answer：Okay, so I have this problem where I have three sequences of numbers: ( x_1, x_2, ldots, x_n ); ( y_1, y_2, ldots, y_n ); and ( z_1, z_2, ldots, z_n ). Each of these numbers is either 1 or -1. The conditions given are that the sum of the products ( x_i y_i ) is zero, the sum of ( x_i z_i ) is zero, and the sum of ( y_i z_i ) is also zero. I need to prove that ( n ) must be a multiple of 4.Hmm, let me think about this. Since each ( x_i, y_i, z_i ) is either 1 or -1, their products will also be either 1 or -1. So, each term in the sums ( x_i y_i ), ( x_i z_i ), and ( y_i z_i ) is either 1 or -1. The fact that each of these sums is zero means that there must be an equal number of 1s and -1s in each sum. Therefore, the number of terms, which is ( n ), must be even because you can't have an equal number of 1s and -1s if ( n ) is odd.So, ( n ) has to be even. But the problem is asking to prove that ( n ) is a multiple of 4, not just even. So, being even is a necessary condition, but we need something stronger.Maybe I can think about this in terms of vectors. Each sequence ( x ), ( y ), and ( z ) can be considered as vectors in an ( n )-dimensional space where each component is either 1 or -1. The conditions given are that the dot products between each pair of these vectors are zero. So, ( x cdot y = 0 ), ( x cdot z = 0 ), and ( y cdot z = 0 ).In vector terms, this means that the vectors ( x ), ( y ), and ( z ) are pairwise orthogonal. Now, in an ( n )-dimensional space, you can have at most ( n ) orthogonal vectors. Here, we have three orthogonal vectors, so ( n ) must be at least 3. But that doesn't directly help with proving ( n ) is a multiple of 4.Wait, maybe I should consider the properties of these vectors. Since each vector has components of 1 or -1, their dot product with themselves is ( n ), because each term is 1*1 or (-1)*(-1), which is 1. So, ( x cdot x = y cdot y = z cdot z = n ).Now, if I square the sum ( x cdot y ), which is zero, I get:[(x cdot y)^2 = 0 = left( sum_{i=1}^n x_i y_i right)^2 = sum_{i=1}^n (x_i y_i)^2 + 2 sum_{1 leq i < j leq n} x_i y_i x_j y_j]Since each ( x_i y_i ) is either 1 or -1, ( (x_i y_i)^2 = 1 ), so the first term is ( n ). Therefore:[0 = n + 2 sum_{1 leq i < j leq n} x_i y_i x_j y_j]This simplifies to:[sum_{1 leq i < j leq n} x_i y_i x_j y_j = -frac{n}{2}]But I'm not sure if this is helpful yet. Maybe I should consider the sum of the squares of the dot products or something else.Alternatively, let me think about the number of agreements and disagreements between the vectors. For each pair of vectors, since their dot product is zero, the number of positions where they agree (both 1 or both -1) must equal the number of positions where they disagree (one is 1 and the other is -1). So, for each pair, half of the positions are agreements and half are disagreements.Let me denote:- ( a ) as the number of indices where ( x_i = 1 ) and ( y_i = 1 )- ( b ) as the number of indices where ( x_i = 1 ) and ( y_i = -1 )- ( c ) as the number of indices where ( x_i = -1 ) and ( y_i = 1 )- ( d ) as the number of indices where ( x_i = -1 ) and ( y_i = -1 )Since the dot product ( x cdot y = 0 ), we have:[a - b - c + d = 0]Also, the total number of indices is ( n ), so:[a + b + c + d = n]From these two equations, we can solve for some variables. Let's add the two equations:[(a - b - c + d) + (a + b + c + d) = 0 + n][2a + 2d = n][a + d = frac{n}{2}]Similarly, subtracting the first equation from the second:[(a + b + c + d) - (a - b - c + d) = n - 0][2b + 2c = n][b + c = frac{n}{2}]So, both ( a + d ) and ( b + c ) equal ( frac{n}{2} ). This tells me that the number of agreements and disagreements are each ( frac{n}{2} ).Now, let's consider the third vector ( z ). Similarly, we can define variables for the overlaps between ( z ) and ( x ), and ( z ) and ( y ). Let me denote:- ( e ) as the number of indices where ( x_i = 1 ) and ( z_i = 1 )- ( f ) as the number of indices where ( x_i = 1 ) and ( z_i = -1 )- ( g ) as the number of indices where ( x_i = -1 ) and ( z_i = 1 )- ( h ) as the number of indices where ( x_i = -1 ) and ( z_i = -1 )Similarly, for ( y ) and ( z ):- ( i ) as the number of indices where ( y_i = 1 ) and ( z_i = 1 )- ( j ) as the number of indices where ( y_i = 1 ) and ( z_i = -1 )- ( k ) as the number of indices where ( y_i = -1 ) and ( z_i = 1 )- ( l ) as the number of indices where ( y_i = -1 ) and ( z_i = -1 )But this might get too complicated with so many variables. Maybe there's a smarter way to approach this.Wait, perhaps I can consider the product of all three vectors. Since ( x cdot y = 0 ), ( x cdot z = 0 ), and ( y cdot z = 0 ), maybe there's a relationship when considering all three together.Alternatively, let me think about the sum of the products ( x_i y_i z_i ). Each term is either 1 or -1, so the sum will be an integer. Maybe I can relate this to the given conditions.But I don't have a condition on ( x cdot y cdot z ), so I'm not sure.Wait, another idea: since each pair of vectors is orthogonal, perhaps the set ( {x, y, z} ) forms an orthogonal set. In such a case, the dimension of the space must be at least 3, but we already know that. However, in our case, the vectors are binary (entries are 1 or -1), so maybe there are additional constraints.Let me consider the squares of the vectors. As I thought earlier, each vector has a dot product with itself equal to ( n ). So, ( x cdot x = y cdot y = z cdot z = n ).If I consider the sum ( x + y + z ), what can I say about its dot product with itself?[(x + y + z) cdot (x + y + z) = x cdot x + y cdot y + z cdot z + 2x cdot y + 2x cdot z + 2y cdot z]Substituting the known values:[= n + n + n + 2(0) + 2(0) + 2(0) = 3n]So, the squared norm of ( x + y + z ) is ( 3n ). But since each component of ( x + y + z ) is the sum of three 1s or -1s, each component can be -3, -1, 1, or 3. Therefore, each component squared is 1 or 9. Hence, the total squared norm is the sum of these squares, which must be an integer.But ( 3n ) must be equal to the sum of squares of the components of ( x + y + z ). Each component is either -3, -1, 1, or 3, so each squared component is 1 or 9. Therefore, the total squared norm is ( 3n ), which must be equal to the sum of 1s and 9s.Let me denote ( m ) as the number of components where ( x_i + y_i + z_i = pm 3 ), and ( k ) as the number of components where ( x_i + y_i + z_i = pm 1 ). Then, we have:[9m + k = 3n][m + k = n]From the second equation, ( k = n - m ). Substituting into the first equation:[9m + (n - m) = 3n][8m + n = 3n][8m = 2n][4m = n]So, ( n = 4m ), which means ( n ) is a multiple of 4. That's exactly what we needed to prove!Wait, let me double-check this reasoning. We considered the squared norm of ( x + y + z ) and found that it equals ( 3n ). Then, we expressed this squared norm as the sum of squares of each component, which can only be 1 or 9. By setting up the equations, we found that ( n ) must be four times some integer ( m ). Therefore, ( n ) is indeed a multiple of 4.I think this makes sense. Each component of ( x + y + z ) contributes either 1 or 9 to the total squared norm, and the relationship between these contributions forces ( n ) to be divisible by 4. So, I'm confident that this is the correct approach and that ( n ) must be a multiple of 4.

answer：Okay, so I'm trying to solve this problem where there's a function f(n) with a specific property: f(f(n)) + f(n) = 2n + 3, and we're also given that f(0) = 1. Additionally, we're told that f(2015) = 2016, and we need to find the value of n for which we're finding f(n). Hmm, that last part is a bit confusing. Let me try to parse that.First, let's make sure I understand the problem correctly. We have a function f(n) that satisfies the equation f(f(n)) + f(n) = 2n + 3 for all n, and we know that f(0) = 1. We're also given that f(2015) = 2016, and we need to find the value of n such that f(n) = 2016. Wait, no, actually, the problem says "what is the value of n for which we are finding f(n)?" So, maybe it's asking for n such that f(n) = 2016? Or perhaps it's asking for n such that f(n) = something else? I need to clarify.Wait, the problem says: "If f(2015) = 2016, what is the value of n for which we are finding f(n)?" So, maybe it's asking for n such that f(n) = 2016, which would be n = 2015, since f(2015) = 2016. But that seems too straightforward. Maybe I'm missing something.Alternatively, perhaps the problem is asking for n such that f(n) = 2015, given that f(2015) = 2016. That would make more sense, because then we'd have to use the functional equation to find such an n. Let me assume that's the case.So, let's restate the problem: Given f(f(n)) + f(n) = 2n + 3, f(0) = 1, and f(2015) = 2016, find n such that f(n) = 2015.Alright, let's start by trying to find a pattern or a general form for f(n). We have f(0) = 1. Let's compute f(1) using the functional equation.Let n = 0: f(f(0)) + f(0) = 2*0 + 3 ⇒ f(1) + 1 = 3 ⇒ f(1) = 2.Now, let's find f(2). Let n = 1: f(f(1)) + f(1) = 2*1 + 3 ⇒ f(2) + 2 = 5 ⇒ f(2) = 3.Similarly, n = 2: f(f(2)) + f(2) = 2*2 + 3 ⇒ f(3) + 3 = 7 ⇒ f(3) = 4.Hmm, I see a pattern here: f(0) = 1, f(1) = 2, f(2) = 3, f(3) = 4. It seems like f(n) = n + 1 for these values. Let's test this hypothesis.Assume f(k) = k + 1 for some integer k ≥ 0. Then, f(f(k)) = f(k + 1) = (k + 1) + 1 = k + 2. Plugging into the functional equation: f(f(k)) + f(k) = (k + 2) + (k + 1) = 2k + 3, which matches the right-hand side. So, by induction, f(n) = n + 1 for all n ≥ 0.Wait, but we're given that f(2015) = 2016, which fits this pattern since 2015 + 1 = 2016. So, f(n) = n + 1 for all n, including n = 2015.But the problem is asking for the value of n for which we're finding f(n). If f(n) = n + 1, then f(n) = 2016 implies n + 1 = 2016 ⇒ n = 2015. But that's the n we already know, since f(2015) = 2016.Wait, maybe I misinterpreted the question. Perhaps it's asking for n such that f(n) = 2015. Let's check.If f(n) = n + 1, then f(n) = 2015 ⇒ n + 1 = 2015 ⇒ n = 2014.But the problem states f(2015) = 2016, so if f(n) = n + 1, then f(2014) = 2015. Therefore, n = 2014.But let me double-check this using the functional equation to ensure consistency.Given f(n) = n + 1, let's verify f(f(n)) + f(n) = 2n + 3.f(f(n)) = f(n + 1) = (n + 1) + 1 = n + 2.So, f(f(n)) + f(n) = (n + 2) + (n + 1) = 2n + 3, which matches the given equation. Therefore, f(n) = n + 1 is indeed the correct function.Thus, if f(n) = 2015, then n = 2014.Wait, but the problem says "If f(2015) = 2016, what is the value of n for which we are finding f(n)?" So, perhaps it's asking for n such that f(n) = 2016, which would be n = 2015. But that's already given. Alternatively, if it's asking for n such that f(n) = 2015, then n = 2014.I think the confusion arises from the wording. Let me read it again: "If f(2015) = 2016, what is the value of n for which we are finding f(n)?" It seems like it's asking for n such that f(n) = 2016, which is 2015, but that's already provided. Alternatively, perhaps it's asking for n such that f(n) = 2015, which would be 2014.Given that f(n) = n + 1, and f(2015) = 2016, then f(2014) = 2015. Therefore, the value of n for which f(n) = 2015 is 2014.So, the answer is n = 2014.

answer：Okay, so I need to help Howard figure out how much money he made and spent over the weekend. Let me try to understand the problem step by step.First, Howard started with 26 last week. Then, over the weekend, he decided to do two things to make extra money: wash windows and walk dogs. From washing windows, he made W dollars, and from walking dogs, he made D dollars. So, his total earnings from these activities would be W + D.But, he also spent some money. He bought supplies for washing windows, which cost him X dollars, and he bought dog treats for the dogs he walked, which cost him Y dollars. So, his total expenses are X + Y.After all this earning and spending, he ended up with 52. So, I need to create a system of equations that represents this situation.Let me think about how to set this up. The initial amount of money he had was 26. Then he earned W and D, so that's adding W + D to his initial amount. But then he spent X and Y, so that's subtracting X + Y from his total. And after all that, he has 52 left.So, the equation would be:26 + W + D - X - Y = 52Is that right? Let me double-check. He starts with 26, adds his earnings from washing windows (W) and walking dogs (D), then subtracts his expenses for supplies (X) and dog treats (Y), and ends up with 52. Yeah, that seems to make sense.But the problem says to create a system of equations with W, D, X, and Y as unknown variables. So, I need more than one equation, but I only have one equation so far. Hmm, maybe I need to think if there are any other relationships or constraints given in the problem.Looking back, the problem doesn't provide any additional information or constraints about W, D, X, or Y individually. It just gives the total after all transactions. So, with the information given, I can only form one equation.Therefore, the system of equations would consist of just this one equation:26 + W + D - X - Y = 52I don't think there's a way to create more equations without additional information. So, this should be the system that represents the situation.

answer：First, I recognize that the group initially consists of 10 persons, and their average weight is denoted as ( A ). Therefore, the total weight of the group before the new person arrives is ( 10A ).When the new person replaces the individual weighing 65 kg, the total weight of the group changes. The new total weight is calculated by removing the weight of the person who left (65 kg) and adding the weight of the new person (97 kg). This results in a new total weight of ( 10A - 65 + 97 ).Simplifying this, the new total weight becomes ( 10A + 32 ) because ( 97 - 65 = 32 ).Next, the new average weight of the group, denoted as ( A' ), is the new total weight divided by the number of people, which remains 10. Therefore, the new average weight is:[A' = frac{10A + 32}{10} = A + 3.2]To find the increase in average weight, I subtract the original average weight (( A )) from the new average weight (( A' )):[A' - A = (A + 3.2) - A = 3.2]Thus, the average weight increased by 3.2 kg.

answer：First, I need to determine the total number of cans collected, which is 85 cans.LaDonna picked up 25 cans. This information is provided directly.Next, I need to find out how many cans Prikya picked up. The problem states that Prikya picked up twice as many cans as LaDonna. Therefore, I will multiply the number of cans LaDonna picked up by 2:[ text{Prikya's cans} = 2 times 25 = 50 , text{cans} ]Finally, to find out how many cans Yoki picked up, I will subtract the number of cans picked up by LaDonna and Prikya from the total number of cans:[ text{Yoki's cans} = 85 - 25 - 50 = 10 , text{cans} ]Therefore, Yoki picked up boxed{10} cans.

answer：First, I need to evaluate the expression (-|-8|). The absolute value of (-8) is (8), so (-|-8| = -8).Next, I compare (-8) with (-6). On the number line, (-8) is to the left of (-6), which means (-8) is less than (-6).Therefore, the correct comparison is (-|-8| lt -6).