💡First, I need to determine the number of first-year students in the club. I know that there are 12 more first-year students than non-first-year students, and the total number of people in the club is 128.I'll start by letting ( x ) represent the number of non-first-year students. Since there are 12 more first-year students, the number of first-year students can be expressed as ( x + 12 ).The total number of students in the club is the sum of first-year and non-first-year students. Therefore, I can set up the equation:[x + (x + 12) = 128]Combining like terms, the equation simplifies to:[2x + 12 = 128]Next, I'll solve for ( x ) by subtracting 12 from both sides:[2x = 116]Then, I'll divide both sides by 2 to find the value of ( x ):[x = 58]This means there are 58 non-first-year students. To find the number of first-year students, I'll add 12 to the number of non-first-year students:[58 + 12 = 70]Therefore, there are 70 first-year students in the club.

💡Alright, so I have this problem about hyperbolas. It's divided into two parts. Let me try to tackle them one by one.**Part 1: Finding the Equation of Hyperbola C**Okay, the problem says that hyperbola C has a real axis length of 4. I remember that the real axis length is the distance between the two vertices of the hyperbola, which is 2a. So if the real axis length is 4, that means 2a = 4, so a = 2. Got that.Next, it says that hyperbola C shares a common focus with another hyperbola given by the equation (x²)/2 - (y²)/3 = 1. I need to find the foci of this given hyperbola to figure out where the foci of hyperbola C are.For a hyperbola of the form (x²)/a² - (y²)/b² = 1, the distance to the foci from the center is given by c, where c² = a² + b². So for the given hyperbola, a² is 2 and b² is 3. Therefore, c² = 2 + 3 = 5, so c = sqrt(5). That means the foci are at (±sqrt(5), 0).Since hyperbola C shares these foci, its foci are also at (±sqrt(5), 0). Now, I need to find the equation of hyperbola C. I know that for hyperbola C, a = 2, and c = sqrt(5). So, using the relationship c² = a² + b² for hyperbolas, I can solve for b².Plugging in the known values: c² = a² + b² => 5 = 4 + b² => b² = 1. So, b = 1.Now, since the hyperbola shares the same foci and is centered at the origin (since the foci are on the x-axis), the equation of hyperbola C should be (x²)/4 - (y²)/1 = 1, which simplifies to (x²)/4 - y² = 1.Wait, let me double-check that. The standard form is (x²)/a² - (y²)/b² = 1, so with a² = 4 and b² = 1, yes, that should be correct.**Part 2: Finding the Minimum Value of |PM|**Alright, now I need to find the minimum distance from point M(5, 0) to any point P on hyperbola C. So, point M is at (5, 0), and P is (x, y) on the hyperbola C.First, let's recall the distance formula. The distance between two points (x1, y1) and (x2, y2) is sqrt[(x2 - x1)² + (y2 - y1)²]. So, in this case, |PM| = sqrt[(x - 5)² + (y - 0)²] = sqrt[(x - 5)² + y²].But since P lies on hyperbola C, which is (x²)/4 - y² = 1, I can express y² in terms of x². From the equation, y² = (x²)/4 - 1.So, substituting y² into the distance formula, we get:|PM| = sqrt[(x - 5)² + ((x²)/4 - 1)].Let me simplify this expression step by step.First, expand (x - 5)²:(x - 5)² = x² - 10x + 25.Now, add ((x²)/4 - 1) to this:x² - 10x + 25 + (x²)/4 - 1 = (x² + (x²)/4) - 10x + (25 - 1) = (5x²)/4 - 10x + 24.So, |PM| = sqrt[(5x²)/4 - 10x + 24].Hmm, that looks a bit complicated. Maybe I can factor out some terms or complete the square to make it easier to find the minimum.Let me factor out 5/4 from the quadratic expression inside the square root:(5x²)/4 - 10x + 24 = (5/4)(x² - 8x) + 24.Wait, let me check that:(5/4)(x² - 8x) = (5x²)/4 - 10x. Yes, that's correct. So, the expression becomes:(5/4)(x² - 8x) + 24.Now, to complete the square for x² - 8x, I need to add and subtract (8/2)² = 16.So, x² - 8x = (x² - 8x + 16) - 16 = (x - 4)² - 16.Substituting back into the expression:(5/4)[(x - 4)² - 16] + 24 = (5/4)(x - 4)² - (5/4)*16 + 24.Calculate the constants:(5/4)*16 = 20, so:(5/4)(x - 4)² - 20 + 24 = (5/4)(x - 4)² + 4.So, now the expression inside the square root is (5/4)(x - 4)² + 4.Therefore, |PM| = sqrt[(5/4)(x - 4)² + 4].Now, to find the minimum value of |PM|, I need to minimize this expression. Since the square root function is a monotonically increasing function, the minimum of |PM| occurs at the minimum of the expression inside the square root.So, let's denote f(x) = (5/4)(x - 4)² + 4.To find the minimum of f(x), we can look at the expression. It's a quadratic function in terms of (x - 4), which is always non-negative because it's squared. The minimum occurs when (x - 4)² is minimized, which is when x = 4.So, the minimum value of f(x) is when x = 4:f(4) = (5/4)(0) + 4 = 4.Therefore, the minimum value of |PM| is sqrt(4) = 2.Wait, but hold on. I need to make sure that x = 4 is actually on hyperbola C. Because if x = 4 is not on the hyperbola, then the minimum might occur at another point.Looking back at hyperbola C: (x²)/4 - y² = 1. When x = 4, plugging into the equation:(16)/4 - y² = 1 => 4 - y² = 1 => y² = 3 => y = ±sqrt(3). So, yes, x = 4 is on the hyperbola, and y is real. So, the point (4, sqrt(3)) and (4, -sqrt(3)) are on the hyperbola.Therefore, the minimum distance |PM| is indeed 2.Wait, let me double-check the calculations just to be sure.Starting from |PM| = sqrt[(x - 5)² + y²], and y² = (x²)/4 - 1.So, |PM|² = (x - 5)² + (x²)/4 - 1.Expanding (x - 5)²: x² - 10x + 25.Adding (x²)/4 - 1: x² - 10x + 25 + x²/4 - 1 = (5x²)/4 - 10x + 24.Yes, that's correct.Then, factoring out 5/4: (5/4)(x² - 8x) + 24.Completing the square: (5/4)[(x - 4)² - 16] + 24 = (5/4)(x - 4)² - 20 + 24 = (5/4)(x - 4)² + 4.So, |PM| = sqrt[(5/4)(x - 4)² + 4].The minimum occurs when (x - 4)² is zero, so x = 4, giving |PM| = sqrt(4) = 2.Yes, that seems correct.Alternatively, I could use calculus to find the minimum. Let's try that approach too.Let me define f(x) = |PM|² = (x - 5)² + y² = (x - 5)² + (x²)/4 - 1.So, f(x) = x² - 10x + 25 + x²/4 - 1 = (5x²)/4 - 10x + 24.To find the minimum, take the derivative f'(x):f'(x) = (10x)/4 - 10 = (5x)/2 - 10.Set f'(x) = 0:(5x)/2 - 10 = 0 => (5x)/2 = 10 => x = (10 * 2)/5 = 4.So, x = 4 is the critical point. To confirm it's a minimum, check the second derivative:f''(x) = 5/2, which is positive, so it's a minimum.Therefore, the minimum value of |PM|² is f(4) = (5*(4)^2)/4 - 10*4 + 24 = (5*16)/4 - 40 + 24 = 20 - 40 + 24 = 4.Thus, |PM| = sqrt(4) = 2.Yep, same result. So, I'm confident that the minimum value is 2.**Summary**1. The equation of hyperbola C is (x²)/4 - y² = 1.2. The minimum value of |PM| is 2.**Final Answer**1. The equation of hyperbola ( C ) is (boxed{dfrac{x^2}{4} - y^2 = 1}).2. The minimum value of ( |PM| ) is (boxed{2}).

💡Okay, so I have this problem where I'm given positive numbers p, q, a, b, c, and it's stated that p is not equal to q. It says that p, a, q form a geometric sequence, and p, b, c, q form an arithmetic sequence. Then, I need to figure out what the quadratic equation bx² - 2ax + c = 0 does. The options are about whether it has real roots, equal roots, same sign roots, or opposite sign roots.Alright, let's break this down step by step. First, let's recall what a geometric sequence is. In a geometric sequence, each term is multiplied by a common ratio. So, if p, a, q are in geometric progression, that means a is the geometric mean of p and q. So, mathematically, that should be a² = p*q. Yeah, that makes sense because in a geometric sequence, the square of the middle term is equal to the product of the first and the third terms.Next, the arithmetic sequence part. An arithmetic sequence has a common difference between consecutive terms. So, if p, b, c, q are in arithmetic progression, that means the difference between each term is the same. Let's denote the common difference as d. So, starting from p, the next term is p + d = b, then b + d = c, and c + d = q. So, we can write:b = p + dc = b + d = p + 2dq = c + d = p + 3dSo, from this, we can solve for d. Since q = p + 3d, that means d = (q - p)/3. Therefore, we can express b and c in terms of p and q.So, substituting d back into b and c:b = p + (q - p)/3 = (3p + q - p)/3 = (2p + q)/3Similarly, c = p + 2*(q - p)/3 = (3p + 2q - 2p)/3 = (p + 2q)/3So, now we have expressions for a, b, and c in terms of p and q.Given that a² = p*q, so a = sqrt(p*q). But since all numbers are positive, we don't have to worry about the negative square root.Now, let's look at the quadratic equation: bx² - 2ax + c = 0.To determine the nature of the roots, we can compute the discriminant. The discriminant D is given by D = (coefficient of x)² - 4*(coefficient of x²)*(constant term). So, in this case:D = (-2a)² - 4*b*c = 4a² - 4bcWe can factor out the 4:D = 4(a² - bc)We already know that a² = p*q, so let's substitute that in:D = 4(p*q - b*c)Now, let's compute bc. We have b = (2p + q)/3 and c = (p + 2q)/3. So, multiplying them together:b*c = [(2p + q)/3] * [(p + 2q)/3] = (2p + q)(p + 2q)/9Let's expand the numerator:(2p + q)(p + 2q) = 2p*p + 2p*2q + q*p + q*2q = 2p² + 4pq + pq + 2q² = 2p² + 5pq + 2q²So, bc = (2p² + 5pq + 2q²)/9Therefore, D = 4(p*q - (2p² + 5pq + 2q²)/9)Let me compute this step by step. First, let's write p*q as (9pq)/9 to have a common denominator:D = 4[(9pq)/9 - (2p² + 5pq + 2q²)/9] = 4[(9pq - 2p² - 5pq - 2q²)/9]Simplify the numerator:9pq - 5pq = 4pqSo, numerator becomes: -2p² + 4pq - 2q²Factor out a -2:-2(p² - 2pq + q²)Notice that p² - 2pq + q² is (p - q)².So, numerator is -2(p - q)²Therefore, D = 4*(-2(p - q)²)/9 = (-8(p - q)²)/9Since p ≠ q, (p - q)² is positive, so D is negative because of the negative sign.So, discriminant D is negative, which means the quadratic equation has no real roots.Therefore, the correct answer should be option A: has no real roots.Wait, let me just double-check my calculations to make sure I didn't make any mistakes.Starting from D = 4(a² - bc). a² is pq, bc is (2p + q)(p + 2q)/9. Expanding that gives 2p² + 5pq + 2q², which is correct. Then, subtracting that from pq, which is 9pq/9, so 9pq - 2p² - 5pq - 2q² = -2p² + 4pq - 2q², which factors to -2(p² - 2pq + q²) = -2(p - q)². Multiply by 4 and divide by 9, so D = (-8/9)(p - q)². Since p ≠ q, this is negative. So, yes, discriminant is negative, no real roots. So, answer is A.I think that's solid. I don't see any errors in the steps.**Final Answer**boxed{A}

💡Okay, so I have this problem where I'm given that a and b are positive numbers, and their sum is 1. I need to prove two inequalities: the first one is that 1/a plus 1/b is at least 4, and the second one is that the sum of (a + 1/a) squared and (b + 1/b) squared is at least 25/2. Hmm, let me try to figure this out step by step.Starting with part (I): I need to show that 1/a + 1/b is greater than or equal to 4. Since a and b are positive and add up to 1, I wonder if I can use some inequality principles here. Maybe the AM-HM inequality? I remember that the harmonic mean is involved when dealing with reciprocals.Wait, the harmonic mean of two numbers a and b is 2ab/(a + b). Since a + b is 1, the harmonic mean would be 2ab. The arithmetic mean of a and b is (a + b)/2, which is 1/2. I also recall that the arithmetic mean is always greater than or equal to the harmonic mean. So, 1/2 ≥ 2ab. That would mean that ab ≤ 1/4.But how does that help me with 1/a + 1/b? Let me see. 1/a + 1/b can be written as (b + a)/ab, which is 1/ab since a + b = 1. So, 1/a + 1/b = 1/ab. From earlier, I know that ab ≤ 1/4, so 1/ab ≥ 4. That makes sense! So, 1/a + 1/b is indeed at least 4. Cool, that wasn't too bad.Now, moving on to part (II): I need to prove that (a + 1/a)^2 + (b + 1/b)^2 is at least 25/2. Hmm, this seems a bit more complex. Let me expand these squares first to see if that helps.Expanding (a + 1/a)^2 gives me a^2 + 2*(a)*(1/a) + (1/a)^2, which simplifies to a^2 + 2 + 1/a^2. Similarly, (b + 1/b)^2 expands to b^2 + 2 + 1/b^2. So, adding these together, I get a^2 + b^2 + 4 + 1/a^2 + 1/b^2.So, the expression becomes a^2 + b^2 + 4 + 1/a^2 + 1/b^2. Hmm, I need to find a lower bound for this. Maybe I can find the minimum value of a^2 + b^2 and 1/a^2 + 1/b^2 separately.I know from part (I) that 1/a + 1/b is at least 4, but I need 1/a^2 + 1/b^2. Maybe I can use the Cauchy-Schwarz inequality here. The Cauchy-Schwarz inequality states that (x1^2 + x2^2)(y1^2 + y2^2) ≥ (x1y1 + x2y2)^2. If I let x1 = 1/a, x2 = 1/b, y1 = y2 = 1, then it becomes (1/a^2 + 1/b^2)(1 + 1) ≥ (1/a + 1/b)^2. So, 2*(1/a^2 + 1/b^2) ≥ (1/a + 1/b)^2.From part (I), I know that 1/a + 1/b ≥ 4, so (1/a + 1/b)^2 ≥ 16. Therefore, 2*(1/a^2 + 1/b^2) ≥ 16, which implies that 1/a^2 + 1/b^2 ≥ 8.Okay, so now I have that 1/a^2 + 1/b^2 is at least 8. What about a^2 + b^2? I remember that for any two numbers, a^2 + b^2 is at least (a + b)^2 / 2 by the Cauchy-Schwarz inequality or the QM-AM inequality. Since a + b = 1, this gives a^2 + b^2 ≥ (1)^2 / 2 = 1/2.So, putting it all together: a^2 + b^2 + 4 + 1/a^2 + 1/b^2 ≥ 1/2 + 4 + 8 = 1/2 + 12 = 12.5, which is 25/2. So, the entire expression is at least 25/2. That seems to work!Wait, let me double-check. If I use the Cauchy-Schwarz inequality for 1/a^2 + 1/b^2, I get that it's at least 8. And a^2 + b^2 is at least 1/2. Adding them together with the 4 gives 1/2 + 4 + 8 = 12.5, which is indeed 25/2. So, that seems correct.But just to be thorough, let me consider when equality holds. For part (I), equality holds when a = b = 1/2. Let me plug that into part (II). If a = b = 1/2, then (a + 1/a)^2 + (b + 1/b)^2 becomes 2*(1/2 + 2)^2 = 2*(5/2)^2 = 2*(25/4) = 25/2. So, equality holds when a = b = 1/2, which is consistent.Hmm, I think that's solid. I used known inequalities like AM-HM and Cauchy-Schwarz to break down the problem into parts I could handle. It's interesting how these inequalities can be combined to solve more complex problems. I should remember these techniques for future problems.Another thought: Maybe I could have approached part (II) differently, like using the result from part (I) directly. Let me see. If I consider (a + 1/a)^2 + (b + 1/b)^2, maybe I can relate it to (1/a + 1/b)^2 somehow. Expanding the squares, as I did before, gives me a^2 + b^2 + 4 + 1/a^2 + 1/b^2. I already have 1/a^2 + 1/b^2 ≥ 8 and a^2 + b^2 ≥ 1/2, so adding those gives me the necessary bound.Alternatively, maybe I could use the fact that (a + 1/a) is minimized when a = 1, but since a + b = 1, a can't be 1 unless b is 0, which isn't allowed. So, the minimum must occur somewhere else, probably when a = b = 1/2, as we saw earlier.Yeah, I think my approach is correct. I'll stick with it. It's always good to verify the equality condition to make sure everything lines up, and in this case, it does. So, I feel confident about my solution.**Final Answer**(I) boxed{4}(II) boxed{dfrac{25}{2}}

💡Alright, so I have this problem about a right-angled triangle where the lengths of the three sides form a geometric sequence. I need to figure out which of the given options (A, B, C, D) is correct. Let me try to break this down step by step.First, let me recall what a geometric sequence is. In a geometric sequence, each term after the first is found by multiplying the previous term by a constant called the common ratio, denoted usually by 'r'. So, if the sides of the triangle are in a geometric sequence, I can denote them as 'a', 'ar', and 'ar²', where 'a' is the first term and 'r' is the common ratio.Since it's a right-angled triangle, the Pythagorean theorem must hold. That is, the sum of the squares of the two shorter sides equals the square of the longest side (the hypotenuse). So, if I assume that 'a' is the shortest side, then 'ar' would be the next side, and 'ar²' would be the hypotenuse. Therefore, the Pythagorean theorem would be:a² + (ar)² = (ar²)²Let me write that out:a² + (ar)² = (ar²)²Simplifying each term:a² + a²r² = a²r⁴Now, I can factor out a² from each term:a²(1 + r²) = a²r⁴Since a² is common on both sides, I can divide both sides by a² (assuming a ≠ 0, which makes sense because side lengths can't be zero):1 + r² = r⁴So, now I have the equation:r⁴ - r² - 1 = 0Hmm, this looks like a quartic equation, but it's actually a quadratic in terms of r². Let me set y = r² to make it easier:y² - y - 1 = 0Now, I can solve this quadratic equation for y using the quadratic formula:y = [1 ± √(1 + 4)] / 2 = [1 ± √5] / 2Since y = r² and r² must be positive (because it's a ratio of lengths), I discard the negative solution:y = [1 + √5] / 2So, r² = [1 + √5] / 2Therefore, r = √([1 + √5] / 2)Okay, so now I have the common ratio 'r'. Let me see if I can simplify this expression or relate it to any known ratios.Wait, [1 + √5]/2 is actually the golden ratio, often denoted by the Greek letter phi (φ). So, r² = φ, and r = √φ.But maybe I don't need to get into the golden ratio here. Let me see what the sides look like.The sides are a, ar, ar². So, if I let a = 1 for simplicity, the sides would be 1, r, and r².But let me check if this makes sense. If a = 1, then the sides are 1, r, and r², and they satisfy the Pythagorean theorem:1² + r² = (r²)²Which simplifies to:1 + r² = r⁴Which is exactly the equation I had before, so that's consistent.Now, looking at the options given:A: The ratio of the lengths of the three sides is 3:4:5B: The ratio of the lengths of the three sides is 3:√3:1C: The sine of the larger acute angle is (√5 - 1)/2D: The sine of the smaller acute angle is (√5 - 1)/2First, let's check options A and B, which are about the ratio of the sides.Option A says 3:4:5. That's a well-known Pythagorean triple, but does it form a geometric sequence?Let's see. In a geometric sequence, the ratio between consecutive terms is constant. So, 4/3 ≈ 1.333, and 5/4 = 1.25. These are not equal, so 3:4:5 is not a geometric sequence. Therefore, option A is incorrect.Option B says 3:√3:1. Let's check if this is a geometric sequence.First, let's see the ratios between consecutive terms.√3 / 3 = 1/√3 ≈ 0.5771 / √3 ≈ 0.577So, the ratios are equal, which means 3:√3:1 is indeed a geometric sequence with a common ratio of 1/√3.But wait, in a right-angled triangle, the hypotenuse is the longest side. In this ratio, 3 is the largest, followed by √3, then 1. But in a right-angled triangle, the hypotenuse is the longest side, so if 3 is the hypotenuse, then the other sides would be √3 and 1. Let's check if this satisfies the Pythagorean theorem.1² + (√3)² = 1 + 3 = 4But the hypotenuse squared would be 3² = 9, which is not equal to 4. Therefore, this ratio does not satisfy the Pythagorean theorem. So, option B is also incorrect.Okay, so options A and B are out. Now, let's look at options C and D, which are about the sine of the larger and smaller acute angles, respectively.First, let's recall that in a right-angled triangle, the two acute angles are complementary, meaning they add up to 90 degrees. The larger acute angle is opposite the longer leg, and the smaller acute angle is opposite the shorter leg.Given that the sides are in a geometric sequence, let's denote them as a, ar, ar², with ar² being the hypotenuse.So, the legs are a and ar, and the hypotenuse is ar².Let me denote the smaller acute angle as θ, opposite the shorter leg 'a', and the larger acute angle as φ, opposite the longer leg 'ar'.Then, sin(θ) = opposite/hypotenuse = a / ar² = 1 / r²Similarly, sin(φ) = opposite/hypotenuse = ar / ar² = 1 / rSo, sin(θ) = 1 / r² and sin(φ) = 1 / rFrom earlier, we have r² = [1 + √5]/2, so r = √([1 + √5]/2)Therefore, sin(θ) = 1 / r² = 2 / (1 + √5)Let me rationalize the denominator:2 / (1 + √5) = [2(1 - √5)] / [(1 + √5)(1 - √5)] = [2(1 - √5)] / (1 - 5) = [2(1 - √5)] / (-4) = [2(√5 - 1)] / 4 = (√5 - 1)/2So, sin(θ) = (√5 - 1)/2Similarly, sin(φ) = 1 / r = 1 / √([1 + √5]/2) = √(2 / (1 + √5)) = √(2(1 - √5)/ (1 - 5)) = √(2(√5 - 1)/4) = √( (√5 - 1)/2 )But let's see if this simplifies to something familiar.Alternatively, since θ and φ are complementary, sin(φ) = cos(θ). Let's compute cos(θ):cos(θ) = adjacent/hypotenuse = ar / ar² = 1 / rWait, that's the same as sin(φ). So, sin(φ) = 1 / rBut we already have sin(θ) = (√5 - 1)/2So, sin(φ) = 1 / r = √(2 / (1 + √5)) = √(2(√5 - 1)/4) = √( (√5 - 1)/2 )But let's compute this numerically to see if it matches any known values.First, compute r² = [1 + √5]/2 ≈ (1 + 2.236)/2 ≈ 1.618So, r ≈ √1.618 ≈ 1.272Therefore, sin(θ) = 1 / r² ≈ 1 / 1.618 ≈ 0.618And sin(φ) = 1 / r ≈ 1 / 1.272 ≈ 0.786Now, (√5 - 1)/2 ≈ (2.236 - 1)/2 ≈ 1.236/2 ≈ 0.618, which matches sin(θ)So, sin(θ) = (√5 - 1)/2, which is approximately 0.618, and sin(φ) ≈ 0.786Looking back at the options:C: The sine of the larger acute angle is (√5 - 1)/2D: The sine of the smaller acute angle is (√5 - 1)/2From our calculations, sin(θ) = (√5 - 1)/2, where θ is the smaller acute angle, and sin(φ) ≈ 0.786, which is larger than sin(θ). Therefore, option D is correct, and option C is incorrect.So, the correct answer is D.

💡Alright, so I've got this problem here about a square, some black regions, and dropping a coin. I need to find the probability that the coin covers any part of a black region. Let me try to break this down step by step.First, let me visualize the setup. There's a square with a side length of 10 units. Inside this square, there's a white circular region with a radius of 3 units, centered right at the middle of the square. Then, there are four black squares, each with a side length of 1 unit, placed at each of the four corners of the main square. So, each black square is pretty small, just 1x1 units, and they're right at the corners.Now, we're dropping a circular coin with a diameter of 2 units onto this square. Since the diameter is 2 units, the radius of the coin is 1 unit. The question is asking for the probability that when we drop this coin randomly onto the square, it will cover any part of a black region.Okay, so probability is generally the ratio of the favorable area to the total possible area. In this case, the favorable area would be the regions where dropping the coin would result in it covering a black square, and the total possible area is where the center of the coin can land such that the entire coin is within the main square.Let me start by figuring out the total area where the center of the coin can land. Since the coin has a radius of 1 unit, the center of the coin must be at least 1 unit away from the edges of the main square to ensure the entire coin is within the square. So, the main square is 10 units on each side, but the center of the coin can only be within an 8x8 square in the middle (because we subtract 1 unit from each side). So, the total area where the center can land is 8 * 8 = 64 square units.Next, I need to figure out the area where the center of the coin can land such that it covers any part of a black region. Each black square is at a corner, so let's focus on one corner first and then multiply by four since they're all symmetrical.Each black square is 1x1 units. To cover any part of this black square, the center of the coin must be within a certain distance from the black square. Since the coin has a radius of 1 unit, the center can be up to 1 unit away from the black square in any direction and still cover part of it.So, for each black square, the area where the center of the coin can land to cover it is a square of side length 3 units (1 unit for the black square itself plus 1 unit on each side). Wait, actually, since the black square is at the corner, the area around it where the coin's center can land to cover it is a square of side length 3 units, but only in the corner. So, each black square contributes a 3x3 area where the coin's center can land to cover it.But wait, the main square is 10 units, and the center of the coin must be within 1 unit of the black square to cover it. So, for each black square at the corner, the area where the center can land is a square of side length 3 units, but only the part that's within the main square. Since the black square is at the corner, the 3x3 area extends 1 unit beyond the black square into the main square.However, I need to make sure that this 3x3 area doesn't overlap with the areas from the other black squares. Since the black squares are at the corners, their respective 3x3 areas don't overlap with each other. So, each black square contributes a unique 3x3 area where the coin's center can land to cover it.Therefore, for four black squares, the total area where the center can land to cover any black region is 4 * (3 * 3) = 36 square units.Wait a minute, but the total area where the center can land is 64 square units, and the area covering the black regions is 36 square units. So, the probability would be 36/64, which simplifies to 9/16.But hold on, I need to make sure that these 3x3 areas don't extend beyond the main square. Since the main square is 10 units, and each 3x3 area is only extending 1 unit beyond the black square, which is at the corner, so the 3x3 area is entirely within the main square. So, that's fine.Also, I need to consider if the coin's center can be anywhere else that might cover the black square. For example, if the coin is partially overlapping the black square from the side or the top. But since the black square is only 1x1, and the coin has a radius of 1, the center of the coin just needs to be within 1 unit of the black square in any direction.But since the black square is at the corner, the area where the center can land to cover it is a square of 3x3, as I thought earlier. So, I think my calculation is correct.Therefore, the probability that the coin covers any part of a black region is 9/16.