💡To solve the problem, we start by defining the price of the item as ( x ) coins and the number of people as ( n ).1. **First Condition:** - If each person contributes 8 coins, there will be 3 coins left. - This can be expressed as: [ 8n = x + 3 ] - Dividing both sides by 8 gives: [ n = frac{x + 3}{8} ]2. **Second Condition:** - If each person contributes 7 coins, there will be a shortage of 4 coins. - This can be expressed as: [ 7n = x - 4 ] - Dividing both sides by 7 gives: [ n = frac{x - 4}{7} ]3. **Equating the Two Expressions for ( n ):** - Since both expressions equal ( n ), we set them equal to each other: [ frac{x + 3}{8} = frac{x - 4}{7} ]This equation represents the relationship between the price of the item and the number of people based on the given conditions.

💡Okay, so I have this problem here: In the range from 0 to 2π, the angle that has the same terminal side as -π/3 is (    ). The options are A: π/3, B: 2π/3, C: 4π/3, and D: 5π/3. Hmm, I need to figure out which one of these is the correct answer.First, I remember that angles in standard position are measured from the positive x-axis. Positive angles go counterclockwise, and negative angles go clockwise. So, -π/3 is a negative angle, which means it's measured clockwise from the positive x-axis.Let me visualize the unit circle. Starting at the positive x-axis, if I rotate π/3 radians clockwise, where do I end up? π/3 is 60 degrees, right? So, rotating 60 degrees clockwise from the positive x-axis would place the terminal side in the fourth quadrant. That makes sense because negative angles go clockwise.Now, the question is asking for an angle between 0 and 2π that has the same terminal side as -π/3. So, essentially, I need to find a positive angle that points in the same direction as -π/3. I think the way to do this is to add 2π to -π/3 because adding a full rotation (2π) should bring me back to the same terminal side. Let me calculate that:-π/3 + 2π = (-π/3) + (6π/3) = (5π)/3.So, 5π/3 is the positive angle that has the same terminal side as -π/3. Let me check the options: A is π/3, which is in the first quadrant, not the fourth. B is 2π/3, which is in the second quadrant. C is 4π/3, which is in the third quadrant. D is 5π/3, which is in the fourth quadrant. That matches with where -π/3 would point.Wait, just to make sure, let me think about another way. If I have an angle of -π/3, that's the same as rotating 60 degrees below the positive x-axis. So, in the fourth quadrant. The positive angle that points there would be 360 degrees minus 60 degrees, which is 300 degrees. Converting 300 degrees to radians, since 180 degrees is π, so 300 degrees is (300/180)π = (5/3)π. Yep, that's 5π/3. So that's consistent.I think that makes sense. So, the angle between 0 and 2π that has the same terminal side as -π/3 is 5π/3. So, the correct answer should be D.Just to recap, angles that differ by full rotations (2π) have the same terminal side. So, adding 2π to -π/3 gives me 5π/3, which is within the range of 0 to 2π. That's why it's the correct answer.I don't think I made any mistakes here. It seems straightforward once I remember that adding 2π to a negative angle gives the positive equivalent in the same position. So, yeah, I'm confident that D is the right choice.

💡Alright, so I've got this multiple-choice question with four options, A through D. I need to figure out which ones are correct. Let's take them one by one and see.Starting with **Option A**:It says, "Given (x^{2} - 2x + 1 = 0), then the algebraic expression ((x - 1)^{2} + x(x - 4) + (x - 2)(x + 2) = 0)."First, let's solve the quadratic equation (x^{2} - 2x + 1 = 0). This looks familiar; it's a perfect square. So, ((x - 1)^{2} = 0), which means (x = 1).Now, substitute (x = 1) into the expression:[(1 - 1)^{2} + 1(1 - 4) + (1 - 2)(1 + 2)]Calculating each part:- ((1 - 1)^{2} = 0)- (1(1 - 4) = 1(-3) = -3)- ((1 - 2)(1 + 2) = (-1)(3) = -3)Adding them up: (0 - 3 - 3 = -6).But the expression is supposed to equal 0. So, this isn't correct. Option A is **incorrect**.Moving on to **Option B**:"Given (x^{2} - 3x + 1 = 0), then (x^{3} + frac{1}{x^{3}} - 3 = 15)."Okay, starting with the equation (x^{2} - 3x + 1 = 0). Maybe I can find (x + frac{1}{x}) from this.Divide both sides by (x) (assuming (x neq 0)):[x - 3 + frac{1}{x} = 0 implies x + frac{1}{x} = 3]Let me denote (y = x + frac{1}{x}). Then, (y = 3).I need to find (x^{3} + frac{1}{x^{3}}). There's a formula for this:[(x + frac{1}{x})^{3} = x^{3} + 3x + frac{3}{x} + frac{1}{x^{3}} = x^{3} + frac{1}{x^{3}} + 3(x + frac{1}{x})]So,[y^{3} = x^{3} + frac{1}{x^{3}} + 3y]Plugging in (y = 3):[27 = x^{3} + frac{1}{x^{3}} + 9]Subtract 9 from both sides:[x^{3} + frac{1}{x^{3}} = 18]Now, the expression in the option is (x^{3} + frac{1}{x^{3}} - 3 = 18 - 3 = 15). That matches. So, Option B is **correct**.Next, **Option C**:"If (a = frac{1}{20}x + 20), (b = frac{1}{20}x + 19), (c = frac{1}{20}x + 21), then (a^{2} + b^{2} + c^{2} - ab - bc - ac = 3)."Hmm, this looks like it might involve some algebraic manipulation. Let me see.First, let's compute the differences between the variables:- (a - b = (frac{1}{20}x + 20) - (frac{1}{20}x + 19) = 1)- (b - c = (frac{1}{20}x + 19) - (frac{1}{20}x + 21) = -2)- (c - a = (frac{1}{20}x + 21) - (frac{1}{20}x + 20) = 1)So, (a - b = 1), (b - c = -2), (c - a = 1).Now, the expression (a^{2} + b^{2} + c^{2} - ab - bc - ac) can be rewritten using the identity:[a^{2} + b^{2} + c^{2} - ab - bc - ac = frac{1}{2}[(a - b)^{2} + (b - c)^{2} + (c - a)^{2}]]Plugging in the differences:[frac{1}{2}[1^{2} + (-2)^{2} + 1^{2}] = frac{1}{2}[1 + 4 + 1] = frac{1}{2}[6] = 3]So, the expression equals 3, which matches the statement. Option C is **correct**.Finally, **Option D**:"Given that the lengths of the two legs of a right triangle are the roots of the equation (2x^{2} - 8x + 7 = 0), then the length of the hypotenuse of this right triangle is 9."Alright, let's find the roots of the quadratic equation (2x^{2} - 8x + 7 = 0). Using the quadratic formula:[x = frac{8 pm sqrt{64 - 56}}{4} = frac{8 pm sqrt{8}}{4} = frac{8 pm 2sqrt{2}}{4} = frac{4 pm sqrt{2}}{2} = 2 pm frac{sqrt{2}}{2}]So, the roots are (2 + frac{sqrt{2}}{2}) and (2 - frac{sqrt{2}}{2}).These are the lengths of the legs. To find the hypotenuse, we need to compute (sqrt{m^{2} + n^{2}}), where (m) and (n) are the roots.First, let's find (m + n) and (mn) from the quadratic equation.From (2x^{2} - 8x + 7 = 0), the sum of roots (m + n = frac{8}{2} = 4), and the product (mn = frac{7}{2}).Now, (m^{2} + n^{2} = (m + n)^{2} - 2mn = 16 - 7 = 9).Therefore, the hypotenuse is (sqrt{9} = 3).But the option says the hypotenuse is 9. That's not correct. So, Option D is **incorrect**.**Summary:**- A: Incorrect- B: Correct- C: Correct- D: IncorrectSo, the correct options are B and C.

💡First, I need to determine Walt's current age based on the information provided about the music teacher and the physics teacher.Let's denote Walt's current age as ( W ).**Music Teacher:**- The music teacher is three times as old as Walt now.- In 12 years, the music teacher will be twice as old as Walt.From the first statement:[text{Music Teacher's current age} = 3W]From the second statement:[3W + 12 = 2(W + 12)]Solving this equation:[3W + 12 = 2W + 24][3W - 2W = 24 - 12][W = 12]**Physics Teacher:**- The physics teacher is four times as old as Walt now.- When the physics teacher retires in 15 years, he will be thrice as old as Walt.From the first statement:[text{Physics Teacher's current age} = 4W]From the second statement:[4W + 15 = 3(W + 15)]Solving this equation:[4W + 15 = 3W + 45][4W - 3W = 45 - 15][W = 30]There is an inconsistency between the two results. According to the music teacher's information, Walt is 12 years old, but according to the physics teacher's information, Walt is 30 years old. This suggests that there is a mistake in the information provided, as both conditions cannot be true simultaneously with a single value for Walt's age.

💡Okay, so I have this problem with two parts. Let me try to figure out each step carefully.First, the problem is about the Cartesian coordinate plane with a polar coordinate system. The origin O is the pole, and the positive x-axis is the polar axis. There's a curve C with the polar equation ρ = 4 cos θ. Then there's a line l with parametric equations:x = -3 + (√3 / 2) ty = (1/2) twhere t is the parameter.Part 1 asks for the polar coordinates of the projection point P of the pole (which is the origin O) onto the line l.Alright, so I need to find the projection of the origin onto line l. The projection point P is the closest point on line l to the origin. So, essentially, I need to find the point P on line l such that the vector OP is perpendicular to the direction vector of line l.First, let me write down the parametric equations of line l:x = -3 + (√3 / 2) ty = 0 + (1/2) tSo, the direction vector of line l can be found from the coefficients of t. That would be ( √3 / 2, 1/2 ). But maybe it's easier to use the standard form of the line.Alternatively, I can eliminate the parameter t to get the Cartesian equation of line l.From the parametric equations:x = -3 + (√3 / 2) t => x + 3 = (√3 / 2) t => t = (2 / √3)(x + 3)Similarly, y = (1/2) t => t = 2ySo, setting them equal:(2 / √3)(x + 3) = 2ySimplify:( x + 3 ) / √3 = yMultiply both sides by √3:x + 3 = √3 yBring all terms to one side:x - √3 y + 3 = 0So, the equation of line l is x - √3 y + 3 = 0.Now, to find the projection of the origin O(0,0) onto this line.The formula for the projection of a point (x0, y0) onto the line ax + by + c = 0 is given by:P = ( (b(bx0 - ay0) - ac ) / (a² + b² ), (a(-bx0 + ay0) - bc ) / (a² + b² ) )But maybe it's easier to use the formula for the foot of the perpendicular from a point to a line.Alternatively, since we have the line in the form x - √3 y + 3 = 0, we can write it as:x = √3 y - 3So, the direction vector of the line is ( √3, 1 ), because the slope is 1 / √3, so rise over run is 1 over √3, meaning for every √3 units in x, y increases by 1.Wait, actually, the direction vector can be found from the coefficients of t in the parametric equations. Earlier, we had direction vector ( √3 / 2, 1/2 ). So, that's a direction vector, but it's a unit vector? Let me check.The magnitude of ( √3 / 2, 1/2 ) is sqrt( ( (√3 / 2)^2 + (1/2)^2 ) ) = sqrt( (3/4 + 1/4) ) = sqrt(1) = 1. So yes, it's a unit direction vector.But maybe I can use the standard formula for projection.Given a line in the form ax + by + c = 0, the projection of the origin onto this line is given by:( -a c / (a² + b² ), -b c / (a² + b² ) )Wait, let me verify that.Actually, the formula for the foot of the perpendicular from the origin to the line ax + by + c = 0 is:( -a c / (a² + b² ), -b c / (a² + b² ) )But wait, in our case, the line is x - √3 y + 3 = 0, so a = 1, b = -√3, c = 3.So, plugging into the formula:x = -a c / (a² + b² ) = -1 * 3 / (1 + 3) = -3 / 4y = -b c / (a² + b² ) = -(-√3) * 3 / (1 + 3) = (√3 * 3) / 4 = 3√3 / 4So, the projection point P is ( -3/4, 3√3 / 4 )Wait, but let me think again. The formula I used is for the foot of the perpendicular from the origin to the line ax + by + c = 0. But in our case, the line is x - √3 y + 3 = 0, so c = 3, which is not zero. So, is the formula still applicable?Wait, actually, the formula is for the foot of the perpendicular from the origin to the line ax + by + c = 0. So, if the line is ax + by + c = 0, then yes, the foot is at ( -a c / (a² + b² ), -b c / (a² + b² ) )But in our case, the line is x - √3 y + 3 = 0, so a = 1, b = -√3, c = 3.So, plugging in:x = -a c / (a² + b² ) = -1 * 3 / (1 + 3) = -3/4y = -b c / (a² + b² ) = -(-√3) * 3 / (1 + 3) = (√3 * 3)/4 = 3√3 / 4So, yes, the projection point P is ( -3/4, 3√3 / 4 )But wait, let me confirm this by another method.Another way is to parametrize the line l and find the value of t where the vector OP is perpendicular to the direction vector of l.So, the parametric equations are:x = -3 + (√3 / 2 ) ty = 0 + (1/2 ) tSo, any point P on line l can be written as ( -3 + (√3 / 2 ) t, (1/2 ) t )The vector OP is ( -3 + (√3 / 2 ) t, (1/2 ) t )The direction vector of line l is ( √3 / 2, 1 / 2 )For OP to be perpendicular to the direction vector, their dot product must be zero.So,( -3 + (√3 / 2 ) t ) * ( √3 / 2 ) + ( (1/2 ) t ) * ( 1 / 2 ) = 0Let me compute this:First term: ( -3 + (√3 / 2 ) t ) * ( √3 / 2 ) = -3 * √3 / 2 + ( (√3 / 2 ) t ) * ( √3 / 2 ) = - (3√3)/2 + ( 3 / 4 ) tSecond term: ( (1/2 ) t ) * ( 1 / 2 ) = (1/4 ) tSo, adding both terms:- (3√3)/2 + (3/4 ) t + (1/4 ) t = 0Combine like terms:- (3√3)/2 + (4/4 ) t = 0 => - (3√3)/2 + t = 0So, t = (3√3)/2Now, plug this value of t back into the parametric equations to find P:x = -3 + (√3 / 2 ) * (3√3 / 2 ) = -3 + ( (√3 * 3√3 ) / 4 ) = -3 + ( 9 / 4 ) = (-12/4 + 9/4 ) = (-3/4 )y = (1/2 ) * (3√3 / 2 ) = (3√3 ) / 4So, P is ( -3/4, 3√3 / 4 ), which matches the earlier result.Now, to convert this point into polar coordinates.Polar coordinates are given by ( ρ, θ ), where ρ is the distance from the origin, and θ is the angle from the positive x-axis.Compute ρ:ρ = sqrt( x² + y² ) = sqrt( ( -3/4 )² + ( 3√3 / 4 )² ) = sqrt( 9/16 + 27/16 ) = sqrt( 36/16 ) = sqrt( 9/4 ) = 3/2Compute θ:tan θ = y / x = ( 3√3 / 4 ) / ( -3/4 ) = -√3So, tan θ = -√3But since the point P is in the second quadrant (x is negative, y is positive), θ is in the second quadrant.The reference angle for tan θ = √3 is π/3, so θ = π - π/3 = 2π/3Therefore, the polar coordinates of P are ( 3/2, 2π/3 )So, that's part 1 done.Now, part 2: Suppose that M and N are moving points on curve C and line l, respectively. Find the minimum value of |MN|.Curve C is given by ρ = 4 cos θ. Let me recall that in polar coordinates, ρ = 4 cos θ is a circle with radius 2, centered at (2, 0) in Cartesian coordinates.Yes, because ρ = 4 cos θ can be rewritten as ρ² = 4ρ cos θ, which is x² + y² = 4x, so x² - 4x + y² = 0, completing the square: (x - 2)^2 + y^2 = 4. So, it's a circle with center at (2, 0) and radius 2.Line l is given by the parametric equations, which we already converted to Cartesian form: x - √3 y + 3 = 0.So, we have a circle centered at (2, 0) with radius 2, and a line l: x - √3 y + 3 = 0.We need to find the minimum distance between a point M on the circle and a point N on the line l.Wait, but actually, M is on the circle and N is on the line, so |MN| is the distance between M and N, and we need the minimum of this distance as M and N vary over the circle and the line.But actually, since M and N are independent points on the circle and the line, the minimum distance |MN| is the minimum distance between any point on the circle and any point on the line.But that's equivalent to the minimum distance between the circle and the line.But wait, the minimum distance between a circle and a line can be found by computing the distance from the center of the circle to the line, and then subtracting the radius if the line does not intersect the circle.Wait, actually, if the distance from the center to the line is greater than the radius, then the minimum distance between the circle and the line is the distance from the center to the line minus the radius.If the distance is less than or equal to the radius, then the minimum distance is zero, because the line intersects the circle.So, let's compute the distance from the center of the circle (2, 0) to the line l: x - √3 y + 3 = 0.The formula for the distance from a point (x0, y0) to the line ax + by + c = 0 is |a x0 + b y0 + c| / sqrt(a² + b² )So, in our case, a = 1, b = -√3, c = 3.So, distance d = |1*2 + (-√3)*0 + 3| / sqrt(1 + 3) = |2 + 0 + 3| / 2 = |5| / 2 = 5/2So, the distance from the center (2,0) to the line l is 5/2.The radius of the circle is 2.Since 5/2 = 2.5 > 2, the line does not intersect the circle, so the minimum distance between the circle and the line is d - r = 5/2 - 2 = 1/2.Therefore, the minimum value of |MN| is 1/2.Wait, let me make sure.Alternatively, we can parametrize points on the circle and points on the line and find the minimum distance.But that might be more complicated. The method I used is standard: compute the distance from the center to the line, subtract the radius, and that's the minimum distance between the circle and the line.Yes, that makes sense.So, the minimum value of |MN| is 1/2.**Final Answer**1. The polar coordinates of point P are boxed{left( dfrac{3}{2}, dfrac{2}{3}pi right)}.2. The minimum value of |MN| is boxed{dfrac{1}{2}}.

💡Okay, so I have this problem where a company wants to invest in 3 different projects across 4 cities: Beijing, Shanghai, Lanzhou, and Yinchuan. The catch is that they can't invest more than 2 projects in the same city. I need to figure out how many different investment plans there are. Hmm, let's break this down step by step.First, I need to understand what exactly is being asked. The company has 3 projects, and they want to distribute these projects among 4 cities, but no city can have more than 2 projects. So, essentially, I need to count all the possible ways to assign these 3 projects to the 4 cities under the given constraint.I think the best way to approach this is by considering the different scenarios that satisfy the condition of no more than 2 projects in any city. Since there are only 3 projects, the possible distributions of projects across the cities can be categorized into two main cases:1. **Case 1: All three projects are in different cities.** In this case, each project is assigned to a unique city, and no city has more than one project. Since there are 4 cities and 3 projects, this should be a straightforward permutation problem.2. **Case 2: Two projects are in one city, and the third project is in another city.** Here, one city gets two projects, and another city gets the remaining one project. This means we have to choose which city gets two projects, which city gets one project, and then assign the specific projects to these cities.By calculating the number of possible investment plans for each case and then adding them together, I should get the total number of different investment plans.**Case 1: All three projects in different cities**Since there are 4 cities and we need to assign each of the 3 projects to a different city, this is essentially a permutation problem where we are selecting 3 cities out of 4 and assigning the projects to them. The number of ways to do this is given by the permutation formula:[P(n, k) = frac{n!}{(n - k)!}]Where ( n ) is the total number of items (cities), and ( k ) is the number of items to choose (projects). Plugging in the numbers:[P(4, 3) = frac{4!}{(4 - 3)!} = frac{24}{1} = 24]So, there are 24 ways to assign the 3 projects to 3 different cities.**Case 2: Two projects in one city, and one project in another city**In this scenario, we need to choose which city will have two projects and which city will have one project. Since there are 4 cities, the number of ways to choose the city with two projects is 4. Once we've chosen that city, we have 3 remaining cities to choose from for the city that will have one project. So, the number of ways to choose the cities is:[4 times 3 = 12]However, we also need to consider the number of ways to assign the specific projects to these cities. We have 3 projects, and we need to choose 2 of them to assign to the first city. The number of ways to choose 2 projects out of 3 is given by the combination formula:[C(n, k) = frac{n!}{k!(n - k)!}]Plugging in the numbers:[C(3, 2) = frac{3!}{2!(3 - 2)!} = frac{6}{2 times 1} = 3]So, there are 3 ways to choose which 2 projects go to the first city, and the remaining project automatically goes to the second city.Therefore, the total number of ways for this case is:[12 times 3 = 36]**Total Number of Investment Plans**Now, to find the total number of different investment plans, we add the number of plans from both cases:[24 + 36 = 60]So, there are 60 different investment plans that satisfy the condition of no more than 2 projects in any city.**Verification**Just to make sure I didn't make a mistake, let's verify the calculations.For Case 1, assigning 3 projects to 3 different cities out of 4:[P(4, 3) = 4 times 3 times 2 = 24]That seems correct.For Case 2, choosing 1 city out of 4 to have 2 projects, then 1 city out of the remaining 3 to have 1 project, and then choosing 2 projects out of 3:[4 times 3 times 3 = 36]Yes, that also seems correct.Adding them together:[24 + 36 = 60]Looks good. I think I've covered all possible scenarios without overcounting or undercounting.**Alternative Approach**Another way to think about this problem is by considering it as a distribution problem where we are distributing 3 distinct projects into 4 distinct cities with the constraint that no city has more than 2 projects.The formula for distributing ( n ) distinct objects into ( k ) distinct boxes with a maximum of ( m ) objects per box is given by:[sum_{i=0}^{k} S(n, i) times i! times C(k, i)]Where ( S(n, i) ) is the Stirling numbers of the second kind, but this might be more complicated than necessary for this problem.Alternatively, since the numbers are small, it's easier to break it down into cases as I did before.**Conclusion**After carefully considering both cases and verifying the calculations, I'm confident that the total number of different investment plans is 60.