 - l)/2

Multiply both sides by 2:

3l*(3l - 1) = 13l^2 - l

Expand the left side:

9l^2 - 3l = 13l^2 - l

Bring all terms to one side:

9l^2 - 3l - 13l^2 + l = 0

Combine like terms:

-4l^2 - 2l = 0

Factor out -2l:

-2l(2l + 1) = 0

So, l = 0 or l = -1/2

But l can't be negative, so l = 0

If l = 0, then r = 2*0 = 0, which doesn't make sense because there are players.

Hmm, maybe I made a mistake somewhere. Let me double-check my steps.

Starting from:

C(n, 2) = l*r + C(l, 2) + C(r, 2)

With r = 2l, n = l + r = 3l

So:

3l*(3l - 1)/2 = 2l*l + l(l-1)/2 + 2l*2l

Simplify right side:

2l^2 + l(l-1)/2 + 4l^2 = 2l^2 + (l^2 - l)/2 + 4l^2 = (4l^2 + l^2 - l + 8l^2)/2 = (13l^2 - l)/2

Set equal to left side:

3l*(3l - 1)/2 = (13l^2 - l)/2

Multiply both sides by 2:

3l*(3l - 1) = 13l^2 - l

Expand:

9l^2 - 3l = 13l^2 - l

Bring all terms to one side:

9l^2 - 3l - 13l^2 + l = 0 → -4l^2 - 2l = 0 → -2l(2l + 1) = 0

Again, l = 0 or l = -1/2

This suggests that l must be zero, but that leads to no players, which can't be right.

Maybe I need to consider that some matches might not involve both groups, but according to the problem, every participant plays every other participant exactly once, so there are no matches without both groups.

Alternatively, perhaps the assumption that C(n, 2) = l*r + C(l, 2) + C(r, 2) is incorrect.

Wait, actually, in a tournament where every player plays every other player exactly once, the total number of matches is indeed C(n, 2). However, when dividing players into left-handed and right-handed, the matches can be categorized based on the handedness of the players involved.

Specifically:

- Matches between two left-handed players: C(l, 2)

- Matches between two right-handed players: C(r, 2)

- Matches between one left-handed and one right-handed player: l*r

And the sum of these should equal the total number of matches, which is C(n, 2).

So, C(l, 2) + C(r, 2) + l*r = C(n, 2)

Given that r = 2l and n = l + r = 3l, we can plug in:

C(l, 2) + C(r, 2) + l*r = 3l*(3l - 1)/2

Which simplifies to:

[l(l-1)/2] + [r(r-1)/2] + l*r = [3l*(3l - 1)]/2

Substitute r = 2l:

[l(l-1)/2] + [2l*(2l - 1)/2] + l*2l = [3l*(3l - 1)]/2

Simplify:

[l(l-1)/2] + [2l(2l - 1)]/2 + 2l^2 = [9l^2 - 3l]/2

Multiply through by 2 to eliminate denominators:

l(l-1) + 2l(2l - 1) + 4l^2 = 9l^2 - 3l

Expand:

l^2 - l + 4l^2 - 2l + 4l^2 = 9l^2 - 3l

Combine like terms:

9l^2 - 3l = 9l^2 - 3l

This is always true, meaning that for any positive integer l, this equation holds, given the constraints.

But earlier, solving for l led to l = 0 or l = -1/2, which don't make sense in this context.

Perhaps there's another way to approach this problem.

Let's consider the total number of games won by left-handed players and right-handed players.

Each game contributes one win to either the left-handed or right-handed category.

So, the total number of wins is equal to the total number of games, which is C(n, 2) = 3l*(3l - 1)/2.

According to the problem, the number of wins by left-handed players is 40% more than the number of wins by right-handed players.

Let’s denote the number of wins by right-handed players as w_r and by left-handed players as w_l.

Then, w_l = 1.4 * w_r

And w_l + w_r = C(n, 2) = 3l*(3l - 1)/2

Substitute w_l:

1.4*w_r + w_r = 2.4*w_r = 3l*(3l - 1)/2

Therefore, w_r = [3l*(3l - 1)/2] / 2.4 = [3l*(3l - 1)/2] * [5/12] = [15l*(3l - 1)] / 24 = [5l*(3l - 1)] / 8

Similarly, w_l = 1.4 * w_r = 1.4 * [5l*(3l - 1)] / 8 = [7l*(3l - 1)] / 8

But w_l + w_r should equal C(n, 2):

[5l*(3l - 1)] / 8 + [7l*(3l - 1)] / 8 = [12l*(3l - 1)] / 8 = [3l*(3l - 1)] / 2

Which equals C(n, 2), so this checks out.

But I need to find the value of n such that this makes sense.

Alternatively, perhaps I can consider the ratio of wins directly.

Since w_l = 1.4*w_r, and w_l + w_r = C(n, 2), I can express everything in terms of w_r.

From w_l = 1.4*w_r, and w_l + w_r = C(n, 2), substituting gives:

1.4*w_r + w_r = 2.4*w_r = C(n, 2)

Therefore, w_r = C(n, 2) / 2.4

Similarly, w_l = 1.4 * C(n, 2) / 2.4 = 1.4 / 2.4 * C(n, 2) = 7/12 * C(n, 2)

Now, the number of wins by left-handed players is w_l, and similarly for right-handed.

But I'm still stuck with variables here.

Maybe I need to consider the distribution of wins differently.

Let’s think about the number of matches involving left-handed and right-handed players.

The number of matches between left-handed players is C(l, 2) = l(l-1)/2

Similarly, the number of matches between right-handed players is C(r, 2) = r(r-1)/2

And the number of matches between left-handed and right-handed players is l*r

Now, in matches between left-handed players, all wins go to left-handed players.

In matches between right-handed players, all wins go to right-handed players.

In matches between left-handed and right-handed players, wins can go to either side.

But according to the problem, the total number of wins by left-handed players is 40% more than those by right-handed players.

So, wins by left-handed players come from:

- Wins in matches between left-handed players: all l(l-1)/2 wins

- Wins in matches between left-handed and right-handed players: l*r wins

Similarly, wins by right-handed players come from:

- Wins in matches between right-handed players: all r(r-1)/2 wins

- Wins in matches between left-handed and right-handed players: l*r wins

Wait, but in matches between left-handed and right-handed players, wins can go to either side.

However, the problem states that the number of games won by left-handed players is 40% more than those won by right-handed players.

This implies that in addition to their wins in intra-group matches, left-handed players get an extra 40% of wins from the inter-group matches compared to right-handed players.

This is getting complicated. Maybe I need to set up equations based on the wins.

Let’s denote:

w_ll: wins by left-handed players in matches between left-handed players

w_lr: wins by left-handed players in matches between left-handed and right-handed players

w_rl: wins by right-handed players in matches between left-handed and right-handed players

w_rr: wins by right-handed players in matches between right-handed players

Then:

w_l = w_ll + w_lr

w_r = w_rr + w_rl

And according to the problem, w_l = 1.4 * w_r

Also, the total number of wins is w_l + w_r = C(n, 2)

We also know that w_ll + w_rr = C(l, 2) = l(l-1)/2

And w_lr + w_rl = l*r

Additionally, w_lr + w_rl = w_rr + w_ll

Because the total wins are equal to the total matches.

Wait, no. The total wins are equal to the total matches, which is C(n, 2), but wins are distributed among left-handed and right-handed players.

I need to relate these variables to the given conditions.

This seems too vague. Maybe I should try plugging in the multiple-choice options and see which one fits.

The options are:

(A) 45

(B) 66

(C) 48

(D) 36

(E) 15

First, recall that C(n, 2) = n(n-1)/2

Let's check which of these numbers can be expressed as n(n-1)/2 for some positive integer n.

Option (A): 45 → n(n-1)/2 = 45 → n(n-1) = 90 → n ≈ sqrt(90) ≈ 9.5, not an integer.

Option (B): 66 → n(n-1)/2 = 66 → n(n-1) = 132 → n ≈ sqrt(132) ≈ 11.5, not an integer.

Option (C): 48 → n(n-1)/2 = 48 → n(n-1) = 96 → n ≈ sqrt(96) ≈ 9.8, close to 10.

Option (D): 36 → n(n-1)/2 = 36 → n(n-1) = 72 → n ≈ sqrt(72) ≈ 8.5, not an integer.

Option (E): 15 → n(n-1)/2 = 15 → n(n-1) = 30 → n ≈ sqrt(30) ≈ 5.5, not an integer.

So, only option (C) 48 corresponds to n(n-1)/2 for an integer n.

Therefore, the total number of games played is 48.

**Final Answer**

\[ \boxed{48} \]