answer：First, I need to determine the individual shares of A, B, C, and D based on the given proportion of 2:3:4:6. To do this, I'll introduce a common variable, x, to represent the multiple of each share.Next, I'll set up an equation based on the information that D receives 700 more than C. This will allow me to solve for the value of x.Once I have the value of x, I can calculate B's share before any tax deductions.After determining B's share, I need to check if it meets the minimum wage requirement of 1000. If it does, I'll then apply the 15% tax rate to the portion of B's salary that exceeds 1500, if applicable.Finally, I'll calculate B's share after the tax deduction and ensure that the final amount complies with the minimum wage requirement.

answer：Alright, so I have this problem where I need to find the center of a circle that passes through the point (0,2) and is tangent to the parabola y = x² at the point (1,1). Hmm, okay, let me think about how to approach this.First, I remember that if a circle is tangent to a curve at a point, then the tangent line to the curve at that point is also the tangent line to the circle. So, maybe I should start by finding the equation of the tangent line to the parabola at (1,1).To find the tangent line, I need the derivative of the parabola y = x². The derivative dy/dx is 2x, which gives the slope of the tangent at any point x. At x = 1, the slope m is 2*1 = 2. So, the slope of the tangent line at (1,1) is 2.Now, using the point-slope form of a line, which is y - y₁ = m(x - x₁), where (x₁, y₁) is the point of tangency, I can write the equation of the tangent line as:y - 1 = 2(x - 1)Simplifying this, I get:y = 2x - 2 + 1y = 2x - 1Okay, so the tangent line is y = 2x - 1. Now, since the circle is tangent to the parabola at (1,1), this line should also be tangent to the circle. That means the distance from the center of the circle to this tangent line should be equal to the radius of the circle.Let me denote the center of the circle as (a, b). The distance from (a, b) to the tangent line y = 2x - 1 can be calculated using the formula for the distance from a point to a line:Distance = |2a - b - 1| / sqrt(2² + (-1)²) = |2a - b - 1| / sqrt(5)This distance should be equal to the radius of the circle. Also, since the circle passes through the point (0,2), the distance from (a, b) to (0,2) is also the radius. So, I can set up the equation:sqrt((a - 0)² + (b - 2)²) = |2a - b - 1| / sqrt(5)Let me square both sides to eliminate the square roots and absolute value:(a)² + (b - 2)² = (2a - b - 1)² / 5Multiplying both sides by 5 to eliminate the denominator:5a² + 5(b - 2)² = (2a - b - 1)²Expanding both sides:Left side: 5a² + 5(b² - 4b + 4) = 5a² + 5b² - 20b + 20Right side: (2a - b - 1)² = 4a² + b² + 1 - 4ab - 4a + 2bSo, putting it all together:5a² + 5b² - 20b + 20 = 4a² + b² + 1 - 4ab - 4a + 2bLet me bring all terms to the left side:5a² + 5b² - 20b + 20 - 4a² - b² - 1 + 4ab + 4a - 2b = 0Simplify term by term:(5a² - 4a²) + (5b² - b²) + (-20b - 2b) + (20 - 1) + 4ab + 4a = 0Which simplifies to:a² + 4b² - 22b + 19 + 4ab + 4a = 0Hmm, this seems a bit complicated. Maybe I made a mistake in expanding or simplifying. Let me double-check.Wait, actually, when expanding the right side, I think I might have messed up the signs. Let me re-examine the expansion of (2a - b - 1)².(2a - b - 1)² = (2a)² + (-b)² + (-1)² + 2*(2a)*(-b) + 2*(2a)*(-1) + 2*(-b)*(-1)= 4a² + b² + 1 - 4ab - 4a + 2bYes, that seems correct. So, the right side is indeed 4a² + b² + 1 - 4ab - 4a + 2b.So, bringing everything to the left:5a² + 5b² - 20b + 20 - 4a² - b² - 1 + 4ab + 4a - 2b = 0Simplify:(5a² - 4a²) = a²(5b² - b²) = 4b²(-20b - 2b) = -22b(20 - 1) = 19Then, +4ab and +4a.So, the equation becomes:a² + 4b² - 22b + 19 + 4ab + 4a = 0Hmm, this still looks a bit messy. Maybe there's another approach. Since the circle passes through (1,1) and (0,2), the center (a,b) must be equidistant from both points. So, the distance from (a,b) to (1,1) equals the distance from (a,b) to (0,2).Let me write that equation:sqrt((a - 1)² + (b - 1)²) = sqrt((a - 0)² + (b - 2)²)Squaring both sides:(a - 1)² + (b - 1)² = a² + (b - 2)²Expanding both sides:Left side: a² - 2a + 1 + b² - 2b + 1 = a² + b² - 2a - 2b + 2Right side: a² + b² - 4b + 4Subtracting right side from left side:(a² + b² - 2a - 2b + 2) - (a² + b² - 4b + 4) = 0Simplify:-2a - 2b + 2 + 4b - 4 = 0Which simplifies to:-2a + 2b - 2 = 0Divide both sides by -2:a - b + 1 = 0So, a = b - 1Okay, so now I have a relationship between a and b: a = b - 1. Maybe I can substitute this into the earlier equation I had.Going back to the equation I derived earlier:a² + 4b² - 22b + 19 + 4ab + 4a = 0Since a = b - 1, let's substitute a with (b - 1):(b - 1)² + 4b² - 22b + 19 + 4*(b - 1)*b + 4*(b - 1) = 0Let me expand each term:(b - 1)² = b² - 2b + 14*(b - 1)*b = 4b² - 4b4*(b - 1) = 4b - 4Now, substitute back:(b² - 2b + 1) + 4b² - 22b + 19 + (4b² - 4b) + (4b - 4) = 0Combine like terms:b² + 4b² + 4b² = 9b²-2b -22b -4b +4b = (-2 -22 -4 +4)b = (-24)b1 + 19 -4 = 16So, the equation becomes:9b² -24b +16 = 0This is a quadratic in terms of b. Let me try to solve it:9b² -24b +16 = 0Using the quadratic formula:b = [24 ± sqrt(24² - 4*9*16)] / (2*9)Calculate discriminant:24² = 5764*9*16 = 576So, sqrt(576 - 576) = sqrt(0) = 0Thus, b = 24 / 18 = 4/3 ≈ 1.333...Since the discriminant is zero, there's only one solution: b = 4/3.Now, since a = b -1, then a = 4/3 -1 = 1/3 ≈ 0.333...So, the center of the circle is at (1/3, 4/3).Wait, but let me verify this because earlier I had a different approach where I considered the tangent line and the distance from the center to the tangent line equals the radius. Maybe I should check if this center satisfies that condition.The center is (1/3, 4/3). The radius should be the distance from (1/3, 4/3) to (0,2):sqrt((1/3 - 0)² + (4/3 - 2)²) = sqrt((1/3)² + (-2/3)²) = sqrt(1/9 + 4/9) = sqrt(5/9) = sqrt(5)/3Now, the distance from the center to the tangent line y = 2x -1 should also be sqrt(5)/3.Using the distance formula:|2*(1/3) - (4/3) -1| / sqrt(2² + (-1)²) = |(2/3 -4/3 -3/3)| / sqrt(5) = |(-5/3)| / sqrt(5) = (5/3)/sqrt(5) = (5)/(3sqrt(5)) = sqrt(5)/3Yes, that matches. So, the radius is indeed sqrt(5)/3, and the distance from the center to the tangent line is the same. Therefore, the center (1/3, 4/3) seems correct.Wait a minute, earlier when I tried another method, I got a different result. Let me check that again.In the initial approach, I considered the tangent line at (1,1) with slope 2, then found the equation of the tangent line as y = 2x -1. Then, I set up the distance from the center (a,b) to this line equal to the radius, which is also the distance from (a,b) to (0,2).After setting up the equations, I ended up with a quadratic equation leading to a = 1/2 and b = 3/2. But now, using another method, I got a =1/3 and b=4/3. There's a discrepancy here, which means I must have made a mistake somewhere.Let me go back to the first method.In the first method, I found the tangent line correctly as y = 2x -1. Then, I set up the distance from (a,b) to this line equal to the radius, which is also the distance from (a,b) to (0,2). So, I wrote:sqrt(a² + (b - 2)²) = |2a - b -1| / sqrt(5)Then, squared both sides:a² + (b - 2)² = (2a - b -1)² /5Multiplying both sides by 5:5a² + 5(b² -4b +4) = (4a² + b² +1 -4ab -4a +2b)Expanding:5a² +5b² -20b +20 =4a² +b² +1 -4ab -4a +2bBringing all terms to the left:5a² -4a² +5b² -b² -20b -2b +20 -1 +4ab +4a=0Which simplifies to:a² +4b² -22b +19 +4ab +4a=0Then, using the fact that the center is equidistant from (1,1) and (0,2), I found that a = b -1.Substituting a = b -1 into the equation:(b -1)² +4b² -22b +19 +4(b -1)b +4(b -1)=0Expanding:b² -2b +1 +4b² -22b +19 +4b² -4b +4b -4=0Wait, hold on, in the previous calculation, I think I made a mistake in expanding 4(b -1)b. It should be 4b² -4b, not 4b² -4b + something else.Wait, let's re-examine the expansion:4*(b -1)*b =4b² -4b4*(b -1)=4b -4So, substituting back:(b² -2b +1) +4b² -22b +19 +4b² -4b +4b -4=0Now, combine like terms:b² +4b² +4b² =9b²-2b -22b -4b +4b = (-2 -22 -4 +4)b = (-24)b1 +19 -4=16So, 9b² -24b +16=0Which is the same quadratic as before, leading to b=4/3 and a=1/3.But in the initial approach, I thought I got a=1/2 and b=3/2. Wait, maybe I confused the two methods.Wait, in the initial problem statement, the user provided a solution where they found the slope of the tangent line as 1, leading to a=1/2 and b=3/2. But in reality, the slope of the tangent line at (1,1) is 2, not 1. So, that initial solution was incorrect because it miscalculated the slope.Therefore, my second method, where I correctly calculated the slope as 2, leading to a=1/3 and b=4/3, is the correct one.But let me double-check everything again to be sure.1. Find the tangent line at (1,1) on y=x². The derivative is 2x, so at x=1, slope m=2. Equation: y=2x -1.2. The center (a,b) must be such that the distance from (a,b) to (0,2) equals the distance from (a,b) to the tangent line y=2x -1.3. Also, since the circle passes through (1,1), the distance from (a,b) to (1,1) must equal the radius.4. So, setting up the equations: - Distance from (a,b) to (0,2): sqrt(a² + (b-2)²) - Distance from (a,b) to (1,1): sqrt((a-1)² + (b-1)²) - Distance from (a,b) to the line y=2x -1: |2a - b -1| / sqrt(5)5. Equate the first and third distances: sqrt(a² + (b-2)²) = |2a - b -1| / sqrt(5) Squaring both sides: a² + (b-2)² = (2a - b -1)² /5 Multiply both sides by 5: 5a² +5(b² -4b +4) =4a² +b² +1 -4ab -4a +2b Simplify: 5a² +5b² -20b +20 =4a² +b² +1 -4ab -4a +2b Bring all terms to left: a² +4b² -22b +19 +4ab +4a=06. Also, equate the distances from (a,b) to (1,1) and (0,2): sqrt((a-1)² + (b-1)²) = sqrt(a² + (b-2)²) Squaring both sides: (a-1)² + (b-1)² =a² + (b-2)² Expand: a² -2a +1 +b² -2b +1 =a² +b² -4b +4 Simplify: -2a -2b +2 = -4b +4 -2a +2b -2=0 a = b -17. Substitute a = b -1 into the earlier equation: (b -1)² +4b² -22b +19 +4(b -1)b +4(b -1)=0 Expand: b² -2b +1 +4b² -22b +19 +4b² -4b +4b -4=0 Combine like terms: 9b² -24b +16=0 Solve: b=(24±sqrt(576-576))/18=24/18=4/3 So, b=4/3, a=1/38. Verify the radius: Distance from (1/3,4/3) to (0,2): sqrt((1/3)^2 + (4/3 -2)^2)=sqrt(1/9 +4/9)=sqrt(5/9)=sqrt(5)/3 Distance to tangent line: |2*(1/3) -4/3 -1| /sqrt(5)=|2/3 -4/3 -3/3|=|-5/3| /sqrt(5)=5/(3sqrt(5))=sqrt(5)/3 So, both distances are equal, confirming the radius.Therefore, the center is at (1/3,4/3). But wait, in the initial problem, the user's solution had a different answer. Let me check that again.The user's solution started by finding the tangent line with slope m=1, which is incorrect because the derivative at (1,1) is 2, not 1. So, their initial step was wrong, leading to an incorrect center at (1/2,3/2). Therefore, my corrected solution with a=1/3 and b=4/3 is the accurate one.I think I've thoroughly checked all steps, and everything seems consistent. So, the center of the circle is (1/3,4/3).

answer：First, I need to determine the total number of possible 5-card hands from a standard deck of 52 cards. This can be calculated using the combination formula ( binom{52}{5} ), which equals 2,598,960.Next, to calculate the number of possible flush hands, I consider that a flush consists of 5 cards all from the same suit. There are 4 suits in the deck, so I need to choose one suit, which can be done in 4 ways.For each chosen suit, I need to select 5 cards out of the 13 available in that suit. This is calculated using the combination formula ( binom{13}{5} ), which equals 1,287.Therefore, the total number of flush hands is the product of the number of ways to choose the suit and the number of ways to choose the cards within that suit: ( 4 times 1,287 = 5,148 ).Finally, to find the probability of being dealt a flush, I divide the number of flush hands by the total number of possible 5-card hands: ( frac{5,148}{2,598,960} ). Simplifying this fraction gives the probability of being dealt a flush.

answer：Okay, so I've got this statistics problem about a medical study looking at the relationship between a local disease and hygiene habits. They surveyed 100 people with the disease (case group) and 100 without (control group). The data is in a table showing how many in each group have "not good enough" or "good enough" hygiene.Part 1 asks if we can be 99% confident that there's a difference in hygiene habits between the two groups. I think this is a hypothesis test situation. Maybe using a chi-square test? They gave a formula for K², which looks like the chi-square statistic. The table also provides critical values for different significance levels, including 0.010, which corresponds to 99% confidence.First, I need to set up the observed data. The table is:| | Not Good Enough | Good Enough ||--------------|-----------------|-------------|| Case Group | 40 | 60 || Control Group| 10 | 90 |I should calculate the expected frequencies under the null hypothesis that there's no association between disease and hygiene. To do that, I need the totals for each row and column.Total for Case Group: 40 + 60 = 100Total for Control Group: 10 + 90 = 100Total for Not Good Enough: 40 + 10 = 50Total for Good Enough: 60 + 90 = 150Overall total: 100 + 100 = 200Now, expected frequency for each cell is (row total * column total) / overall total.For Case Group and Not Good Enough: (100 * 50) / 200 = 25Case Group and Good Enough: (100 * 150) / 200 = 75Control Group and Not Good Enough: (100 * 50) / 200 = 25Control Group and Good Enough: (100 * 150) / 200 = 75So the expected table is:| | Not Good Enough | Good Enough ||--------------|-----------------|-------------|| Case Group | 25 | 75 || Control Group| 25 | 75 |Now, calculate the chi-square statistic using the formula:K² = n(ad - bc)² / [(a+b)(c+d)(a+c)(b+d)]Where a=40, b=60, c=10, d=90, n=200.First, compute ad - bc: 40*90 - 60*10 = 3600 - 600 = 3000Then, square that: 3000² = 9,000,000Multiply by n: 200 * 9,000,000 = 1,800,000,000Now, compute the denominator: (a+b)(c+d)(a+c)(b+d) = (100)(100)(50)(150) = 100*100=10,000; 50*150=7,500; so 10,000*7,500=75,000,000So K² = 1,800,000,000 / 75,000,000 = 24Now, compare this to the critical value for 99% confidence, which is 6.635. Since 24 > 6.635, we reject the null hypothesis. So, yes, we can be 99% confident there's a difference.Moving on to Part 2, which is about calculating a risk level indicator R. It's defined as the ratio of probabilities P(B|A)/P(not B|A) and P(B|not A)/P(not B|not A). They want us to prove that R can be expressed in terms of conditional probabilities given B and not B.Starting with R = [P(B|A)/P(not B|A)] * [P(B|not A)/P(not B|not A)]I need to show this is equal to [P(A|B)/P(not A|B)] * [P(not A|not B)/P(A|not B)]Let me recall Bayes' theorem and properties of conditional probabilities. Maybe express everything in terms of joint probabilities.P(B|A) = P(A and B)/P(A)Similarly, P(not B|A) = P(A and not B)/P(A)So P(B|A)/P(not B|A) = [P(A and B)/P(A)] / [P(A and not B)/P(A)] = P(A and B)/P(A and not B)Similarly, P(B|not A)/P(not B|not A) = P(not A and B)/P(not A and not B)So R = [P(A and B)/P(A and not B)] * [P(not A and B)/P(not A and not B)]Now, let's look at the expression we need to prove:[P(A|B)/P(not A|B)] * [P(not A|not B)/P(A|not B)]Expressed in terms of joint probabilities:P(A|B) = P(A and B)/P(B)P(not A|B) = P(not A and B)/P(B)So P(A|B)/P(not A|B) = [P(A and B)/P(B)] / [P(not A and B)/P(B)] = P(A and B)/P(not A and B)Similarly, P(not A|not B) = P(not A and not B)/P(not B)P(A|not B) = P(A and not B)/P(not B)So P(not A|not B)/P(A|not B) = [P(not A and not B)/P(not B)] / [P(A and not B)/P(not B)] = P(not A and not B)/P(A and not B)Multiplying these two ratios:[P(A and B)/P(not A and B)] * [P(not A and not B)/P(A and not B)] = [P(A and B) * P(not A and not B)] / [P(not A and B) * P(A and not B)]Wait, this seems similar to R's expression earlier. Let me check:Earlier, R was [P(A and B)/P(A and not B)] * [P(not A and B)/P(not A and not B)] = [P(A and B) * P(not A and B)] / [P(A and not B) * P(not A and not B)]But the expression from the proof is [P(A and B) * P(not A and not B)] / [P(not A and B) * P(A and not B)]Hmm, seems like there's a discrepancy. Maybe I made a mistake in the proof steps.Wait, let's go back. The original R is [P(B|A)/P(not B|A)] * [P(B|not A)/P(not B|not A)]Expressed as:[P(A and B)/P(A)] / [P(A and not B)/P(A)] * [P(not A and B)/P(not A)] / [P(not A and not B)/P(not A)] = [P(A and B)/P(A and not B)] * [P(not A and B)/P(not A and not B)]Which is the same as [P(A and B) * P(not A and B)] / [P(A and not B) * P(not A and not B)]But the expression we need to prove is [P(A|B)/P(not A|B)] * [P(not A|not B)/P(A|not B)] = [P(A and B)/P(not A and B)] * [P(not A and not B)/P(A and not B)] = [P(A and B) * P(not A and not B)] / [P(not A and B) * P(A and not B)]So, actually, R is equal to [P(A and B) * P(not A and not B)] / [P(not A and B) * P(A and not B)]But in the problem statement, they have R as [P(A|B)/P(not A|B)] * [P(not A|not B)/P(A|not B)] which is equal to [P(A and B)/P(not A and B)] * [P(not A and not B)/P(A and not B)] = same as above.Therefore, R is equal to [P(A|B)/P(not A|B)] * [P(not A|not B)/P(A|not B)]So the proof is correct.Now, part (ii) asks to estimate R using the survey data.First, estimate P(A|B) and P(A|not B).From the data:P(A|B) = number of cases with not good enough hygiene / total cases = 40/100 = 0.4P(A|not B) = number of controls with not good enough hygiene / total controls = 10/100 = 0.1Similarly, P(not A|B) = 1 - P(A|B) = 0.6P(not A|not B) = 1 - P(A|not B) = 0.9Now, plug these into the expression for R:R = [P(A|B)/P(not A|B)] * [P(not A|not B)/P(A|not B)] = [0.4/0.6] * [0.9/0.1] = (2/3) * 9 = 6So the estimated R is 6.I think that's it. I should double-check my calculations to make sure I didn't make any arithmetic errors.

answer：First, I need to determine the total area of the 6x6 grid. Since each small square measures 1 cm by 1 cm, the entire grid has an area of 6 cm multiplied by 6 cm, which equals 36 cm².Next, I'll calculate the area of the unshaded circles. There is one large circle with a radius of 1.5 cm. The area of a circle is given by the formula πr², so the area of the large circle is π times (1.5)², which equals 2.25π cm².There are also five smaller circles, each with a radius of 0.5 cm. The area of one small circle is π times (0.5)², which equals 0.25π cm². Since there are five of these small circles, their combined area is 5 times 0.25π, resulting in 1.25π cm².Adding the areas of the large and small circles together gives a total unshaded area of 2.25π + 1.25π, which equals 3.5π cm².Finally, to find the area of the visible shaded region, I subtract the total unshaded area from the total grid area: 36 cm² minus 3.5π cm². This gives the visible shaded area as 36 - 3.5π cm².Therefore, the values of C and D are 36 and 3.5, respectively. Adding these together, C + D equals 36 + 3.5, which is 39.5.

answer：Alright, so I have this problem here: Let P(x) and Q(x) be non-constant polynomials with integer coefficients. It's given that the polynomial P(x)Q(x) - 2009 has at least 25 distinct integer roots. I need to prove that the degree of each polynomial P(x) and Q(x) is greater than 2.Hmm, okay. Let me try to unpack this step by step. First, since P(x) and Q(x) are polynomials with integer coefficients, their product P(x)Q(x) will also have integer coefficients. Then, subtracting 2009 from that product, we get another polynomial with integer coefficients: P(x)Q(x) - 2009.It's given that this polynomial has at least 25 distinct integer roots. That means there are 25 different integer values, say a1, a2, ..., a25, such that when we plug each ai into P(x)Q(x) - 2009, the result is zero. So, for each ai, P(ai)Q(ai) = 2009.Now, since P(ai) and Q(ai) are both integers (because P and Q have integer coefficients and ai are integers), their product is 2009. So, P(ai) and Q(ai) must be integer divisors of 2009.Let me factorize 2009 to find its divisors. 2009 divided by 7 is 287, and 287 divided by 7 is 41. So, 2009 = 7^2 * 41. Therefore, the positive divisors of 2009 are 1, 7, 49, 41, 287, and 2009. Including the negative divisors, we have -1, -7, -49, -41, -287, and -2009. So, in total, there are 12 integer divisors of 2009.Now, for each ai, P(ai) and Q(ai) must be among these 12 divisors. Since there are 25 different ai's, but only 12 possible pairs of divisors (since P(ai) and Q(ai) multiply to 2009), by the Pigeonhole Principle, at least one of these divisor pairs must repeat multiple times.Wait, actually, each ai gives a pair (P(ai), Q(ai)) such that their product is 2009. So, the number of possible distinct pairs is equal to the number of distinct factorizations of 2009 into two integers. Since 2009 has 12 divisors, the number of such pairs is 12, considering both positive and negative divisors.But since we have 25 roots, which is more than 12, by the Pigeonhole Principle, at least one of these pairs must occur at least ceiling(25/12) = 3 times. So, there exists at least one pair (d, e) such that d*e = 2009, and for at least 3 different ai's, P(ai) = d and Q(ai) = e.Therefore, P(x) - d has at least 3 distinct integer roots, and Q(x) - e has at least 3 distinct integer roots. Since a non-constant polynomial of degree n can have at most n roots, this implies that the degree of P(x) - d is at least 3, and similarly for Q(x) - e.But P(x) - d is just a shift of P(x) by a constant, so it doesn't change the degree. Therefore, the degree of P(x) must be at least 3, and similarly, the degree of Q(x) must be at least 3.Wait, but the problem says "greater than 2," which is the same as "at least 3." So, that seems to be the conclusion.But let me double-check if there's any possibility that one of the polynomials could have degree exactly 2. Suppose, for contradiction, that P(x) has degree 2. Then, P(x) - d is also degree 2. But a degree 2 polynomial can have at most 2 roots. However, we have at least 3 roots for P(x) - d, which is a contradiction. Therefore, P(x) must have degree greater than 2. Similarly, Q(x) must have degree greater than 2.So, putting it all together, the degrees of both P(x) and Q(x) must be greater than 2.