💡Okay, so I have this equation to solve: 16 raised to the power of (2y minus 3) equals (1/4) raised to the power of (y plus 2). Hmm, that looks a bit complicated, but maybe I can simplify it by expressing both sides with the same base. I remember that 16 is a power of 4 because 4 squared is 16. So, 16 can be written as 4². Similarly, 1/4 is the same as 4 to the power of -1 because 4⁻¹ is 1/4. That should help me rewrite the equation with the same base, which is 4.Let me try that. So, replacing 16 with 4², the left side becomes (4²) raised to the (2y - 3). On the right side, replacing 1/4 with 4⁻¹, it becomes (4⁻¹) raised to the (y + 2). Now, using the exponent rule that (a^b)^c equals a^(b*c), I can simplify both sides. On the left side, that would be 4^(2*(2y - 3)) and on the right side, it's 4^(-1*(y + 2)). So, simplifying the exponents, the left side becomes 4^(4y - 6) and the right side becomes 4^(-y - 2). Now that both sides have the same base, 4, I can set the exponents equal to each other because if a^b = a^c, then b must equal c. So, I set 4y - 6 equal to -y - 2.Now, I need to solve for y. Let me write that equation out: 4y - 6 = -y - 2. To get all the y terms on one side, I'll add y to both sides. That gives me 4y + y - 6 = -2, which simplifies to 5y - 6 = -2.Next, I'll add 6 to both sides to isolate the term with y. That gives me 5y = 4.Finally, I'll divide both sides by 5 to solve for y, which gives me y = 4/5.Wait, let me check my steps again to make sure I didn't make a mistake. Starting from the beginning, expressing both sides with base 4 seems correct. Then, applying the exponent rule correctly, I think I did that right. Setting the exponents equal: 4y - 6 = -y - 2. Adding y to both sides: 5y - 6 = -2. Adding 6: 5y = 4. Dividing by 5: y = 4/5. Hmm, that seems consistent.But wait, in the initial problem, the left side was 16^(2y - 3) and the right side was (1/4)^(y + 2). Let me plug y = 4/5 back into the original equation to verify.Calculating the left side: 16^(2*(4/5) - 3) = 16^(8/5 - 15/5) = 16^(-7/5). Calculating the right side: (1/4)^(4/5 + 2) = (1/4)^(4/5 + 10/5) = (1/4)^(14/5).Now, 16 is 4², so 16^(-7/5) is (4²)^(-7/5) = 4^(-14/5). Similarly, (1/4)^(14/5) is 4^(-14/5). So both sides equal 4^(-14/5), which means y = 4/5 is indeed the correct solution. I think I did everything right this time.

💡First, I need to determine how much the distributor receives after the online store deducts its 20% commission from the price the buyer sees, which is 25.50.To calculate the commission amount:20% of 25.50 is 0.20 multiplied by 25.50, which equals 5.10.Next, I subtract the commission from the price to find out how much the distributor actually receives:25.50 minus 5.10 equals 20.40.The distributor purchases the product from the producer at 17 per item. To find the profit, I subtract the cost from the revenue:20.40 minus 17.00 equals 3.40.Finally, to determine the profit percentage, I divide the profit by the cost and multiply by 100:3.40 divided by 17.00 equals 0.2, and multiplying by 100 gives 20%.Therefore, the distributor aims to maintain a 20% profit margin on the cost of the item.

💡Alright, so I've got this problem here with four statements, and I need to figure out which ones are incorrect. Let's take it step by step.First, let's look at statement ①: "-a is always negative." Hmm, okay. So, if I have a number 'a', and I put a negative sign in front of it, is that always negative? Well, that depends on what 'a' is, right? If 'a' is positive, then yes, -a would be negative. But what if 'a' is negative to begin with? For example, if a = -5, then -a would be -(-5) which is 5, which is positive. So, -a isn't always negative; it depends on the value of 'a'. So, statement ① seems incorrect.Moving on to statement ②: "If |a| = |b|, then a = b." Okay, absolute value of 'a' equals absolute value of 'b'. Does that mean 'a' equals 'b'? Well, not necessarily. Because absolute value measures the distance from zero, regardless of direction. So, if |a| = |b|, 'a' could be equal to 'b', or 'a' could be equal to '-b'. For example, if a = 3 and b = -3, then |a| = |b| = 3, but a ≠ b. So, statement ② is also incorrect.Now, statement ③: "A rational number is either an integer or a fraction." Alright, so what's a rational number? A rational number is any number that can be expressed as the quotient of two integers, where the denominator isn't zero. That includes integers because, for example, 5 can be written as 5/1, which is a fraction. It also includes fractions like 1/2 or 3/4. So, this statement seems correct because it accurately describes what a rational number is.Finally, statement ④: "A rational number is either positive or negative." Hmm, so is every rational number either positive or negative? Well, zero is a rational number, right? And zero isn't positive or negative. So, this statement is missing out on the possibility of zero. Therefore, statement ④ is incorrect.So, summarizing:- Statement ①: Incorrect- Statement ②: Incorrect- Statement ③: Correct- Statement ④: IncorrectThat means three statements are incorrect. So, the answer should be option C: 3 statements.

💡Okay, so I have this calculus problem here, and I need to figure out the limit as k approaches 0 of [f(x₀ - k) - f(x₀)] divided by 2k. They also told me that the second derivative of f at x₀ is 2. Hmm, okay. Let me think about how to approach this.First, I remember that limits involving functions and their derivatives often relate to the definition of the derivative itself. The derivative of a function at a point is the limit of the difference quotient as the change in x approaches zero. In this case, the expression given is similar to a difference quotient but with a negative k and a factor of 2 in the denominator.Let me write down the definition of the first derivative to see if that helps. The derivative f'(x₀) is equal to the limit as k approaches 0 of [f(x₀ + k) - f(x₀)] / k. That's the standard definition. But in our problem, we have f(x₀ - k) instead of f(x₀ + k). Maybe I can manipulate that.If I consider f(x₀ - k), that's like shifting x₀ by -k. So, maybe I can rewrite the expression in terms of a positive k. Let me set h = -k. Then, as k approaches 0, h also approaches 0. Substituting, the expression becomes [f(x₀ + h) - f(x₀)] / (-2h). So, that's equal to -1/2 times [f(x₀ + h) - f(x₀)] / h. Wait, that looks familiar. The limit as h approaches 0 of [f(x₀ + h) - f(x₀)] / h is just f'(x₀). So, putting it all together, the original limit is equal to -1/2 times f'(x₀). So, the limit is -f'(x₀)/2.But the problem gives me f''(x₀) = 2. Hmm, how does that relate to f'(x₀)? I know that the second derivative is the derivative of the first derivative, so f''(x₀) = d/dx [f'(x)] evaluated at x₀. But I don't have information about f'(x₀) directly. Is there a way to find f'(x₀) from the given information?Wait, maybe I'm overcomplicating things. The problem only asks for the limit, which we've expressed in terms of f'(x₀). So, unless there's more information, I can't find the exact value of f'(x₀). But hold on, the answer choices are numerical, so maybe I can express the limit in terms of f''(x₀) somehow.Wait, let me think again. If f''(x₀) = 2, that tells me about the concavity of the function at that point, but not directly about the first derivative. Unless there's some relation I'm missing. Maybe I need to use Taylor series expansion or something?Let me try expanding f(x₀ - k) using Taylor series around x₀. The Taylor expansion of f(x₀ - k) up to the second order would be f(x₀) - k f'(x₀) + (k² / 2) f''(x₀) + higher-order terms. So, f(x₀ - k) ≈ f(x₀) - k f'(x₀) + (k² / 2) f''(x₀).Subtracting f(x₀) from both sides, we get f(x₀ - k) - f(x₀) ≈ -k f'(x₀) + (k² / 2) f''(x₀). Then, dividing by 2k, we have [f(x₀ - k) - f(x₀)] / (2k) ≈ [-k f'(x₀) + (k² / 2) f''(x₀)] / (2k).Simplifying the right-hand side, we get [-f'(x₀)/2 + (k / 4) f''(x₀)]. Now, taking the limit as k approaches 0, the term with k in it will go to zero, leaving us with -f'(x₀)/2.So, the limit is -f'(x₀)/2. But we still don't know f'(x₀). Wait, the problem only gives us f''(x₀) = 2. Is there a way to find f'(x₀) from that? Or maybe I made a mistake earlier.Wait, let me go back. When I did the substitution h = -k, I ended up with the limit being -f'(x₀)/2. So, unless f'(x₀) is related to f''(x₀), which it isn't directly, unless we have more information about the function f.Wait, hold on. Maybe I misapplied the Taylor series. Let me double-check that. The expansion of f(x₀ - k) is f(x₀) - k f'(x₀) + (k² / 2) f''(x₀) - ... So, that seems correct.Subtracting f(x₀) gives -k f'(x₀) + (k² / 2) f''(x₀). Dividing by 2k gives (-f'(x₀)/2 + (k / 4) f''(x₀)). So, as k approaches 0, the second term goes away, leaving -f'(x₀)/2.So, unless f'(x₀) is given, or can be inferred from f''(x₀), which is 2, I can't find the exact value. But the answer choices are numbers, so maybe I need to make an assumption or perhaps I missed something.Wait, maybe the problem is expecting me to recognize that the limit is related to the derivative, but since f''(x₀) is given, maybe it's a trick question where the limit is actually -1, because f''(x₀) is 2, so f'(x₀) is 2? Wait, no, that doesn't make sense because f''(x₀) is the derivative of f'(x), not f'(x₀) itself.Wait, hold on. Maybe I need to use the definition of the second derivative. The second derivative is the limit as k approaches 0 of [f'(x₀ + k) - f'(x₀)] / k. But I don't have information about f'(x₀ + k). Hmm.Alternatively, maybe I can use the expression for the second derivative in terms of the function values. There's a formula for the second derivative using central differences: f''(x₀) ≈ [f(x₀ + k) - 2f(x₀) + f(x₀ - k)] / k². But in our case, we have [f(x₀ - k) - f(x₀)] / (2k). That's different.Wait, maybe I can relate the two. Let me write down the central difference formula for f''(x₀):f''(x₀) ≈ [f(x₀ + k) - 2f(x₀) + f(x₀ - k)] / k².Given that f''(x₀) = 2, we can write:2 ≈ [f(x₀ + k) - 2f(x₀) + f(x₀ - k)] / k².But our limit is [f(x₀ - k) - f(x₀)] / (2k). Let me see if I can express [f(x₀ - k) - f(x₀)] in terms of the central difference formula.From the central difference formula:f(x₀ + k) - 2f(x₀) + f(x₀ - k) ≈ 2k².So, rearranging, f(x₀ - k) ≈ 2f(x₀) - f(x₀ + k) + 2k².But I'm not sure if that helps. Alternatively, maybe I can express [f(x₀ - k) - f(x₀)] as - [f(x₀) - f(x₀ - k)].Wait, let me think about the first derivative again. The first derivative is the limit as k approaches 0 of [f(x₀ + k) - f(x₀)] / k. Similarly, it's also equal to the limit as k approaches 0 of [f(x₀) - f(x₀ - k)] / k. So, that means [f(x₀) - f(x₀ - k)] / k approaches f'(x₀) as k approaches 0.So, [f(x₀ - k) - f(x₀)] / k is equal to - [f(x₀) - f(x₀ - k)] / k, which approaches -f'(x₀). Therefore, [f(x₀ - k) - f(x₀)] / (2k) approaches -f'(x₀)/2.So, the limit is -f'(x₀)/2. But we don't know f'(x₀). Hmm, unless f'(x₀) is related to f''(x₀). But f''(x₀) is the derivative of f'(x), so unless we have more information about f'(x) near x₀, we can't determine f'(x₀) from f''(x₀).Wait, but the answer choices are numbers, so maybe I need to make an assumption or perhaps the problem is designed so that f'(x₀) is equal to f''(x₀), but that doesn't make sense because f''(x₀) is the rate of change of f'(x). Unless f'(x) is linear, but we don't know that.Wait, maybe I made a mistake earlier in interpreting the problem. Let me read it again: "If f''(x₀) = 2, then lim_{k→0} [f(x₀ - k) - f(x₀)] / (2k) equals (　　)".So, the limit is expressed in terms of f''(x₀). But from my earlier reasoning, the limit is -f'(x₀)/2. So, unless f'(x₀) is given or can be inferred, I can't get a numerical answer. But the answer choices are numbers, so perhaps I need to reconsider.Wait, maybe I need to use the second derivative in some way. Let me think about expanding f(x₀ - k) using the second derivative. Earlier, I did a Taylor expansion up to the second order, which gave me f(x₀ - k) ≈ f(x₀) - k f'(x₀) + (k² / 2) f''(x₀). So, subtracting f(x₀), we have f(x₀ - k) - f(x₀) ≈ -k f'(x₀) + (k² / 2) f''(x₀). Dividing by 2k, we get (-f'(x₀)/2 + (k / 4) f''(x₀)). So, as k approaches 0, the term with k goes away, leaving -f'(x₀)/2.But again, without knowing f'(x₀), I can't find the numerical value. Wait, unless f'(x₀) is zero? But that's not given. Hmm.Wait, maybe I need to consider higher-order terms. If I include the third derivative term, perhaps I can relate it to f''(x₀). But that seems complicated and probably not necessary for this problem.Wait, let me think differently. Maybe the limit is actually the definition of the derivative, but in reverse. The expression [f(x₀ - k) - f(x₀)] / (2k) is similar to the derivative, but with a negative step and a factor of 2. So, as k approaches 0, this limit should be equal to -f'(x₀)/2.But since f''(x₀) is given, maybe f'(x₀) is related to f''(x₀) in some way. For example, if f''(x) is constant, then f'(x) would be linear, and f(x) would be quadratic. So, if f''(x₀) = 2, then f'(x) = 2x + C, where C is a constant. But without knowing C, we can't determine f'(x₀). Hmm.Wait, unless the function is symmetric or something. But I don't think that's given. Maybe I'm overcomplicating this. Let me go back to the original limit.The limit is [f(x₀ - k) - f(x₀)] / (2k) as k approaches 0. Let me write this as (-1/2) * [f(x₀) - f(x₀ - k)] / k. So, that's (-1/2) times the difference quotient for f'(x₀). Therefore, the limit is (-1/2) * f'(x₀).So, the limit is -f'(x₀)/2. But we don't know f'(x₀). However, the answer choices are numbers, so maybe f'(x₀) is 2? But why? Because f''(x₀) is 2. But f''(x₀) is the derivative of f'(x) at x₀, not f'(x₀) itself.Wait, unless f'(x) is equal to f''(x) at x₀, which would mean f'(x₀) = f''(x₀) = 2. But that's not necessarily true. For example, if f(x) = x², then f'(x) = 2x and f''(x) = 2. At x₀ = 0, f'(0) = 0 and f''(0) = 2. So, in that case, f'(x₀) is 0, not 2.Wait, so in that example, f'(x₀) is 0, and f''(x₀) is 2. So, the limit would be -0/2 = 0. But 0 isn't one of the answer choices. Hmm, that's confusing.Wait, maybe I need to choose a different function where f'(x₀) is related to f''(x₀). For example, if f(x) = e^{2x}, then f'(x) = 2e^{2x}, and f''(x) = 4e^{2x}. So, at x₀, f''(x₀) = 4e^{2x₀} = 2, so e^{2x₀} = 0.5, so x₀ = (ln 0.5)/2. Then f'(x₀) = 2e^{2x₀} = 2*(0.5) = 1. So, in this case, f'(x₀) = 1, so the limit would be -1/2, which is option D. But in the previous example, the limit was 0, which isn't an option. So, this is conflicting.Wait, maybe I need to think differently. Perhaps the limit is actually related to the second derivative. Let me see. If I take the limit as k approaches 0 of [f(x₀ - k) - f(x₀)] / (2k), which is equal to (-1/2) f'(x₀). But if I also consider the second derivative, which is the limit as k approaches 0 of [f'(x₀ + k) - f'(x₀)] / k. But I don't have information about f'(x₀ + k).Alternatively, maybe I can use the fact that f''(x₀) = 2 to find f'(x₀) by integrating. If f''(x) = 2, then f'(x) = 2x + C. But without knowing f'(x) at some point, I can't determine C. So, unless x₀ is 0, which isn't specified, I can't find C.Wait, maybe the problem assumes that f'(x₀) = f''(x₀), which would make f'(x₀) = 2. Then the limit would be -2/2 = -1, which is option A. But that's an assumption, and I don't think that's necessarily true.Alternatively, maybe the problem is designed so that the limit is -1, regardless of f'(x₀). But that doesn't make sense because the limit depends on f'(x₀).Wait, maybe I need to use L’Hospital’s Rule. Let me see. If I rewrite the limit as [f(x₀ - k) - f(x₀)] / (2k). As k approaches 0, both numerator and denominator approach 0, so it's a 0/0 indeterminate form. Therefore, I can apply L’Hospital’s Rule by taking derivatives of numerator and denominator with respect to k.So, derivative of numerator with respect to k is f’(x₀ - k)*(-1) - 0 = -f’(x₀ - k). Derivative of denominator is 2. So, the limit becomes lim_{k→0} [-f’(x₀ - k)] / 2. As k approaches 0, x₀ - k approaches x₀, so this limit is -f’(x₀)/2.Again, same result. So, unless f’(x₀) is known, I can't get a numerical answer. But the answer choices are numbers, so maybe f’(x₀) is 2? But why?Wait, maybe I need to consider that f''(x₀) = 2 implies that f’(x) is increasing at x₀ with a rate of 2. But that doesn't give me f’(x₀) directly.Wait, unless the function is such that f’(x₀) = f''(x₀), which would make f’(x₀) = 2. But that's not a general rule. For example, in the function f(x) = x², f''(x) = 2, but f’(x) = 2x, which is not equal to 2 unless x = 1.Wait, maybe the problem is designed in such a way that f’(x₀) = f''(x₀). So, f’(x₀) = 2. Then, the limit is -2/2 = -1, which is option A.Alternatively, maybe I'm missing something in the problem statement. Let me read it again: "If f''(x₀) = 2, then lim_{k→0} [f(x₀ - k) - f(x₀)] / (2k) equals (　　)".So, the problem gives f''(x₀) = 2 and asks for the limit, which we've determined is -f’(x₀)/2. But without knowing f’(x₀), we can't find the numerical value. However, the answer choices are numbers, so perhaps the problem assumes that f’(x₀) = 2, making the limit -1.Alternatively, maybe the problem is testing the understanding that the limit is -f’(x₀)/2, and since f''(x₀) = 2, perhaps f’(x₀) is related in a way that makes the limit -1. But I'm not sure.Wait, maybe I can think of it differently. If f''(x₀) = 2, then near x₀, f'(x) ≈ f'(x₀) + 2(x - x₀). So, if I take the limit as k approaches 0, f'(x₀ - k) ≈ f'(x₀) + 2(-k). But I don't see how that helps.Wait, going back to L’Hospital’s Rule, the limit is -f’(x₀)/2. So, unless f’(x₀) is 2, the limit is -1. But why would f’(x₀) be 2? Maybe the problem assumes that f’(x₀) = f''(x₀), but that's not a standard assumption.Alternatively, maybe the problem is designed so that the limit is -1, regardless of f’(x₀), but that doesn't make sense because the limit depends on f’(x₀).Wait, maybe I made a mistake in the Taylor expansion. Let me double-check. The expansion of f(x₀ - k) is f(x₀) - k f’(x₀) + (k² / 2) f''(x₀) - ... So, subtracting f(x₀), we get -k f’(x₀) + (k² / 2) f''(x₀). Dividing by 2k, we get (-f’(x₀)/2 + (k / 4) f''(x₀)). So, as k approaches 0, the limit is -f’(x₀)/2.So, unless f’(x₀) is 2, the limit is -1. But why would f’(x₀) be 2? Maybe the problem assumes that f’(x₀) = f''(x₀), which would make f’(x₀) = 2. Then, the limit is -1.Alternatively, maybe the problem is designed to test the understanding that the limit is -f’(x₀)/2, and since f''(x₀) is given, perhaps f’(x₀) is 2, making the limit -1.But I'm not entirely sure. Maybe I should look for another approach. Let me consider the definition of the second derivative. The second derivative can be expressed as the limit as k approaches 0 of [f(x₀ + k) - 2f(x₀) + f(x₀ - k)] / k². Given that f''(x₀) = 2, we have:2 = lim_{k→0} [f(x₀ + k) - 2f(x₀) + f(x₀ - k)] / k².But our limit is [f(x₀ - k) - f(x₀)] / (2k). Let me see if I can relate these two expressions.From the central difference formula, we have:f''(x₀) = lim_{k→0} [f(x₀ + k) - 2f(x₀) + f(x₀ - k)] / k².So, 2 = lim_{k→0} [f(x₀ + k) - 2f(x₀) + f(x₀ - k)] / k².Let me rearrange this:[f(x₀ + k) - 2f(x₀) + f(x₀ - k)] = 2k² + o(k²).So, [f(x₀ - k) - f(x₀)] = - [f(x₀) - f(x₀ - k)] = - [k f’(x₀) - (k² / 2) f''(x₀) + ...].Wait, that's similar to the Taylor expansion. So, [f(x₀ - k) - f(x₀)] ≈ -k f’(x₀) + (k² / 2) f''(x₀).So, [f(x₀ - k) - f(x₀)] / (2k) ≈ (-k f’(x₀) + (k² / 2) f''(x₀)) / (2k) = (-f’(x₀)/2 + (k / 4) f''(x₀)).As k approaches 0, the term with k goes away, leaving -f’(x₀)/2.So, again, the limit is -f’(x₀)/2. But without knowing f’(x₀), I can't find the numerical value. However, the answer choices are numbers, so maybe f’(x₀) is 2, making the limit -1.Alternatively, maybe the problem is designed to recognize that the limit is -1, assuming f’(x₀) = 2. But I'm not sure if that's a valid assumption.Wait, maybe I can think of a specific function where f''(x₀) = 2 and compute the limit to see what it is. Let's take f(x) = x². Then, f''(x) = 2 for all x, so f''(x₀) = 2. Then, f'(x) = 2x. So, at x₀, f'(x₀) = 2x₀. Then, the limit is -f'(x₀)/2 = -x₀.But unless x₀ is 1, the limit isn't necessarily -1. For example, if x₀ = 0, the limit is 0. If x₀ = 1, the limit is -1. So, unless x₀ is 1, the limit isn't -1. But the problem doesn't specify x₀, just that f''(x₀) = 2.Wait, maybe the problem assumes that x₀ is such that f'(x₀) = 2, making the limit -1. But that's an assumption.Alternatively, maybe the problem is designed to recognize that the limit is -1, regardless of f'(x₀), but that doesn't make sense because the limit depends on f'(x₀).Wait, maybe I'm overcomplicating this. Let me go back to the original limit:lim_{k→0} [f(x₀ - k) - f(x₀)] / (2k).This is equal to (-1/2) * lim_{k→0} [f(x₀) - f(x₀ - k)] / k, which is (-1/2) * f'(x₀).So, the limit is -f'(x₀)/2. But since f''(x₀) = 2, which is the derivative of f'(x) at x₀, we can write f''(x₀) = lim_{h→0} [f'(x₀ + h) - f'(x₀)] / h = 2.But that doesn't directly give us f'(x₀). Unless we have more information about f'(x) near x₀, we can't determine f'(x₀).Wait, maybe the problem is designed to recognize that the limit is -1, assuming that f'(x₀) = 2. So, the answer is A: -1.But I'm not entirely sure. Maybe I should go with that, given the answer choices.

💡Alright, so I've got this problem about a 100x100 board where each cell is colored either black or white. All the cells on the boundary are black, and there are no monochromatic 2x2 squares. I need to prove that there's at least one 2x2 square that's colored in a checkerboard pattern. Hmm, okay, let's break this down.First off, the board is pretty large—100x100. That's a lot of cells! But maybe the size isn't too important right now. The key points are that all boundary cells are black, and there are no monochromatic 2x2 squares. So, every 2x2 square must have at least one black and one white cell. But the problem wants me to show that there's at least one 2x2 square that's a perfect checkerboard—meaning it alternates colors like a chessboard.Let me think about what a checkerboard pattern looks like. It's a 2x2 square where the top-left and bottom-right are one color, and the top-right and bottom-left are the other color. So, it's not monochromatic, obviously, but it's also not just any mix of colors—it's got a specific alternating pattern.Since all the boundary cells are black, that means the first and last rows and columns are all black. So, the cells adjacent to the edges are all black. That might influence the coloring of the cells just inside the boundary.Now, the problem says there are no monochromatic 2x2 squares. So, every 2x2 square must have at least one black and one white cell. But we need to show that at least one of these squares is a perfect checkerboard.Maybe I can approach this by contradiction. Suppose that there are no checkerboard 2x2 squares. Then, every 2x2 square must have either three blacks and one white or three whites and one black. But wait, if all boundary cells are black, then the cells adjacent to the boundary are also black. So, the first few cells inside the boundary are black as well.Wait, but if I have a 2x2 square that's adjacent to the boundary, it's going to have three black cells and one white cell, right? Because the boundary cells are all black. So, maybe that's the case for squares near the boundary.But as we move further into the board, away from the boundary, the coloring could be more mixed. But the problem states that there are no monochromatic 2x2 squares, so every 2x2 square must have at least one black and one white cell.If I assume that there are no checkerboard patterns, then every 2x2 square must have either three blacks and one white or three whites and one black. But given that the boundary is all black, the number of black cells is going to be higher near the edges.But how does this help me? Maybe I can count the number of color changes or something like that. Let's think about the number of times the color changes from black to white or white to black in the rows and columns.If I consider a single row, starting from the left boundary, which is black. The next cell could be black or white. If it's black, then the color hasn't changed. If it's white, then it's a color change. Similarly, moving along the row, each time the color changes, that's a separator between black and white regions.Similarly, in columns, starting from the top boundary, which is black, the next cell could be black or white, and so on.Now, if I consider the entire board, the number of color changes in each row and each column must be even. Why? Because the first and last cells in each row and column are black, so the number of times the color changes back and forth must be even to end up back at black.So, for each row, the number of color changes is even, and the same for each column. Therefore, the total number of color changes in the entire board is even.But how does this relate to the 2x2 squares? Well, each 2x2 square contributes to the color changes in its row and column. If a 2x2 square has a checkerboard pattern, it contributes two color changes in both its row and column. If it has three of one color and one of the other, it contributes one color change in both its row and column.Wait, let me think about that again. If a 2x2 square has a checkerboard pattern, then in each row of the square, the colors alternate, so there's one color change per row. Similarly, in each column, there's one color change per column. So, overall, for the entire board, each checkerboard 2x2 square contributes two color changes (one in each row and column).On the other hand, if a 2x2 square has three of one color and one of the other, then in each row, there's either one color change or none, depending on the arrangement. Similarly, in the columns. So, overall, such a square might contribute one or zero color changes.But I'm not sure if this line of reasoning is leading me anywhere. Maybe I need to think differently.Let me consider the entire board and count the number of 2x2 squares. There are 99x99 = 9801 such squares. Each of these squares must have at least one black and one white cell, but we're assuming none of them are checkerboards.So, each 2x2 square must have either three blacks and one white or three whites and one black. Now, let's think about the total number of black and white cells in the entire board.The boundary cells are all black, so there are 4*100 - 4 = 396 boundary cells (subtracting the four corners which are counted twice). So, there are 396 black cells on the boundary. The remaining cells are 100*100 - 396 = 9604 cells, which can be either black or white.Now, if we consider the entire board, the total number of black cells is at least 396, but could be more. Similarly, the number of white cells is at most 9604.But how does this relate to the 2x2 squares? Maybe I can use some kind of averaging argument. If each 2x2 square has either three blacks and one white or three whites and one black, then the total number of black cells can be expressed in terms of the number of 2x2 squares.Wait, but each cell is part of multiple 2x2 squares. Specifically, each cell (except those on the last row and column) is part of four 2x2 squares. So, if I sum over all 2x2 squares, the total number of black cells counted across all squares would be 4 times the number of black cells in the entire board minus the black cells on the last row and column.But this seems complicated. Maybe there's a better way.Let me think about the parity of the number of black cells. Since all boundary cells are black, and the number of boundary cells is even (396), the total number of black cells is even plus the number of black cells inside the boundary. But I'm not sure if that helps.Wait, maybe I can use the fact that the number of color changes is even. If I consider the entire board, the number of color changes in rows and columns must be even. But if there are no checkerboard 2x2 squares, then each 2x2 square contributes an odd number of color changes, which might lead to a contradiction.Hmm, I'm not entirely sure. Maybe I need to think about the dual graph of the board, where each cell is a vertex and edges connect adjacent cells. Then, the coloring corresponds to a 2-coloring of the vertices. But I'm not sure if that helps either.Wait, another idea: if there are no checkerboard 2x2 squares, then the coloring must be such that every 2x2 square has three of one color and one of the other. This might imply that the coloring is "mostly" one color, but given that the boundary is all black, maybe the interior is mostly white? But I don't know.Alternatively, maybe I can use induction. Start with a smaller board and see if the property holds, then try to generalize. But a 100x100 board is quite large, so induction might not be straightforward.Wait, another thought: if there are no checkerboard 2x2 squares, then the coloring must be such that in every 2x2 square, the majority color is either black or white. But since the boundary is all black, maybe this creates a sort of "wave" of black cells inward, but I'm not sure.I'm getting stuck here. Maybe I need to look for a different approach. Let's think about the fact that there are no monochromatic 2x2 squares. So, every 2x2 square has at least one black and one white cell. But we need to show that at least one of them is a perfect checkerboard.Suppose, for contradiction, that there are no checkerboard 2x2 squares. Then, every 2x2 square has either three blacks and one white or three whites and one black.Now, consider the entire board. Each cell is part of multiple 2x2 squares. Specifically, each cell (except those on the last row and column) is part of four 2x2 squares. So, if I sum over all 2x2 squares, the total number of black cells counted across all squares would be 4 times the number of black cells in the entire board minus the black cells on the last row and column.But I'm not sure if this helps. Maybe I can consider the total number of black cells in all 2x2 squares. If each 2x2 square has either three blacks and one white or three whites and one black, then the total number of black cells across all squares is either 3 or 1 per square.But there are 9801 squares, so the total number of black cells counted across all squares is between 9801 and 29403. But the actual number of black cells in the entire board is 396 plus some number inside.Wait, but each black cell is counted multiple times, depending on how many 2x2 squares it's part of. So, maybe I can set up an equation.Let B be the total number of black cells in the entire board. Then, the total number of black cells counted across all 2x2 squares is 4*(B - 99 - 99) + 2*(99 + 99) + 396. Wait, that seems complicated.Alternatively, maybe I can use the fact that each black cell is counted in four 2x2 squares, except those on the edges. So, the total number of black cells across all 2x2 squares is 4*(B - 2*99) + 3*(2*99) + 2*99.Wait, I'm not sure. Maybe I need to think differently.Let me consider the dual problem: instead of counting black cells, count the number of color changes. Since the number of color changes is even, and each checkerboard 2x2 square contributes two color changes, while other squares contribute one or three, maybe I can get a contradiction.But I'm not sure. Maybe I need to think about the fact that if there are no checkerboard squares, then the coloring must be such that every 2x2 square has a majority of one color, which might lead to a contradiction with the boundary conditions.Wait, another idea: consider the four corners of the board. They are all black. Now, look at the cells adjacent to the corners. Since the boundary is all black, the cells adjacent to the corners are also black. So, the first few cells inside the boundary are black.But then, moving inward, if every 2x2 square has three blacks and one white, then the next layer inward would have more blacks than whites. But this might create a situation where the entire board is mostly black, but I'm not sure.Alternatively, maybe I can use the fact that the number of black cells must be even or odd, but I'm not sure.Wait, maybe I can use the fact that the number of color changes is even. If I assume there are no checkerboard squares, then each 2x2 square contributes an odd number of color changes, which might make the total number of color changes odd, contradicting the fact that it must be even.But I'm not sure if that's true. Let me think: if a 2x2 square has three blacks and one white, how many color changes does it contribute? In each row, there's one color change, and in each column, there's one color change. So, total color changes per such square is two, which is even.Wait, but if there are no checkerboard squares, then every 2x2 square contributes two color changes. So, the total number of color changes would be 2*9801 = 19602, which is even. But earlier, I thought the total number of color changes must be even because the first and last cells in each row and column are black. So, that doesn't lead to a contradiction.Hmm, maybe I need to think differently.Wait, another approach: consider the fact that the boundary is all black, so the first and last rows and columns are all black. Now, look at the second row. Since the first row is all black, the second row must have at least one white cell to avoid a monochromatic 2x2 square. Similarly, the second column must have at least one white cell.But if the second row has a white cell, then the cell below it in the third row must be black to avoid a monochromatic 2x2 square. Similarly, the cell to the right of the white cell in the second row must be black.Wait, this seems like it's creating a checkerboard pattern. If the second row has a white cell, then the third row must have a black cell in the same column, and the cell to the right of the white cell must be black. This could propagate a checkerboard pattern inward.But I'm not sure if this necessarily leads to a checkerboard 2x2 square. Maybe it does, but I'm not entirely certain.Alternatively, maybe I can use the fact that the coloring must alternate in some way to avoid monochromatic squares, which would lead to a checkerboard pattern.Wait, another idea: consider the dual graph of the board, where each cell is a vertex and edges connect adjacent cells. Then, the coloring corresponds to a 2-coloring of the vertices. If there are no monochromatic 2x2 squares, then every 2x2 square has both colors, which is already satisfied by the checkerboard pattern.But I'm not sure if that helps.Wait, maybe I can use the fact that the boundary is all black, so the first row is all black. Then, the second row must have at least one white cell. Let's say the second row has a white cell at position (2, k). Then, the cell below it at (3, k) must be black to avoid a monochromatic 2x2 square with the black cells at (1, k) and (2, k).Similarly, the cell to the right of (2, k) at (2, k+1) must be black to avoid a monochromatic 2x2 square with the black cells at (2, k) and (2, k+1).But then, the cell at (3, k+1) must be white to avoid a monochromatic 2x2 square with the black cells at (3, k) and (2, k+1).This seems like it's creating a checkerboard pattern starting from the white cell at (2, k). So, maybe this forces a checkerboard pattern in the vicinity.But does this necessarily lead to a full checkerboard 2x2 square? It seems like it does, because the cells around (2, k) would alternate colors, forming a checkerboard.Wait, let me visualize this. If (2, k) is white, then (3, k) is black, (2, k+1) is black, and (3, k+1) is white. So, the 2x2 square formed by (2, k), (2, k+1), (3, k), (3, k+1) is a checkerboard.Yes! So, if there's a white cell in the second row, it forces the cells around it to alternate colors, creating a checkerboard 2x2 square.But wait, the problem states that there are no monochromatic 2x2 squares, but it doesn't say anything about checkerboard squares. So, if we have a white cell in the second row, it necessarily creates a checkerboard square, which is what we wanted to prove.Therefore, there must be at least one checkerboard 2x2 square on the board.I think that makes sense. By assuming that there's a white cell in the second row (which must exist to avoid a monochromatic square with the black boundary), we end up creating a checkerboard pattern in that area. So, such a square must exist.

💡First, I note that there are initially 10 students on the bus.At the first stop, 3 students get off, reducing the number of students to 7. Then, 2 students get on, bringing the total to 9 students.Moving to the second stop, 1 student gets off, leaving 8 students on the bus. After that, 4 students get on, increasing the total to 12 students.Finally, at the third stop, 2 students get off, resulting in 10 students remaining on the bus. Then, 3 students get on, bringing the final count to 13 students.Therefore, after all stops, there are 13 students on the bus.