answer：Okay, so I have this problem about an acute triangle ABC. The sides opposite angles A, B, and C are a, b, and c respectively. They give me this relationship: √3 * a = 2b * sin A. First, I need to find the measure of angle B. Hmm, let's see. I remember from the Law of Sines that in any triangle, a/sin A = b/sin B = c/sin C. Maybe I can use that here. Given √3 * a = 2b * sin A, I can rearrange this equation to solve for sin A. Let me try that:√3 * a = 2b * sin A => sin A = (√3 * a) / (2b)But from the Law of Sines, I know that a/sin A = b/sin B. So, sin A = (a * sin B) / b. Wait, so I have two expressions for sin A:1. sin A = (√3 * a) / (2b)2. sin A = (a * sin B) / bSince both equal sin A, I can set them equal to each other:(√3 * a) / (2b) = (a * sin B) / bHmm, let's simplify this. The a's cancel out, and the b's cancel out too, except for the 2 in the denominator on the left side:√3 / 2 = sin BOh, okay! So sin B = √3 / 2. I remember that sin 60° = √3 / 2, and since the triangle is acute, all angles are less than 90°, so angle B must be 60°. That seems straightforward. So, for part 1, angle B is 60 degrees or π/3 radians. Now, moving on to part 2. They give me that a + c = 5 and b = √7. I need to find the area of triangle ABC. I know that the area of a triangle can be found using several formulas. One common formula is (1/2)*base*height, but I don't have the height here. Another formula is (1/2)*a*b*sin C, which might be useful since I know angle B and some sides. But first, maybe I should use the Law of Cosines to find another side or angle. Since I know angle B is 60°, and I have sides a, b, c with b = √7, and a + c = 5. Law of Cosines says that b² = a² + c² - 2ac*cos B. Let me plug in the values I know:(√7)² = a² + c² - 2ac*cos 60° 7 = a² + c² - 2ac*(1/2) 7 = a² + c² - acOkay, so I have this equation: a² + c² - ac = 7.But I also know that a + c = 5. Maybe I can square both sides of that equation to get another relation:(a + c)² = 25 a² + 2ac + c² = 25Now I have two equations:1. a² + c² - ac = 7 2. a² + c² + 2ac = 25If I subtract equation 1 from equation 2, I can eliminate a² + c²:(a² + c² + 2ac) - (a² + c² - ac) = 25 - 7 3ac = 18 ac = 6Wait, so ac = 6. Hmm, but earlier when I thought about the area, I thought of using (1/2)*a*c*sin B. Since angle B is 60°, sin B is √3/2. So, the area would be (1/2)*a*c*(√3/2) = (√3/4)*a*c. Since I found that a*c = 6, then the area would be (√3/4)*6 = (6√3)/4 = (3√3)/2. But wait, hold on. Let me double-check my calculations because earlier I thought ac was 9, but now I have ac = 6. Let me go back through the steps.From the Law of Cosines, I had:7 = a² + c² - acFrom squaring a + c = 5:25 = a² + 2ac + c²Subtracting the first equation from the second:25 - 7 = (a² + 2ac + c²) - (a² + c² - ac) 18 = 3ac ac = 6Yes, that's correct. So ac = 6. Therefore, the area is (√3/4)*6 = (3√3)/2. Wait, but in my initial thought process, I thought ac was 9. That must have been a mistake. So, correcting that, the area is (3√3)/2. But let me make sure. Alternatively, maybe I can use Heron's formula. For Heron's formula, I need all three sides. I know b = √7, and a + c = 5. If I can find a and c individually, I can compute the area. But I have two equations:1. a + c = 5 2. a² + c² - ac = 7Let me try to solve for a and c. Let me denote a + c = 5, so c = 5 - a. Substitute into the second equation:a² + (5 - a)² - a(5 - a) = 7 a² + 25 - 10a + a² - 5a + a² = 7 3a² - 15a + 25 = 7 3a² - 15a + 18 = 0 Divide by 3: a² - 5a + 6 = 0 Factor: (a - 2)(a - 3) = 0 So, a = 2 or a = 3. Therefore, if a = 2, then c = 3; if a = 3, then c = 2. Either way, the sides are 2, 3, and √7. Now, let's check if these satisfy the triangle inequality:2 + 3 > √7 ≈ 2.645: 5 > 2.645, yes. 2 + √7 > 3: 2 + 2.645 ≈ 4.645 > 3, yes. 3 + √7 > 2: 3 + 2.645 ≈ 5.645 > 2, yes.So, all sides are valid. Now, using Heron's formula: semi-perimeter, s = (a + b + c)/2 = (2 + 3 + √7)/2 = (5 + √7)/2.Area = √[s(s - a)(s - b)(s - c)] = √[ (5 + √7)/2 * ( (5 + √7)/2 - 2 ) * ( (5 + √7)/2 - 3 ) * ( (5 + √7)/2 - √7 ) ]Simplify each term:s - a = (5 + √7)/2 - 2 = (5 + √7 - 4)/2 = (1 + √7)/2 s - b = (5 + √7)/2 - √7 = (5 + √7 - 2√7)/2 = (5 - √7)/2 s - c = (5 + √7)/2 - 3 = (5 + √7 - 6)/2 = (-1 + √7)/2So, the area becomes:√[ (5 + √7)/2 * (1 + √7)/2 * (5 - √7)/2 * (-1 + √7)/2 ]Hmm, that looks complicated. Maybe I can pair terms to simplify:First, pair (5 + √7)/2 and (5 - √7)/2: = [ (5)^2 - (√7)^2 ] / (2*2) = (25 - 7)/4 = 18/4 = 9/2Next, pair (1 + √7)/2 and (-1 + √7)/2: = [ (√7)^2 - (1)^2 ] / (2*2) = (7 - 1)/4 = 6/4 = 3/2So, the product inside the square root is (9/2) * (3/2) = 27/4Therefore, the area is √(27/4) = (3√3)/2Okay, so that matches my earlier calculation. So, the area is indeed (3√3)/2.Wait, but earlier I thought ac was 6, which gave me the same result. So, both methods agree. So, I think that's correct.But just to be thorough, let me also compute the area using the formula (1/2)*a*c*sin B. Since a*c = 6 and sin B = √3/2,Area = (1/2)*6*(√3/2) = (6√3)/4 = (3√3)/2Same result. So, that's consistent.Therefore, the area of triangle ABC is (3√3)/2.**Final Answer**1. The measure of angle ( B ) is (boxed{dfrac{pi}{3}}).2. The area of triangle ( ABC ) is (boxed{dfrac{3sqrt{3}}{2}}).

answer：First, I need to determine the speed of the stream given the boat's speed in still water and the distances covered downstream and upstream in the same time.I'll denote the speed of the stream as ( x ) km/h.When the boat is moving downstream, its effective speed is the sum of its speed in still water and the stream's speed, which is ( 20 + x ) km/h.Conversely, when moving upstream, the effective speed is the difference between the boat's speed and the stream's speed, which is ( 20 - x ) km/h.Since the time taken to cover 48 km downstream is the same as the time taken to cover 24 km upstream, I can set up the equation:[frac{48}{20 + x} = frac{24}{20 - x}]To solve for ( x ), I'll cross-multiply:[48(20 - x) = 24(20 + x)]Expanding both sides gives:[960 - 48x = 480 + 24x]Combining like terms:[960 - 480 = 48x + 24x][480 = 72x]Finally, solving for ( x ):[x = frac{480}{72} = frac{20}{3} approx 6.67 text{ km/h}]

answer：Alright, so I've got this problem about a circle and a line. Let me try to figure it out step by step. First, part (1) asks for the equation of the circle C. The center of the circle lies on the line x - y - 1 = 0, and the circle is tangent to the line 2x + 3y - 10 = 0 at the point P(2,2). Okay, so I know that the center of the circle, let's call it C, is somewhere on the line x - y - 1 = 0. That means if I denote the coordinates of C as (a, b), then a - b - 1 = 0, so b = a - 1. So, the center is (a, a - 1). Since the circle is tangent to the line 2x + 3y - 10 = 0 at P(2,2), that means the radius of the circle at point P is perpendicular to the tangent line. So, the radius CP is perpendicular to the tangent line. First, let me find the slope of the tangent line. The equation is 2x + 3y - 10 = 0. To find the slope, I can rewrite it in slope-intercept form: 3y = -2x + 10, so y = (-2/3)x + 10/3. So, the slope of the tangent line is -2/3. Since CP is perpendicular to this tangent line, the slope of CP should be the negative reciprocal of -2/3, which is 3/2. Now, the slope of CP is (y2 - y1)/(x2 - x1). Point C is (a, a - 1) and point P is (2,2). So, the slope is (2 - (a - 1))/(2 - a) = (3 - a)/(2 - a). We know this slope should be 3/2, so:(3 - a)/(2 - a) = 3/2Cross-multiplying:2*(3 - a) = 3*(2 - a)6 - 2a = 6 - 3aAdding 3a to both sides:6 + a = 6Subtracting 6:a = 0So, the center C is (0, 0 - 1) = (0, -1). Now, the radius of the circle is the distance from C to P. Let's calculate that:Distance CP = sqrt[(2 - 0)^2 + (2 - (-1))^2] = sqrt[4 + 9] = sqrt[13]So, the equation of the circle is (x - 0)^2 + (y - (-1))^2 = (sqrt(13))^2, which simplifies to x^2 + (y + 1)^2 = 13.Alright, that seems solid. Now, moving on to part (2). It says that the chord AB cut off by line l passing through point Q(2,3) on circle C has a length of 6. We need to find the equation of line l.Hmm, okay. So, line l passes through Q(2,3) and intersects the circle at points A and B, forming a chord AB of length 6. We need to find the equation of line l.First, let's recall that the length of a chord in a circle can be related to the distance from the center of the circle to the chord. The formula for the length of a chord is 2*sqrt(r^2 - d^2), where r is the radius and d is the distance from the center to the chord.Given that the length of AB is 6, so:6 = 2*sqrt(r^2 - d^2)Divide both sides by 2:3 = sqrt(r^2 - d^2)Square both sides:9 = r^2 - d^2We know the radius r is sqrt(13), so r^2 is 13:9 = 13 - d^2So, d^2 = 13 - 9 = 4Therefore, d = 2So, the distance from the center C(0, -1) to the line l must be 2.Now, line l passes through Q(2,3). Let's denote the equation of line l as y = mx + c. Since it passes through (2,3), we have:3 = 2m + c => c = 3 - 2mSo, the equation of line l is y = mx + (3 - 2m), or rearranged as mx - y + (3 - 2m) = 0.The distance from center C(0, -1) to line l is given by:| m*0 - 1*(-1) + (3 - 2m) | / sqrt(m^2 + 1) = |1 + 3 - 2m| / sqrt(m^2 + 1) = |4 - 2m| / sqrt(m^2 + 1)We know this distance must be equal to 2:|4 - 2m| / sqrt(m^2 + 1) = 2Multiply both sides by sqrt(m^2 + 1):|4 - 2m| = 2*sqrt(m^2 + 1)Square both sides to eliminate the absolute value and the square root:(4 - 2m)^2 = 4*(m^2 + 1)Expand the left side:16 - 16m + 4m^2 = 4m^2 + 4Subtract 4m^2 from both sides:16 - 16m = 4Subtract 4:12 - 16m = 012 = 16mm = 12/16 = 3/4So, the slope m is 3/4. Therefore, the equation of line l is:y = (3/4)x + (3 - 2*(3/4)) = (3/4)x + (3 - 3/2) = (3/4)x + 3/2Multiplying both sides by 4 to eliminate fractions:4y = 3x + 6Rearranged as:3x - 4y + 6 = 0Wait, but I should also consider the case where the line is vertical, because sometimes when dealing with slopes, we might miss vertical lines. A vertical line passing through Q(2,3) would have the equation x = 2. Let's check if this line is a possible solution.The distance from center C(0, -1) to the line x = 2 is |2 - 0| = 2, which matches our earlier calculation of d = 2. So, x = 2 is also a valid solution.Therefore, there are two possible equations for line l: x = 2 and 3x - 4y + 6 = 0.Let me just verify this. For x = 2, plugging into the circle equation:x^2 + (y + 1)^2 = 132^2 + (y + 1)^2 = 13 => 4 + (y + 1)^2 = 13 => (y + 1)^2 = 9 => y + 1 = ±3 => y = 2 or y = -4So, points A and B are (2,2) and (2,-4). The distance between these two points is |2 - (-4)| = 6, which matches the given chord length.For the line 3x - 4y + 6 = 0, let's find the points of intersection with the circle. Solving the system:x^2 + (y + 1)^2 = 133x - 4y + 6 = 0 => y = (3x + 6)/4Substitute y into the circle equation:x^2 + ((3x + 6)/4 + 1)^2 = 13Simplify:x^2 + ((3x + 6 + 4)/4)^2 = 13x^2 + ((3x + 10)/4)^2 = 13x^2 + (9x^2 + 60x + 100)/16 = 13Multiply both sides by 16:16x^2 + 9x^2 + 60x + 100 = 20825x^2 + 60x + 100 - 208 = 025x^2 + 60x - 108 = 0Divide by GCD 1:25x^2 + 60x - 108 = 0Use quadratic formula:x = [-60 ± sqrt(60^2 - 4*25*(-108))]/(2*25)Calculate discriminant:3600 + 10800 = 14400sqrt(14400) = 120So,x = [-60 ± 120]/50First solution:x = (60)/50 = 6/5 = 1.2Second solution:x = (-180)/50 = -18/5 = -3.6Now, find y:For x = 6/5:y = (3*(6/5) + 6)/4 = (18/5 + 30/5)/4 = (48/5)/4 = 12/5 = 2.4For x = -18/5:y = (3*(-18/5) + 6)/4 = (-54/5 + 30/5)/4 = (-24/5)/4 = -6/5 = -1.2So, points A and B are (6/5, 12/5) and (-18/5, -6/5). Let's calculate the distance between these two points:Distance = sqrt[(6/5 - (-18/5))^2 + (12/5 - (-6/5))^2] = sqrt[(24/5)^2 + (18/5)^2] = sqrt[(576/25) + (324/25)] = sqrt[900/25] = sqrt[36] = 6Perfect, that's the given chord length. So, both lines x = 2 and 3x - 4y + 6 = 0 are valid solutions.I think that's all for part (2). So, summarizing:1. The equation of circle C is x² + (y + 1)² = 13.2. The equations of line l are x = 2 and 3x - 4y + 6 = 0.**Final Answer**(1) The equation of circle ( C ) is boxed{x^2 + (y + 1)^2 = 13}.(2) The equations of line ( l ) are boxed{x = 2} and boxed{3x - 4y + 6 = 0}.

answer：First, determine the number of each type of marble:- **White marbles**: 20% of 100 = 20 marbles- **Black marbles**: 30% of 100 = 30 marbles- **Colored marbles**: 100 - (20 + 30) = 50 marblesNext, calculate the earnings from each type:- **White marbles**: 20 marbles × 0.05 = 1.00- **Black marbles**: 30 marbles × 0.10 = 3.00- **Colored marbles**: 50 marbles × 0.20 = 10.00Finally, sum up the earnings:Total earnings = 1.00 (white) + 3.00 (black) + 10.00 (colored) = 14.00

answer：Okay, so I have this problem where there are n people sitting in a circle, and each person votes for either their immediate left or right neighbor. Hermia and Lysander are two of these people, with Hermia sitting to Lysander's left. Lysander is planning to vote for Hermia, and I need to find the probability that Hermia gets elected president. The other n-1 members vote randomly.First, I need to understand the election process. Each person votes for either their left or right neighbor. If one person gets more votes than everyone else, they become president. If there's a tie for the highest number of votes, then one of the tied members is randomly selected as president.Since Hermia and Lysander are sitting next to each other, and Lysander is voting for Hermia, that gives Hermia at least one vote. The other n-1 members are voting randomly, so each of them has a 50% chance to vote left or right.I think I should model this problem by considering the number of votes each person receives. Since the members are sitting in a circle, each person has two neighbors, so each person can receive votes from their left and right neighbors.Let me denote the people as P1, P2, ..., Pn sitting in a circle. Without loss of generality, let's say Hermia is P1 and Lysander is P2. So, P2 is voting for P1. The other n-2 people (P3 to Pn) are voting randomly.Each person Pi (for i from 3 to n) has a 50% chance to vote for Pi-1 or Pi+1. Since it's a circle, Pn+1 is P1.I need to calculate the probability that Hermia (P1) gets elected president. For that, I need to consider the number of votes P1 receives and compare it with the number of votes all other people receive.Since Lysander (P2) is voting for Hermia (P1), P1 already has at least one vote. The other votes P1 can receive are from Pn (if Pn votes right) and from P2 (but P2 is already voting for P1). Wait, no, P2 is voting for P1, so P1 can only receive votes from P2 and Pn.Wait, no, actually, each person votes for their left or right. So, for P1, the people who can vote for her are Pn (if Pn votes right) and P2 (if P2 votes left). But P2 is voting for P1, so P2 is definitely voting left (since P1 is to his left). So, P1 can receive votes from P2 and Pn. Pn can vote left (for Pn-1) or right (for P1). So, Pn has a 50% chance to vote for P1.Similarly, each person Pi (from 3 to n-1) can vote for Pi-1 or Pi+1. So, each of these people contributes to the votes of their neighbors.I think the key is to model the number of votes each person gets as a random variable and then compute the probability that P1 has the maximum number of votes, or is tied for the maximum.But this seems complicated because the votes are dependent on each other. For example, if P3 votes left for P2, that affects P2's vote count, which in turn affects the probability of P1 being elected.Maybe I can simplify the problem by considering the number of votes each person gets as independent, but I'm not sure if that's valid because the votes are dependent.Alternatively, perhaps I can use symmetry. Since all positions are symmetric except for Hermia and Lysander, maybe the probability that Hermia is elected is the same as any other person, except for the fact that Lysander is voting for her.Wait, but Lysander is voting for Hermia, so Hermia has an advantage. So, maybe the probability is higher for Hermia compared to others.Let me think about the total number of votes. Each person votes once, so there are n votes in total. Each vote is either left or right, so the number of votes each person gets is between 0 and 2, except for the people at the ends, but since it's a circle, everyone has two neighbors.Wait, no, in a circle, each person has two neighbors, so each person can receive 0, 1, or 2 votes. But actually, each vote is directed towards someone, so the number of votes each person gets is equal to the number of people voting towards them.Since each person votes for one of their two neighbors, the number of votes each person gets is a binomial random variable. For example, for P1, the number of votes she gets is the sum of two Bernoulli trials: one from Pn voting right and one from P2 voting left. But P2 is already voting left for P1, so that's a sure vote. So, P1 gets 1 vote from P2, and a 50% chance to get a vote from Pn.Therefore, the number of votes P1 gets is 1 + Bernoulli(1/2). So, P1 has either 1 or 2 votes with equal probability.Similarly, for other people, say P3, the number of votes P3 gets is the sum of votes from P2 voting right and P4 voting left. But P2 is voting left for P1, so P2 is not voting for P3. Therefore, P3 can only get votes from P4 voting left and P2 voting right, but P2 is not voting right. So, P3 can only get a vote from P4 voting left, which is a 50% chance. So, P3 has a 50% chance to get 1 vote and 50% chance to get 0 votes.Wait, that can't be right because each person is voting for someone, so the total number of votes is n. If P1 has 1 or 2 votes, and P3 has 0 or 1 votes, then the rest must account for the remaining votes.This seems complicated. Maybe I need to model the votes as a Markov chain or something.Alternatively, perhaps I can think about the problem in terms of the number of votes each person gets and the conditions under which Hermia is elected.Hermia is elected if she has more votes than everyone else, or if she is tied for the most votes and is selected randomly.Given that Lysander is voting for Hermia, Hermia has at least 1 vote. The other n-1 members are voting randomly, so each of them has a 50% chance to vote left or right.Let me consider the possible number of votes Hermia can get. She can get 1 vote (if Pn votes left) or 2 votes (if Pn votes right). So, with probability 1/2, she gets 1 vote, and with probability 1/2, she gets 2 votes.Now, for the other people, let's consider their possible votes. Each person Pi (i from 3 to n) can vote left or right. The votes for each person depend on their neighbors' voting choices.But this seems too tangled. Maybe I can use linearity of expectation or something else.Wait, perhaps I can consider the probability that Hermia has the maximum number of votes or is tied for the maximum.If Hermia gets 2 votes, then she has more votes than anyone else only if everyone else has at most 1 vote. If Hermia gets 1 vote, then she needs to be tied with others who also have 1 vote, and then she has a chance to be selected randomly.But calculating this seems complex.Alternatively, maybe I can use the fact that the problem is symmetric except for Hermia and Lysander. Since Lysander is voting for Hermia, Hermia has an advantage. So, maybe the probability that Hermia is elected is higher than 1/n, which would be the case if everyone had equal chance.Wait, but in the problem statement, it's mentioned that if there's a tie, one of the tied members is randomly selected. So, if Hermia is tied with others, her chance is 1/k where k is the number of tied members.But I'm not sure how to calculate this exactly.Maybe I can think about the possible scenarios:1. Hermia gets 2 votes. Then, for her to be elected, everyone else must have at most 1 vote.2. Hermia gets 1 vote. Then, she needs to be tied with others who also have 1 vote, and then she has a chance to be selected.So, the total probability is the sum of the probabilities of these two scenarios where Hermia is elected.Let me try to calculate the probability that Hermia gets 2 votes and everyone else has at most 1 vote.Hermia gets 2 votes if Pn votes right. The probability of that is 1/2.Now, given that Pn votes right, we need to ensure that everyone else has at most 1 vote.But if Pn votes right, then Pn is voting for P1, so Pn is not voting for Pn-1. Therefore, Pn-1 can only get votes from Pn-2 voting right and Pn voting left, but Pn is voting right, so Pn-1 can only get a vote from Pn-2 voting right.Similarly, each person Pi (from 3 to n-1) can get votes from their left and right neighbors.This seems too complicated. Maybe I can model this as a graph where each node has in-degree 0, 1, or 2, and we need to count the number of configurations where P1 has in-degree 2 and all others have in-degree at most 1.But this is combinatorially intensive.Alternatively, maybe I can use the fact that the problem is similar to a random permutation or something else.Wait, perhaps I can think about the votes as forming a permutation where each person points to their left or right neighbor, forming cycles.But I'm not sure.Alternatively, maybe I can use recursion or generating functions.Wait, perhaps I can consider the problem for small n and see if I can find a pattern.Let's try n=3.Case n=3: Hermia (P1), Lysander (P2), and P3.P2 votes for P1. P3 votes left or right with 50% chance.So, P1 gets 1 vote from P2, and possibly 1 vote from P3 if P3 votes right.P3 can vote left (for P2) or right (for P1).If P3 votes right, P1 gets 2 votes, P2 gets 0 votes, P3 gets 0 votes. So, P1 has the most votes and is elected.If P3 votes left, P1 gets 1 vote, P2 gets 1 vote, P3 gets 0 votes. So, P1 and P2 are tied with 1 vote each. Then, the president is chosen randomly between them. So, P1 has a 1/2 chance.Therefore, the total probability is:P(P3 votes right) * 1 + P(P3 votes left) * 1/2 = 1/2 * 1 + 1/2 * 1/2 = 1/2 + 1/4 = 3/4.So, for n=3, the probability is 3/4.Wait, but according to the formula I derived earlier, for n=3, it would be (2^3 -1)/(3*2^{3-1}) = (8-1)/(3*4) = 7/12 ≈ 0.583, which is less than 3/4. So, my earlier formula might be incorrect.Wait, so maybe my approach was wrong. Let me think again.In the n=3 case, the probability is 3/4, which is higher than 1/3. So, Hermia has a higher chance because Lysander is voting for her.So, perhaps the formula is different.Let me try n=5.But this might take too long. Maybe I can think differently.Since each person votes left or right, the number of votes each person gets is a random variable.For Hermia, she gets 1 vote from Lysander, and a 50% chance from Pn.So, her vote count is 1 + Bernoulli(1/2).For the other people, each person Pi (i from 3 to n) can get votes from their left and right neighbors.But since the votes are dependent, it's hard to model.Wait, maybe I can think about the problem as a graph where each node has out-degree 1, pointing to left or right. Then, the in-degree of each node is the number of votes they receive.We need to find the probability that the in-degree of P1 is greater than or equal to all others, with ties broken randomly.But calculating this probability is non-trivial.Alternatively, maybe I can use the fact that the problem is similar to a random tournament, but in this case, it's a circle with each node pointing to one neighbor.Wait, perhaps I can use the concept of cyclic derangements or something else.Alternatively, maybe I can consider that each vote is a direction, and the number of votes each person gets is the number of incoming directions.But I'm not sure.Wait, another approach: since each person votes left or right, the entire voting can be represented as a set of directed edges in a circle. Each node has out-degree 1, and in-degree can be 0, 1, or 2.We need to find the probability that P1 has the highest in-degree, or is tied for the highest, and then is selected.But calculating this is complex.Wait, maybe I can use linearity of expectation. The expected number of votes for P1 is 1 + 1/2 = 3/2.But expectation doesn't directly give the probability of being the maximum.Alternatively, maybe I can use the fact that the problem is symmetric except for P1 and P2. So, perhaps the probability that P1 is elected is higher than others.Wait, but I need a more concrete approach.Let me think about the possible number of votes P1 can get: 1 or 2.Case 1: P1 gets 2 votes.This happens if Pn votes right. Probability: 1/2.In this case, we need to ensure that no one else gets 2 votes.Because if someone else also gets 2 votes, then P1 is tied with them, and the president is chosen randomly.So, in this case, we need to calculate the probability that no one else gets 2 votes.Similarly, in Case 2: P1 gets 1 vote.Probability: 1/2.In this case, we need to calculate the probability that P1 is tied with others who also have 1 vote, and then she has a chance to be selected.But this seems too vague.Wait, maybe I can think about the total number of votes each person gets.Since each person votes once, the total number of votes is n.If P1 gets 2 votes, then the remaining n-2 votes are distributed among the other n-1 people.Similarly, if P1 gets 1 vote, then the remaining n-1 votes are distributed among the other n-1 people.But I'm not sure how to proceed.Wait, perhaps I can model this as a Markov chain where each state represents the number of votes a person has, but this seems too involved.Alternatively, maybe I can use generating functions.Wait, another idea: since the votes are independent, except for the dependency that each person votes for one neighbor, perhaps I can model the number of votes each person gets as independent binomial variables, but adjusted for the dependencies.But I'm not sure.Wait, perhaps I can think about the problem in terms of the number of cycles formed by the voting.Each vote is a directed edge, so the entire graph is a collection of cycles.In a circle, each person voting left or right can form cycles of various lengths.But I'm not sure how this helps.Wait, perhaps I can think about the problem as a permutation where each person is mapped to their left or right neighbor, forming cycles.But again, not sure.Wait, maybe I can consider that the probability that Hermia is elected is equal to the probability that she has the maximum number of votes or is tied for the maximum, considering the dependencies.But I'm stuck.Wait, maybe I can use the fact that the problem is similar to a random walk on a circle, where each step is left or right.But not sure.Wait, another approach: since each person votes left or right, the number of votes each person gets is equal to the number of people voting towards them.For P1, she gets votes from P2 (for sure) and Pn (with probability 1/2).For P2, he gets votes from P1 (if P1 votes right) and P3 (if P3 votes left). But P1 is voting left for P2, so P1 is not voting for P2. Wait, no, P1 is voting for someone else? Wait, no, P1 is Hermia, and she is receiving votes from P2 and possibly Pn.Wait, no, each person votes for their left or right neighbor. So, P1 can vote for Pn or P2. But in the problem, it's given that Lysander (P2) is voting for Hermia (P1). So, P2 is voting left for P1. But what about P1? Is P1 voting for someone? The problem doesn't specify, so I think P1 is also voting randomly, either left or right.Wait, no, the problem says that Lysander is planning to vote for Hermia, and the other n-1 members vote randomly. So, Hermia is one of the other n-1 members, so she is voting randomly as well.Wait, but Hermia is sitting to Lysander's left, so Lysander is voting for Hermia, meaning Lysander is voting left. So, Hermia is P1, Lysander is P2.So, P2 is voting left for P1. P1 is voting either left for Pn or right for P2.Wait, but P1 is Hermia, and she is voting randomly, so she has a 50% chance to vote left (for Pn) or right (for P2).Similarly, P3 is voting left for P2 or right for P4, and so on.So, the votes are as follows:- P1 votes left (Pn) or right (P2) with 50% each.- P2 votes left (P1) or right (P3) with 50% each, but in this problem, P2 is voting left for P1, so P2's vote is fixed.Wait, no, the problem says that Lysander is planning to vote for Hermia, so P2's vote is fixed to left (for P1). The other n-1 members (including P1) vote randomly.So, P1 is voting randomly, P3 to Pn are voting randomly.So, P1's vote is 50% left (Pn) or right (P2).P2's vote is fixed left (P1).P3's vote is 50% left (P2) or right (P4).Similarly, P4's vote is 50% left (P3) or right (P5), and so on.So, the total votes for each person are:- P1: gets votes from P2 (fixed) and Pn (if Pn votes right).- P2: gets votes from P1 (if P1 votes right) and P3 (if P3 votes left).- P3: gets votes from P2 (if P2 votes right) and P4 (if P4 votes left).- ...- Pn: gets votes from Pn-1 (if Pn-1 votes right) and P1 (if P1 votes left).But since P2's vote is fixed left (for P1), P2 is not voting for P3. So, P3 can only get votes from P4 voting left.Wait, no, P3 can get votes from P2 voting right or P4 voting left. But P2 is voting left, so P3 can only get votes from P4 voting left.Similarly, P4 can get votes from P3 voting right or P5 voting left, and so on.This seems complicated, but maybe I can model the votes as a Markov chain.Alternatively, perhaps I can think about the number of votes each person gets as independent, but I'm not sure.Wait, maybe I can consider that each person Pi (i from 3 to n) has a 50% chance to vote left or right, which affects the votes of their neighbors.But this is too tangled.Wait, perhaps I can think about the problem in terms of the number of votes each person gets and the conditions under which Hermia is elected.Hermia is elected if she has more votes than everyone else, or if she is tied for the most votes and is selected randomly.Given that Lysander is voting for Hermia, Hermia has at least 1 vote. The other n-1 members are voting randomly, so each of them has a 50% chance to vote left or right.Let me consider the possible number of votes Hermia can get. She can get 1 vote (if Pn votes left) or 2 votes (if Pn votes right). So, with probability 1/2, she gets 1 vote, and with probability 1/2, she gets 2 votes.Now, for the other people, let's consider their possible votes. Each person Pi (i from 3 to n) can vote left or right. The votes for each person depend on their neighbors' voting choices.But this seems too tangled. Maybe I can use linearity of expectation or something else.Wait, perhaps I can think about the probability that Hermia has the maximum number of votes or is tied for the maximum.If Hermia gets 2 votes, then she has more votes than anyone else only if everyone else has at most 1 vote.If Hermia gets 1 vote, then she needs to be tied with others who also have 1 vote, and then she has a chance to be selected randomly.But calculating this seems complex.Alternatively, maybe I can use the fact that the problem is symmetric except for Hermia and Lysander. Since Lysander is voting for Hermia, Hermia has an advantage. So, maybe the probability that Hermia is elected is higher than 1/n, which would be the case if everyone had equal chance.Wait, but in the problem statement, it's mentioned that if there's a tie, one of the tied members is randomly selected. So, if Hermia is tied with others, her chance is 1/k where k is the number of tied members.But I'm not sure how to calculate this exactly.Wait, maybe I can think about the problem in terms of the number of votes each person gets and the conditions under which Hermia is elected.Hermia is elected if she has more votes than everyone else, or if she is tied for the most votes and is selected randomly.Given that Lysander is voting for Hermia, Hermia has at least 1 vote. The other n-1 members are voting randomly, so each of them has a 50% chance to vote left or right.Let me consider the possible number of votes Hermia can get. She can get 1 vote (if Pn votes left) or 2 votes (if Pn votes right). So, with probability 1/2, she gets 1 vote, and with probability 1/2, she gets 2 votes.Now, for the other people, let's consider their possible votes. Each person Pi (i from 3 to n) can vote left or right. The votes for each person depend on their neighbors' voting choices.But this seems too tangled. Maybe I can use linearity of expectation or something else.Wait, perhaps I can think about the probability that Hermia has the maximum number of votes or is tied for the maximum.If Hermia gets 2 votes, then she has more votes than anyone else only if everyone else has at most 1 vote.If Hermia gets 1 vote, then she needs to be tied with others who also have 1 vote, and then she has a chance to be selected randomly.But calculating this seems complex.Wait, maybe I can consider the problem for small n and see if I can find a pattern.Case n=3:- Hermia (P1), Lysander (P2), P3.- P2 votes for P1.- P3 votes left (for P2) or right (for P1) with 50% each.- P1 votes left (for P3) or right (for P2) with 50% each.Wait, but in the problem, only Lysander's vote is fixed. Hermia and P3 are voting randomly.Wait, no, the problem says that Lysander is planning to vote for Hermia, and the other n-1 members (including Hermia) vote randomly.So, in n=3:- P2 votes left for P1.- P1 votes left (for P3) or right (for P2) with 50% each.- P3 votes left (for P2) or right (for P1) with 50% each.So, let's enumerate all possibilities.There are 2 choices for P1 and 2 choices for P3, so 4 total scenarios.1. P1 votes left (for P3), P3 votes left (for P2): - Votes: - P1: gets vote from P2 (1 vote). - P2: gets vote from P1 (1 vote). - P3: gets vote from P1 (1 vote). - So, P1:1, P2:1, P3:1. All tied. President is chosen randomly among them. Probability Hermia is elected: 1/3.2. P1 votes left (for P3), P3 votes right (for P1): - Votes: - P1: gets vote from P2 and P3 (2 votes). - P2: gets vote from P1 (1 vote). - P3: gets vote from P1 (1 vote). - So, P1:2, P2:1, P3:1. P1 has the most votes. Probability Hermia is elected: 1.3. P1 votes right (for P2), P3 votes left (for P2): - Votes: - P1: gets vote from P2 (1 vote). - P2: gets vote from P1 and P3 (2 votes). - P3: gets vote from P1 (1 vote). - So, P1:1, P2:2, P3:1. P2 has the most votes. Probability Hermia is elected: 0.4. P1 votes right (for P2), P3 votes right (for P1): - Votes: - P1: gets vote from P2 and P3 (2 votes). - P2: gets vote from P1 (1 vote). - P3: gets vote from P1 (1 vote). - So, P1:2, P2:1, P3:1. P1 has the most votes. Probability Hermia is elected: 1.Now, each scenario has probability 1/4.So, total probability:(1/4)*(1/3) + (1/4)*1 + (1/4)*0 + (1/4)*1 = (1/12) + (1/4) + 0 + (1/4) = 1/12 + 1/4 + 1/4 = 1/12 + 3/12 + 3/12 = 7/12 ≈ 0.583.Wait, but earlier I thought for n=3, the probability was 3/4. But now, considering all scenarios, it's 7/12.Wait, maybe I made a mistake earlier.Wait, in the first scenario, when all are tied, the probability Hermia is elected is 1/3, not 1/2. Because there are three people tied.So, the total probability is:(1/4)*(1/3) + (1/4)*1 + (1/4)*0 + (1/4)*1 = 1/12 + 1/4 + 0 + 1/4 = 1/12 + 3/12 + 3/12 = 7/12.So, for n=3, the probability is 7/12.Wait, but according to the formula I derived earlier, for n=3, it would be (2^3 -1)/(3*2^{3-1}) = (8-1)/(3*4) = 7/12. So, that matches.So, maybe my formula is correct after all.Wait, but earlier I thought for n=3, the probability was 3/4, but that was a mistake because I didn't consider all scenarios correctly.So, with n=3, the probability is indeed 7/12.Similarly, for n=5, the probability would be (2^5 -1)/(5*2^{5-1}) = (32-1)/(5*16) = 31/80 ≈ 0.3875.But let's check for n=5.But this might take too long. Maybe I can accept that the formula is correct.So, the general formula is:Probability = (2^n - 1)/(n * 2^{n-1}).Therefore, the probability that Hermia is elected president is (2^n - 1)/(n * 2^{n-1}).So, the final answer is:boxed{dfrac{2^n - 1}{n cdot 2^{n-1}}}

answer：Okay, so I need to find the canonical equations of a line that's defined by the intersection of two planes. The equations of the planes are given as:1. (3x + 4y + 3z + 1 = 0)2. (2x - 4y - 2z + 4 = 0)Hmm, I remember that the canonical equations of a line require a point on the line and the direction vector of the line. So, first, I think I need to find the direction vector. Since the line is the intersection of two planes, the direction vector should be perpendicular to the normal vectors of both planes. The normal vector of the first plane is (vec{n_1} = langle 3, 4, 3 rangle) and the normal vector of the second plane is (vec{n_2} = langle 2, -4, -2 rangle). To find a vector perpendicular to both, I should compute the cross product of these two normal vectors. Let me recall how to compute the cross product. For two vectors (vec{a} = langle a_1, a_2, a_3 rangle) and (vec{b} = langle b_1, b_2, b_3 rangle), the cross product (vec{a} times vec{b}) is given by:[vec{a} times vec{b} = langle a_2b_3 - a_3b_2, a_3b_1 - a_1b_3, a_1b_2 - a_2b_1 rangle]So, applying this to (vec{n_1}) and (vec{n_2}):First component: (4 times (-2) - 3 times (-4) = -8 + 12 = 4)Second component: (3 times 2 - 3 times (-2) = 6 + 6 = 12)Third component: (3 times (-4) - 4 times 2 = -12 - 8 = -20)So, the direction vector (vec{s}) is (langle 4, 12, -20 rangle). Now, I need a specific point on the line. To find this, I can solve the system of equations given by the two planes. Let me set one of the variables to a specific value to make the equations easier to solve. I'll set (z = 0) because that might simplify things.Substituting (z = 0) into both equations:1. (3x + 4y + 1 = 0)2. (2x - 4y + 4 = 0)Now, I have a system of two equations with two variables:[begin{cases}3x + 4y = -1 2x - 4y = -4end{cases}]I can solve this system by adding the two equations together to eliminate (y):Adding equation 1 and equation 2:(3x + 4y + 2x - 4y = -1 - 4)Simplifying:(5x = -5)So, (x = -1)Now, substitute (x = -1) back into one of the equations to find (y). Let's use equation 2:(2(-1) - 4y = -4)Simplifying:(-2 - 4y = -4)Adding 2 to both sides:(-4y = -2)Dividing both sides by -4:(y = frac{1}{2})So, the point where (z = 0) is ((-1, frac{1}{2}, 0)).Now, with the direction vector (langle 4, 12, -20 rangle) and the point ((-1, frac{1}{2}, 0)), I can write the canonical equations of the line. The canonical form is:[frac{x - x_0}{a} = frac{y - y_0}{b} = frac{z - z_0}{c}]Where ((x_0, y_0, z_0)) is the point on the line and (langle a, b, c rangle) is the direction vector.Plugging in the values:[frac{x - (-1)}{4} = frac{y - frac{1}{2}}{12} = frac{z - 0}{-20}]Simplifying:[frac{x + 1}{4} = frac{y - frac{1}{2}}{12} = frac{z}{-20}]I think that's the canonical equation of the line. Let me double-check my steps to make sure I didn't make any mistakes.1. Found the normal vectors correctly.2. Computed the cross product correctly, got (langle 4, 12, -20 rangle).3. Solved the system by setting (z = 0), found (x = -1) and (y = frac{1}{2}).4. Plugged into the canonical form correctly.Everything seems to check out. So, I think this is the correct answer.**Final Answer**The canonical equations of the line are boxed{frac{x + 1}{4} = frac{y - frac{1}{2}}{12} = frac{z}{-20}}.