💡Alright, so I've got this system of inequalities to solve, and I need to find all the integer solutions. Let me take it step by step because I'm still getting comfortable with inequalities.First, the system is:left{begin{array}{l}frac{x - 1}{2} geq frac{x - 2}{3} 2x - 5 < -3xend{array}right.Okay, so there are two inequalities here. I need to solve each one separately and then find the values of x that satisfy both.Starting with the first inequality:frac{x - 1}{2} geq frac{x - 2}{3}Hmm, fractions can be tricky. Maybe I can eliminate the denominators by finding a common multiple. The denominators are 2 and 3, so the least common multiple is 6. If I multiply both sides of the inequality by 6, that should get rid of the fractions.Multiplying both sides by 6:6 times frac{x - 1}{2} geq 6 times frac{x - 2}{3}Simplifying both sides:3(x - 1) geq 2(x - 2)Alright, now distribute the 3 and the 2:3x - 3 geq 2x - 4Now, I want to get all the x terms on one side and the constants on the other. Let's subtract 2x from both sides:3x - 2x - 3 geq -4Which simplifies to:x - 3 geq -4Now, add 3 to both sides to isolate x:x geq -1Okay, so from the first inequality, x is greater than or equal to -1.Now, moving on to the second inequality:2x - 5 < -3xI need to solve for x here as well. Let's get all the x terms on one side. I'll add 3x to both sides:2x + 3x - 5 < 0Simplifying:5x - 5 < 0Now, add 5 to both sides:5x < 5Finally, divide both sides by 5:x < 1So, from the second inequality, x is less than 1.Now, combining both inequalities:From the first inequality: x ≥ -1From the second inequality: x < 1So, the solution set is all real numbers x such that -1 ≤ x < 1.But the problem asks for all integer solutions. So, let's list the integers within this interval.The integers between -1 and 1 are -1, 0. Wait, 1 is not included because x < 1, so 1 is excluded.Therefore, the integer solutions are -1 and 0.Let me double-check to make sure I didn't make any mistakes.For the first inequality:If x = -1:frac{-1 - 1}{2} = frac{-2}{2} = -1frac{-1 - 2}{3} = frac{-3}{3} = -1So, -1 ≥ -1, which is true.If x = 0:frac{0 - 1}{2} = frac{-1}{2} = -0.5frac{0 - 2}{3} = frac{-2}{3} ≈ -0.666...So, -0.5 ≥ -0.666..., which is true.If x = 1:frac{1 - 1}{2} = 0frac{1 - 2}{3} = frac{-1}{3} ≈ -0.333...So, 0 ≥ -0.333..., which is true, but x = 1 is not included because the second inequality says x < 1.For the second inequality:If x = -1:2(-1) - 5 = -2 - 5 = -7-3(-1) = 3So, -7 < 3, which is true.If x = 0:2(0) - 5 = -5-3(0) = 0So, -5 < 0, which is true.If x = 1:2(1) - 5 = 2 - 5 = -3-3(1) = -3So, -3 < -3 is false because -3 is not less than -3.Therefore, x = 1 is not a solution.So, the integer solutions are indeed -1 and 0.

💡Okay, so I have this trigonometry problem here: Given that sin α minus cos α equals the square root of 2, and α is between 0 and π, I need to find the value of tan α. Hmm, let me think about how to approach this.First, I remember that when dealing with equations involving sine and cosine, sometimes it's helpful to use trigonometric identities to simplify them. One identity that comes to mind is the sine of a difference formula, which is sin(A - B) = sin A cos B - cos A sin B. Maybe I can rewrite sin α - cos α in a form that resembles this identity.Let me see. If I factor out a common term from sin α - cos α, perhaps I can express it as a multiple of sin(α - something). To do that, I think I need to factor out √2 because sin and cos have coefficients of 1, and √2 is a common factor when dealing with such expressions.So, if I factor out √2 from sin α - cos α, it would look like √2 times (sin α / √2 - cos α / √2). That simplifies to √2 times (sin α * (√2/2) - cos α * (√2/2)). Wait, √2/2 is equal to sin π/4 and cos π/4, right? Because sin π/4 is √2/2 and cos π/4 is also √2/2.So, substituting that in, I get √2 times (sin α * sin π/4 - cos α * cos π/4). Now, looking at the expression inside the parentheses, it resembles the sine of a difference formula. Specifically, sin(A - B) = sin A cos B - cos A sin B. But in my case, it's sin α * sin π/4 - cos α * cos π/4, which is actually sin(α - π/4) but with a negative sign because of the order.Wait, let me double-check that. The sine of (α - π/4) would be sin α cos π/4 - cos α sin π/4, which is exactly what I have inside the parentheses. So, sin(α - π/4) equals sin α cos π/4 - cos α sin π/4. Therefore, my expression becomes √2 times sin(α - π/4).So, putting it all together, sin α - cos α equals √2 sin(α - π/4). According to the problem, this equals √2. So, I have √2 sin(α - π/4) equals √2. If I divide both sides by √2, I get sin(α - π/4) equals 1.Now, when does sin θ equal 1? That happens when θ is π/2 plus any multiple of 2π, but since α is between 0 and π, I need to find θ such that α - π/4 equals π/2. So, solving for α, I get α equals π/2 + π/4, which is 3π/4.So, α is 3π/4. Now, I need to find tan α. The tangent of 3π/4. I remember that 3π/4 is in the second quadrant where tangent is negative. Also, tan(π/4) is 1, so tan(3π/4) should be -1.Let me verify that. The tangent function is sin over cos. So, sin(3π/4) is √2/2, and cos(3π/4) is -√2/2. Therefore, tan(3π/4) is (√2/2) divided by (-√2/2), which simplifies to -1. Yep, that checks out.So, putting it all together, starting from sin α - cos α equals √2, I used the sine difference identity to rewrite the equation, solved for α, and then found tan α to be -1.I think that makes sense. I don't see any mistakes in my reasoning. I used the identity correctly, solved for α within the given interval, and correctly evaluated the tangent function at that angle. So, I'm confident that the answer is -1.**Final Answer**The value of tan alpha is boxed{-1}.

💡Alright, so I have this trigonometry problem here. It says that sinθ cosθ equals 1/8, and θ is between π/4 and π/2. I need to find the value of cosθ minus sinθ. Hmm, okay, let me think about how to approach this.First, I remember that sinθ cosθ is related to the double-angle identity. Specifically, sin(2θ) equals 2 sinθ cosθ. So, if sinθ cosθ is 1/8, then sin(2θ) would be 2*(1/8) which is 1/4. That seems useful. So, sin(2θ) = 1/4.Now, θ is between π/4 and π/2. If I multiply that by 2, 2θ would be between π/2 and π. So, 2θ is in the second quadrant where sine is positive, which matches our result that sin(2θ) is 1/4, a positive value. That makes sense.But how does that help me find cosθ - sinθ? Maybe I can square the expression cosθ - sinθ to make use of some identities. Let me try that.So, (cosθ - sinθ)^2 equals cos²θ - 2 sinθ cosθ + sin²θ. I know that cos²θ + sin²θ equals 1, so that simplifies part of it. So, substituting that in, I get 1 - 2 sinθ cosθ. We already know that sinθ cosθ is 1/8, so substituting that in, it becomes 1 - 2*(1/8). Let's calculate that: 2*(1/8) is 1/4, so 1 - 1/4 is 3/4. Therefore, (cosθ - sinθ)^2 equals 3/4.Now, to find cosθ - sinθ, we take the square root of 3/4. The square root of 3 is irrational, so it's √3 over 2. But wait, do we take the positive or negative square root?Looking back at the original problem, θ is between π/4 and π/2. In this interval, cosθ is less than sinθ because as θ increases from π/4 to π/2, sinθ increases and cosθ decreases. So, cosθ - sinθ would be negative in this interval. Therefore, we take the negative square root.So, cosθ - sinθ equals -√3/2. Let me double-check that. If θ is in the first quadrant, both sinθ and cosθ are positive, but since cosθ is less than sinθ here, their difference is negative. Squaring it gave us 3/4, so taking the negative square root makes sense.Wait, just to make sure I didn't make a mistake earlier, let me recap:1. Given sinθ cosθ = 1/8.2. Squared (cosθ - sinθ) to get 1 - 2*(1/8) = 3/4.3. Took the square root, considering the sign based on θ's quadrant.Everything seems to check out. So, yeah, the answer should be -√3/2.**Final Answer**boxed{-frac{sqrt{3}}{2}}

💡Okay, so I have this problem about a geometric sequence where the product of each term and its next term is always a constant, which is called the common product. The sequence is {a_n}, and we're given that a_6 = 2 and the common product is 6. We need to find the value of the product a_1 * a_5 * a_9 * ... * a_{2005} * a_{2009}. The options are A: 2^{502}, B: 3^{502}, C: 2^{503}, D: 3^{503}.Alright, let's break this down step by step. First, I need to understand what a geometric sequence is in this context. Normally, a geometric sequence is one where each term is a constant multiple of the previous term, meaning a_n = a_1 * r^{n-1}, where r is the common ratio. But in this problem, it's defined differently: the product of each term and its next term is constant. So, a_n * a_{n+1} = constant, which is 6 in this case.So, for this sequence, a_1 * a_2 = 6, a_2 * a_3 = 6, a_3 * a_4 = 6, and so on. That means each pair of consecutive terms multiplies to 6. Interesting. So, unlike a standard geometric sequence where each term is a multiple of the previous one, here each term is related to the next term by a product.Given that a_6 = 2, we can work backwards or forwards to find the other terms. Let's see. Since a_5 * a_6 = 6, and a_6 is 2, then a_5 must be 6 / 2 = 3. Similarly, a_4 * a_5 = 6, so a_4 = 6 / a_5 = 6 / 3 = 2. Continuing this pattern, a_3 = 6 / a_4 = 6 / 2 = 3, a_2 = 6 / a_3 = 6 / 3 = 2, and a_1 = 6 / a_2 = 6 / 2 = 3.So, it seems like the terms alternate between 3 and 2. Let's check that. If a_1 = 3, then a_2 = 2, a_3 = 3, a_4 = 2, a_5 = 3, a_6 = 2, and so on. Yes, that makes sense because each pair multiplies to 6. So, the odd terms are 3, and the even terms are 2.Therefore, in the sequence, every term with an odd index is 3, and every term with an even index is 2. So, a_1 = 3, a_3 = 3, a_5 = 3, etc., and a_2 = 2, a_4 = 2, a_6 = 2, etc.Now, the problem asks for the product of a_1 * a_5 * a_9 * ... * a_{2005} * a_{2009}. Let's analyze this. The indices here are 1, 5, 9, ..., 2005, 2009. So, these are terms with indices that are 1 more than a multiple of 4. Wait, let me see: 1, 5, 9,... Each step increases by 4. So, the general term is 1 + 4k, where k starts from 0.So, the indices are 1, 5, 9, ..., 2009. Let's figure out how many terms there are in this product. To find the number of terms, we can consider the sequence of indices: 1, 5, 9, ..., 2009. This is an arithmetic sequence where the first term is 1, the common difference is 4, and the last term is 2009.The formula for the nth term of an arithmetic sequence is a_n = a_1 + (n - 1)d. Here, a_n = 2009, a_1 = 1, d = 4. So, 2009 = 1 + (n - 1)*4. Let's solve for n:2009 = 1 + 4(n - 1)2009 - 1 = 4(n - 1)2008 = 4(n - 1)Divide both sides by 4: 2008 / 4 = n - 1502 = n - 1So, n = 503.Therefore, there are 503 terms in this product.Since each of these terms is a term with an odd index, as we saw earlier, each of these a_{1 + 4k} is 3. So, the product is 3 multiplied by itself 503 times, which is 3^{503}.Looking back at the options, D is 3^{503}, so that should be the answer.But let me double-check to make sure I didn't make any mistakes. First, confirming the terms: a_1 = 3, a_2 = 2, a_3 = 3, a_4 = 2, a_5 = 3, a_6 = 2, etc. So, indeed, every odd term is 3, every even term is 2.Then, the indices we're considering are 1, 5, 9, ..., 2009. Each of these is an odd index, so each term is 3. The number of terms is 503, so the product is 3^{503}. That seems correct.Alternatively, let's think about the sequence in terms of the common product. Since each a_n * a_{n+1} = 6, and a_6 = 2, we can find the previous terms as I did before. So, a_5 = 3, a_4 = 2, a_3 = 3, a_2 = 2, a_1 = 3. So, yes, the pattern holds.Another way to think about it is that the sequence alternates between 3 and 2. So, starting from a_1 = 3, it goes 3, 2, 3, 2, and so on. Therefore, every odd term is 3, every even term is 2.Therefore, when we take the product of a_1, a_5, a_9, ..., a_{2009}, which are all odd terms, each is 3, and there are 503 of them. So, the product is 3^{503}, which is option D.I think that's solid. I don't see any mistakes in the reasoning. So, I'm confident that D is the correct answer.**Final Answer**boxed{D}

💡To determine the largest percentage increase in the number of students taking the AMC 10 at Pythagoras Academy between consecutive years, I will calculate the percentage increase for each year interval.First, I'll calculate the percentage increase from 2003 to 2004:- The number of students increased from 50 to 55.- The increase is 55 - 50 = 5.- The percentage increase is (5 / 50) * 100% = 10%.Next, the percentage increase from 2004 to 2005:- The number of students increased from 55 to 60.- The increase is 60 - 55 = 5.- The percentage increase is (5 / 55) * 100% ≈ 9.09%.Then, the percentage increase from 2005 to 2006:- The number of students increased from 60 to 65.- The increase is 65 - 60 = 5.- The percentage increase is (5 / 60) * 100% ≈ 8.33%.Next, the percentage increase from 2006 to 2007:- The number of students increased from 65 to 75.- The increase is 75 - 65 = 10.- The percentage increase is (10 / 65) * 100% ≈ 15.38%.Finally, the percentage increase from 2007 to 2008:- The number of students increased from 75 to 80.- The increase is 80 - 75 = 5.- The percentage increase is (5 / 75) * 100% ≈ 6.67%.Comparing all these percentage increases, the largest increase is approximately 15.38%, which occurred between the years 2006 and 2007.

💡Okay, so I have this question about the truth values of four types of propositions, and I need to figure out which of the given judgments is correct. The options are A, B, C, and D. Let me try to break this down step by step.First, I need to remember what each of these terms means: proposition, contrapositive, converse, and negation. A proposition is a statement that can be either true or false. The contrapositive of a proposition is formed by negating both the hypothesis and conclusion and then switching their order. The converse is formed by switching the hypothesis and conclusion without negating them. The negation is simply the opposite truth value of the original proposition.Alright, let's go through each option one by one.**Option A: The truth value of a proposition is the same as that of its contrapositive.**Hmm, I think this is correct. I remember that a proposition and its contrapositive are logically equivalent. That means they have the same truth value. For example, if the proposition is "If it rains, then the ground is wet," the contrapositive would be "If the ground is not wet, then it did not rain." Both statements are true under the same conditions. So, A seems correct.**Option B: The truth value of a proposition is the same as that of its converse.**Wait, the converse is switching the hypothesis and conclusion. So, using the same example, the converse would be "If the ground is wet, then it rains." But this isn't necessarily true because the ground could be wet for other reasons, like someone watering their lawn. So, the converse can have a different truth value. Therefore, B is incorrect.**Option C: The truth value of a proposition is the same as that of its negation.**No, the negation of a proposition is the opposite truth value. If the proposition is true, its negation is false, and vice versa. So, they can't have the same truth value. Therefore, C is incorrect.**Option D: The truth value of the converse of a proposition is opposite to that of its negation.**Hmm, let's think. The converse is not necessarily the opposite of the negation. The negation flips the truth value, but the converse is a different statement altogether. For example, if the proposition is true, its negation is false, but the converse could be either true or false independently. So, the truth value of the converse isn't necessarily opposite to the negation. Therefore, D is incorrect.So, after going through each option, only A seems correct. The proposition and its contrapositive have the same truth value, while the converse and negation don't necessarily share the same or opposite truth values as the original proposition.I think I've got it. A is the correct answer.