🧠:<think>First, I need to determine the marked price of each article, which is given as 22.50.Next, I'll calculate the discount amount for one article by applying the 10% discount rate:Discount Amount = 10% of 22.50 = 0.10 × 22.50 = 2.25.Then, I'll find the price of one article after applying the discount:Price After Discount = Marked Price - Discount Amount = 22.50 - 2.25 = 20.25.Since there are two articles, I'll multiply the price after discount by 2 to find the total amount paid:Total Amount Paid = 2 × 20.25 = 40.50.Therefore, the total amount paid for the pair of articles is 40.50.</think>

🧠:<think>Okay, so I have this quadratic function: f(x) = x² + (2a + 3)x + (a² + 1). The problem says that this function only has real zeros, and I need to find the smallest possible value of 'a'. Then, express that value as a fraction p/q where p and q are coprime integers, and find |p| + |q|. Alright, let's start by recalling that for a quadratic equation ax² + bx + c = 0 to have real zeros, the discriminant must be non-negative. The discriminant D is given by D = b² - 4ac. If D is greater than or equal to zero, the equation has real roots.In this case, the quadratic function is f(x) = x² + (2a + 3)x + (a² + 1). So, comparing to the standard form, the coefficients are:- a = 1 (coefficient of x²)- b = (2a + 3) (coefficient of x)- c = (a² + 1) (constant term)So, let's compute the discriminant D:D = b² - 4acD = (2a + 3)² - 4 * 1 * (a² + 1)Let me expand (2a + 3)² first:(2a + 3)² = (2a)² + 2 * 2a * 3 + 3² = 4a² + 12a + 9Now, compute 4ac:4ac = 4 * 1 * (a² + 1) = 4a² + 4So, putting it all together:D = (4a² + 12a + 9) - (4a² + 4)D = 4a² + 12a + 9 - 4a² - 4D = (4a² - 4a²) + 12a + (9 - 4)D = 0 + 12a + 5D = 12a + 5So, the discriminant is 12a + 5. For the quadratic to have real zeros, D must be greater than or equal to zero:12a + 5 ≥ 0Let's solve for 'a':12a + 5 ≥ 012a ≥ -5a ≥ -5/12So, the smallest possible value of 'a' is -5/12. The problem asks for this value in the form p/q where p and q are relatively prime integers. Here, p is -5 and q is 12. Now, we need to find |p| + |q|. The absolute value of p is | -5 | = 5, and the absolute value of q is |12| = 12. So, 5 + 12 = 17.Just to make sure I didn't make any mistakes, let me double-check my steps:1. Identified the quadratic function and its coefficients correctly.2. Calculated the discriminant correctly by expanding (2a + 3)² and subtracting 4ac.3. Simplified the discriminant to 12a + 5.4. Set up the inequality 12a + 5 ≥ 0 and solved for 'a' correctly.5. Expressed the smallest 'a' as -5/12, which is already in simplest terms since 5 and 12 share no common divisors other than 1.6. Calculated |p| + |q| correctly as 5 + 12 = 17.Everything seems to check out. I don't think I made any errors here.Final Answerboxed{17}</think>

🧠:<think>Okay, so I'm trying to figure out how many volunteers should be recruited from the third grade for this book donation activity. The school is using a stratified sampling method based on the proportion of grades. Hmm, stratified sampling means that they're dividing the population into different groups, or strata, and then taking samples from each group proportionally.First, I need to understand the total number of students and how they're distributed across the grades. The high school department has a total of 2040 students. Out of these, 680 are in the first grade, 850 in the second grade, and 510 in the third grade. So, the third grade has the smallest number of students compared to the other grades.The goal is to recruit 12 volunteers in total, and these should be selected proportionally from each grade. That means the number of volunteers from each grade should reflect the size of that grade relative to the total student population.To find out how many volunteers should come from the third grade, I need to calculate the proportion of third-grade students in the entire high school department. The proportion can be calculated by dividing the number of third-grade students by the total number of students.So, the proportion of third-grade students is 510 divided by 2040. Let me do that calculation: 510 ÷ 2040. Hmm, simplifying that, both numbers are divisible by 510. 510 ÷ 510 is 1, and 2040 ÷ 510 is 4. So, the proportion is 1/4.Now, since the total number of volunteers is 12, I need to find out what 1/4 of 12 is. That would be 12 multiplied by 1/4, which equals 3. So, it seems like 3 volunteers should be recruited from the third grade.Wait, let me double-check my calculations to make sure I didn't make a mistake. The total number of students is 2040, and the third grade has 510 students. So, 510 divided by 2040 is indeed 1/4. Multiplying 12 by 1/4 gives 3. Yeah, that seems correct.I guess another way to look at it is by calculating the percentage of third-grade students. If I divide 510 by 2040, I get 0.25, which is 25%. So, 25% of the volunteers should be from the third grade. 25% of 12 is 3. That confirms my earlier calculation.I think I'm confident now that the number of volunteers to be recruited from the third grade is 3.</think>

🧠:<think>Okay, so I have this problem about a moving circle that's externally tangent to one circle and internally tangent to another. The first part is to find the equation of the locus of the center of this moving circle, which is called curve E. The second part is about finding the minimum value of |PO|² + |PF|² where P is a point on E, O is the origin, and F is the right focus of E.Let me start with part (1). I need to figure out the equation of curve E. The moving circle is externally tangent to circle C₁ and internally tangent to circle C₂. So, I should probably write down the equations of C₁ and C₂ first.Circle C₁ is given by x² + y² + 6x + 5 = 0. I can rewrite this in standard form by completing the square. Let's see:x² + 6x + y² + 5 = 0x² + 6x + 9 + y² = -5 + 9(x + 3)² + y² = 4So, C₁ has center at (-3, 0) and radius 2.Similarly, circle C₂ is given by x² + y² - 6x - 91 = 0. Let me complete the square here as well:x² - 6x + y² - 91 = 0x² - 6x + 9 + y² = 91 + 9(x - 3)² + y² = 100So, C₂ has center at (3, 0) and radius 10.Now, the moving circle is externally tangent to C₁ and internally tangent to C₂. Let's denote the center of the moving circle as M(x, y) and its radius as R.When two circles are externally tangent, the distance between their centers is equal to the sum of their radii. So, the distance between M and C₁ should be R + 2.Similarly, when two circles are internally tangent, the distance between their centers is equal to the difference of their radii. So, the distance between M and C₂ should be 10 - R.So, we have two equations:1. Distance from M to C₁: sqrt[(x + 3)² + y²] = R + 22. Distance from M to C₂: sqrt[(x - 3)² + y²] = 10 - RIf I add these two equations together, I get:sqrt[(x + 3)² + y²] + sqrt[(x - 3)² + y²] = (R + 2) + (10 - R) = 12So, the sum of the distances from M to C₁ and C₂ is 12. That sounds like the definition of an ellipse, where the sum of the distances from any point on the ellipse to the two foci is constant.So, curve E is an ellipse with foci at C₁ and C₂, which are (-3, 0) and (3, 0). The major axis length is 12, so 2a = 12, which means a = 6.The distance between the foci is 2c, where c is the distance from the center to each focus. The distance between (-3, 0) and (3, 0) is 6, so 2c = 6, which means c = 3.For an ellipse, we have the relationship a² = b² + c², where b is the semi-minor axis. So, plugging in the known values:a² = 36 = b² + 9So, b² = 36 - 9 = 27Thus, b = sqrt(27) = 3*sqrt(3)Therefore, the equation of the ellipse is:(x²)/(a²) + (y²)/(b²) = 1Which is:(x²)/36 + (y²)/27 = 1So, that should be the equation of curve E.Now, moving on to part (2). We need to find the minimum value of |PO|² + |PF|², where P is a point on E, O is the origin, and F is the right focus of E.First, let's figure out where the foci of E are. Since E is an ellipse with major axis along the x-axis (because the foci are on the x-axis), the foci are located at (±c, 0). We already found that c = 3, so the foci are at (-3, 0) and (3, 0). The right focus F is at (3, 0).So, we need to find the minimum value of |PO|² + |PF|² where P is on E.Let me denote P as (x, y). Then, |PO|² is x² + y², and |PF|² is (x - 3)² + y².So, |PO|² + |PF|² = x² + y² + (x - 3)² + y².Let me expand this:x² + y² + (x² - 6x + 9) + y² = 2x² - 6x + 9 + 2y².So, the expression simplifies to 2x² - 6x + 9 + 2y².Now, since P lies on the ellipse E, we can use the equation of E to express y² in terms of x².From E's equation: (x²)/36 + (y²)/27 = 1Multiply both sides by 27:(3x²)/4 + y² = 27So, y² = 27 - (3x²)/4Plugging this into our expression:2x² - 6x + 9 + 2*(27 - (3x²)/4) = 2x² - 6x + 9 + 54 - (3x²)/2Simplify:2x² - (3x²)/2 = (4x² - 3x²)/2 = x²/2So, x²/2 - 6x + 9 + 54 = x²/2 - 6x + 63So, |PO|² + |PF|² = (x²)/2 - 6x + 63Now, this is a quadratic in x. Let me write it as:(1/2)x² - 6x + 63To find the minimum value, I can complete the square or take the derivative. Since it's a quadratic, completing the square is straightforward.Factor out 1/2 from the x terms:(1/2)(x² - 12x) + 63Complete the square inside the parentheses:x² - 12x = (x - 6)² - 36So, substituting back:(1/2)[(x - 6)² - 36] + 63 = (1/2)(x - 6)² - 18 + 63 = (1/2)(x - 6)² + 45So, |PO|² + |PF|² = (1/2)(x - 6)² + 45This is a quadratic that opens upwards, so the minimum occurs at the vertex. The vertex is at x = 6.Therefore, the minimum value is 45.But wait, let me check if x = 6 is on the ellipse E. The ellipse has a major axis from x = -6 to x = 6, so x = 6 is the rightmost point.So, plugging x = 6 into the ellipse equation:(6²)/36 + y²/27 = 1 => 36/36 + y²/27 = 1 => 1 + y²/27 = 1 => y² = 0 => y = 0So, the point is (6, 0). That makes sense, as it's the rightmost point.Therefore, the minimum value is indeed 45.I think that's it. Let me recap:1. The moving circle's center traces an ellipse with foci at (-3, 0) and (3, 0), major axis 12, so equation is x²/36 + y²/27 = 1.2. The expression |PO|² + |PF|² simplifies to (1/2)(x - 6)² + 45, which is minimized at x = 6, giving the minimum value of 45.Final Answer(1) The equation of curve E is boxed{dfrac{x^2}{36} + dfrac{y^2}{27} = 1}.(2) The minimum value of |PO|^2 + |PF|^2 is boxed{45}.</think>