answer：Alright, so I have this problem about a parabola, and I need to figure out which of the given statements are true. Let me try to break it down step by step.First, the parabola is given by the equation ( C: x^2 = 2py ) where ( p > 0 ). It mentions that the focus ( F ) is at a distance of 4 from the directrix. Hmm, I remember that for a parabola ( x^2 = 4py ), the focus is at ( (0, p) ) and the directrix is the line ( y = -p ). The distance between the focus and the directrix is ( 2p ). So, if this distance is 4, that means ( 2p = 4 ), so ( p = 2 ). Wait, but the equation given is ( x^2 = 2py ), not ( x^2 = 4py ). Let me check that again.If the standard form is ( x^2 = 4py ), then the distance from the focus to the directrix is ( 2p ). So, if the distance is 4, then ( 2p = 4 ) implies ( p = 2 ). Therefore, the equation of the parabola should be ( x^2 = 4py ), which would be ( x^2 = 8y ). But the problem says ( x^2 = 2py ). Maybe I need to adjust for that.Wait, if the standard form is ( x^2 = 4py ), then for ( x^2 = 2py ), it's like ( 4p' = 2p ), so ( p' = p/2 ). So, the focus would be at ( (0, p') = (0, p/2) ), and the directrix is ( y = -p/2 ). The distance between focus and directrix is ( p' + p' = 2p' = p ). So, if the distance is 4, then ( p = 4 ). Therefore, the equation of the parabola is ( x^2 = 2py = 8y ). Okay, so the parabola is ( x^2 = 8y ), focus at ( (0, 2) ) because ( p = 4 ), so ( p/2 = 2 ). Wait, hold on, if ( x^2 = 2py ), then ( 4p' = 2p ) implies ( p' = p/2 ). So, the focus is at ( (0, p') = (0, p/2) ). So, if the distance from focus to directrix is 4, which is ( 2p' = 4 ), so ( p' = 2 ), hence ( p = 4 ). So, the equation is ( x^2 = 8y ), focus at ( (0, 2) ), and directrix ( y = -2 ). Got it.Now, the line ( l ) intersects the parabola at points ( P ) and ( Q ). It says ( overrightarrow{PQ} = 2overrightarrow{PR} ), where ( R(4, 6) ). Hmm, vectors. So, the vector from ( P ) to ( Q ) is twice the vector from ( P ) to ( R ). That means ( R ) is the midpoint of ( PQ ). Because if ( overrightarrow{PQ} = 2overrightarrow{PR} ), then ( Q - P = 2(R - P) ), so ( Q = 2R - P ). Therefore, ( R ) is the midpoint of ( PQ ). So, the coordinates of ( R ) are the averages of the coordinates of ( P ) and ( Q ). So, ( ( (x_P + x_Q)/2, (y_P + y_Q)/2 ) = (4, 6) ). Therefore, ( x_P + x_Q = 8 ) and ( y_P + y_Q = 12 ).Since ( P ) and ( Q ) lie on the parabola ( x^2 = 8y ), their coordinates satisfy ( x_P^2 = 8y_P ) and ( x_Q^2 = 8y_Q ). So, ( y_P = x_P^2 / 8 ) and ( y_Q = x_Q^2 / 8 ). Therefore, ( y_P + y_Q = (x_P^2 + x_Q^2)/8 = 12 ). So, ( x_P^2 + x_Q^2 = 96 ).We also know that ( x_P + x_Q = 8 ). Let me denote ( x_P = a ) and ( x_Q = b ). So, ( a + b = 8 ) and ( a^2 + b^2 = 96 ). I can use these to find ( ab ). Since ( (a + b)^2 = a^2 + 2ab + b^2 ), so ( 64 = 96 + 2ab ), which gives ( 2ab = 64 - 96 = -32 ), so ( ab = -16 ).So, the product of the x-coordinates of ( P ) and ( Q ) is -16. That might be useful later.Now, the line ( l ) passes through ( P ) and ( Q ). Since ( R(4,6) ) is the midpoint, the line ( l ) also passes through ( R ). So, the line ( l ) can be determined by two points: ( R(4,6) ) and, say, ( P(a, a^2/8) ). Let me find the slope of line ( l ).The slope ( m ) is ( (6 - a^2/8) / (4 - a) ). Alternatively, since ( R ) is the midpoint, the slope can also be calculated as ( (y_Q - y_P)/(x_Q - x_P) ). Let me compute that.( y_Q - y_P = (x_Q^2 - x_P^2)/8 = (x_Q - x_P)(x_Q + x_P)/8 ). Since ( x_P + x_Q = 8 ), this becomes ( (x_Q - x_P)(8)/8 = x_Q - x_P ). Therefore, the slope ( m = (y_Q - y_P)/(x_Q - x_P) = (x_Q - x_P)/(x_Q - x_P) = 1 ). So, the slope of line ( l ) is 1.Therefore, the equation of line ( l ) is ( y - 6 = 1*(x - 4) ), which simplifies to ( y = x + 2 ).Now, let's find the points ( P ) and ( Q ) where this line intersects the parabola ( x^2 = 8y ). Substitute ( y = x + 2 ) into the parabola equation:( x^2 = 8(x + 2) )( x^2 = 8x + 16 )( x^2 - 8x - 16 = 0 )This is a quadratic equation in ( x ). Let's solve it:( x = [8 ± sqrt(64 + 64)] / 2 = [8 ± sqrt(128)] / 2 = [8 ± 8*sqrt(2)] / 2 = 4 ± 4*sqrt(2) )So, the x-coordinates of ( P ) and ( Q ) are ( 4 + 4sqrt{2} ) and ( 4 - 4sqrt{2} ). Therefore, their coordinates are:( P(4 + 4sqrt{2}, (4 + 4sqrt{2})^2 / 8 ) ) and ( Q(4 - 4sqrt{2}, (4 - 4sqrt{2})^2 / 8 ) ).Let me compute the y-coordinates:For ( P ):( y_P = (4 + 4sqrt{2})^2 / 8 = (16 + 32sqrt{2} + 32) / 8 = (48 + 32sqrt{2}) / 8 = 6 + 4sqrt{2} )Similarly, for ( Q ):( y_Q = (4 - 4sqrt{2})^2 / 8 = (16 - 32sqrt{2} + 32) / 8 = (48 - 32sqrt{2}) / 8 = 6 - 4sqrt{2} )So, points ( P ) and ( Q ) are ( (4 + 4sqrt{2}, 6 + 4sqrt{2}) ) and ( (4 - 4sqrt{2}, 6 - 4sqrt{2}) ).Now, the problem says that the two tangents to ( C ) passing through ( P ) and ( Q ) intersect at point ( A ). I need to find the coordinates of ( A ).I recall that the equation of the tangent to a parabola ( x^2 = 4py ) at a point ( (x_1, y_1) ) is ( xx_1 = 2p(y + y_1) ). Wait, but our parabola is ( x^2 = 8y ), which is ( x^2 = 4p'y ) with ( 4p' = 8 ), so ( p' = 2 ). Therefore, the equation of the tangent at ( (x_1, y_1) ) is ( xx_1 = 4p'(y + y_1)/2 )... Wait, maybe I should derive it.Given the parabola ( x^2 = 8y ), let's find the equation of the tangent at a general point ( (a, a^2/8) ). The derivative ( dy/dx ) is ( x/4 ), so the slope at ( x = a ) is ( a/4 ). Therefore, the equation of the tangent is:( y - (a^2/8) = (a/4)(x - a) )Simplify:( y = (a/4)x - (a^2)/4 + (a^2)/8 )( y = (a/4)x - (a^2)/8 )So, the tangent equation is ( y = (a/4)x - (a^2)/8 ).Similarly, for point ( Q ) with x-coordinate ( b = 4 - 4sqrt{2} ), the tangent equation is:( y = (b/4)x - (b^2)/8 )So, to find point ( A ), we need to solve these two tangent equations simultaneously.Let me write them out:1. ( y = (a/4)x - (a^2)/8 )2. ( y = (b/4)x - (b^2)/8 )Set them equal:( (a/4)x - (a^2)/8 = (b/4)x - (b^2)/8 )Multiply both sides by 8 to eliminate denominators:( 2a x - a^2 = 2b x - b^2 )Bring all terms to one side:( 2a x - 2b x - a^2 + b^2 = 0 )Factor:( 2(a - b)x + (b^2 - a^2) = 0 )Note that ( b^2 - a^2 = -(a^2 - b^2) = -(a - b)(a + b) ). So,( 2(a - b)x - (a - b)(a + b) = 0 )Factor out ( (a - b) ):( (a - b)(2x - (a + b)) = 0 )Since ( a neq b ) (they are distinct points), we have:( 2x - (a + b) = 0 )So, ( x = (a + b)/2 ). But earlier, we found that ( a + b = 8 ), so ( x = 4 ).Now, substitute ( x = 4 ) back into one of the tangent equations. Let's use the first one:( y = (a/4)(4) - (a^2)/8 = a - (a^2)/8 )But ( a = 4 + 4sqrt{2} ), so:( y = (4 + 4sqrt{2}) - ( (4 + 4sqrt{2})^2 ) / 8 )Compute ( (4 + 4sqrt{2})^2 ):( 16 + 32sqrt{2} + 32 = 48 + 32sqrt{2} )So,( y = 4 + 4sqrt{2} - (48 + 32sqrt{2}) / 8 )Simplify:( y = 4 + 4sqrt{2} - (6 + 4sqrt{2}) = 4 + 4sqrt{2} - 6 - 4sqrt{2} = -2 )Therefore, point ( A ) is ( (4, -2) ).Now, let's check the options one by one.**Option A: ( |AF| = 4sqrt{2} )**First, find the coordinates of ( F ). Earlier, we found that the focus is at ( (0, 2) ). So, ( F(0, 2) ).Point ( A(4, -2) ). The distance ( AF ) is:( |AF| = sqrt{(4 - 0)^2 + (-2 - 2)^2} = sqrt{16 + 16} = sqrt{32} = 4sqrt{2} )So, option A is true.**Option B: ( |PQ| = 12 )**Compute the distance between ( P ) and ( Q ). Points ( P(4 + 4sqrt{2}, 6 + 4sqrt{2}) ) and ( Q(4 - 4sqrt{2}, 6 - 4sqrt{2}) ).Distance ( PQ ):( |PQ| = sqrt{( (4 + 4sqrt{2}) - (4 - 4sqrt{2}) )^2 + ( (6 + 4sqrt{2}) - (6 - 4sqrt{2}) )^2 } )Simplify:( = sqrt{(8sqrt{2})^2 + (8sqrt{2})^2} = sqrt{128 + 128} = sqrt{256} = 16 )So, ( |PQ| = 16 ), not 12. Therefore, option B is false.**Option C: ( PQ perp AF )**First, find the slopes of ( PQ ) and ( AF ).Slope of ( PQ ):We already found earlier that the slope is 1.Slope of ( AF ):Points ( A(4, -2) ) and ( F(0, 2) ).Slope ( m = (2 - (-2))/(0 - 4) = 4 / (-4) = -1 )So, the slope of ( AF ) is -1.Now, check if they are perpendicular. Two lines are perpendicular if the product of their slopes is -1.Slope of ( PQ ) is 1, slope of ( AF ) is -1. Product is ( 1 * (-1) = -1 ). So, yes, they are perpendicular.Therefore, option C is true.**Option D: The circle with diameter ( PQ ) passes through point ( A ).**To check this, we can verify if ( A ) lies on the circle with diameter ( PQ ). The equation of the circle can be found using the midpoint and radius.Midpoint of ( PQ ) is ( R(4, 6) ), which we already know.Radius is half the distance ( PQ ), which is ( 16 / 2 = 8 ).So, the equation of the circle is:( (x - 4)^2 + (y - 6)^2 = 8^2 = 64 )Now, check if ( A(4, -2) ) satisfies this equation:( (4 - 4)^2 + (-2 - 6)^2 = 0 + (-8)^2 = 64 )Yes, it does. Therefore, point ( A ) lies on the circle with diameter ( PQ ). So, option D is true.**Summary:**- A: True- B: False- C: True- D: TrueSo, the correct options are A, C, D.**Final Answer**boxed{ACD}

answer：Okay, so I have this problem about curves and a line in the Cartesian coordinate system. Let me try to understand what it's asking and figure out how to solve it step by step.First, part (1) asks for the polar equation of curve ( C_1 ) and the parametric equations of curve ( C_2 ). Looking at the parametric equations for ( C_1 ):[begin{cases}x = cos theta y = sin thetaend{cases}]Hmm, these look familiar. I remember that parametric equations of the form ( x = cos theta ) and ( y = sin theta ) represent a unit circle centered at the origin. So, ( C_1 ) is a unit circle.To write the polar equation of ( C_1 ), I recall that in polar coordinates, ( x = rho cos theta ) and ( y = rho sin theta ). Since ( x^2 + y^2 = 1 ) for a unit circle, substituting the polar coordinates gives:[rho^2 cos^2 theta + rho^2 sin^2 theta = 1 implies rho^2 (cos^2 theta + sin^2 theta) = 1 implies rho^2 = 1 implies rho = 1]So, the polar equation of ( C_1 ) is simply ( rho = 1 ).Now, for curve ( C_2 ), it's obtained by stretching ( C_1 ) horizontally by ( sqrt{2} ) and vertically by 2. Stretching in the x-direction by a factor of ( sqrt{2} ) means replacing ( x ) with ( frac{x}{sqrt{2}} ) in the equation, and stretching in the y-direction by 2 means replacing ( y ) with ( frac{y}{2} ). Starting from the unit circle equation ( x^2 + y^2 = 1 ), after stretching, the equation becomes:[left( frac{x}{sqrt{2}} right)^2 + left( frac{y}{2} right)^2 = 1 implies frac{x^2}{2} + frac{y^2}{4} = 1]This is the equation of an ellipse. To write the parametric equations for ( C_2 ), I can use the standard parametric form of an ellipse:[begin{cases}x = a cos theta y = b sin thetaend{cases}]where ( a ) and ( b ) are the semi-major and semi-minor axes. From the equation ( frac{x^2}{2} + frac{y^2}{4} = 1 ), we can see that ( a = sqrt{2} ) and ( b = 2 ). So, the parametric equations for ( C_2 ) are:[begin{cases}x = sqrt{2} cos theta y = 2 sin thetaend{cases}]Alright, that takes care of part (1).Moving on to part (2), we need to find a point ( P ) on ( C_2 ) such that the distance from ( P ) to the line ( l ) is minimized, and then compute this minimum distance.First, let me write down the equation of line ( l ):[rho (sqrt{2} cos theta + sin theta) = 4]I need to convert this polar equation into Cartesian form to make it easier to work with. In polar coordinates, ( rho cos theta = x ) and ( rho sin theta = y ). So, substituting these into the equation:[rho sqrt{2} cos theta + rho sin theta = 4 implies sqrt{2} x + y = 4]So, the Cartesian equation of line ( l ) is ( sqrt{2} x + y = 4 ).Now, let's denote a general point ( P ) on ( C_2 ) as ( (sqrt{2} cos theta, 2 sin theta) ). To find the distance from ( P ) to the line ( l ), I can use the formula for the distance from a point ( (x_0, y_0) ) to the line ( ax + by + c = 0 ):[d = frac{|a x_0 + b y_0 + c|}{sqrt{a^2 + b^2}}]First, let's write the equation of line ( l ) in the standard form ( ax + by + c = 0 ):[sqrt{2} x + y - 4 = 0]So, ( a = sqrt{2} ), ( b = 1 ), and ( c = -4 ).Substituting the coordinates of ( P ) into the distance formula:[d(theta) = frac{|sqrt{2} (sqrt{2} cos theta) + 1 (2 sin theta) - 4|}{sqrt{(sqrt{2})^2 + 1^2}} = frac{|2 cos theta + 2 sin theta - 4|}{sqrt{2 + 1}} = frac{|2 cos theta + 2 sin theta - 4|}{sqrt{3}}]Simplify the numerator:[2 cos theta + 2 sin theta - 4 = 2 (cos theta + sin theta) - 4]So, the distance becomes:[d(theta) = frac{|2 (cos theta + sin theta) - 4|}{sqrt{3}}]I can factor out the 2 in the numerator:[d(theta) = frac{2 |(cos theta + sin theta) - 2|}{sqrt{3}}]Hmm, to minimize ( d(theta) ), I need to minimize the expression ( |(cos theta + sin theta) - 2| ). Since the absolute value function is minimized when its argument is as close to zero as possible, I need to find ( theta ) such that ( cos theta + sin theta ) is as close to 2 as possible.But wait, the maximum value of ( cos theta + sin theta ) is ( sqrt{2} ), which is approximately 1.414, and the minimum is ( -sqrt{2} ). So, ( cos theta + sin theta ) can never reach 2. Therefore, the expression ( (cos theta + sin theta) - 2 ) is always negative, so the absolute value becomes:[|(cos theta + sin theta) - 2| = 2 - (cos theta + sin theta)]Therefore, the distance simplifies to:[d(theta) = frac{2 (2 - (cos theta + sin theta))}{sqrt{3}} = frac{4 - 2 (cos theta + sin theta)}{sqrt{3}}]So, to minimize ( d(theta) ), I need to maximize ( cos theta + sin theta ) because it is subtracted. The maximum value of ( cos theta + sin theta ) is ( sqrt{2} ), as I thought earlier. So, the minimum distance occurs when ( cos theta + sin theta = sqrt{2} ).Let me verify that. The expression ( cos theta + sin theta ) can be rewritten using the identity:[cos theta + sin theta = sqrt{2} sin left( theta + frac{pi}{4} right )]Yes, that's correct. So, the maximum value of this expression is ( sqrt{2} ) when ( sin left( theta + frac{pi}{4} right ) = 1 ), which occurs when ( theta + frac{pi}{4} = frac{pi}{2} + 2kpi ), so ( theta = frac{pi}{4} + 2kpi ) for any integer ( k ).Therefore, the minimum distance is:[d_{text{min}} = frac{4 - 2 (sqrt{2})}{sqrt{3}} = frac{4 - 2sqrt{2}}{sqrt{3}}]I can rationalize the denominator if needed:[d_{text{min}} = frac{(4 - 2sqrt{2}) sqrt{3}}{3} = frac{4sqrt{3} - 2sqrt{6}}{3}]So, the minimum distance is ( frac{4sqrt{3} - 2sqrt{6}}{3} ).Now, let's find the point ( P ) on ( C_2 ) where this minimum occurs. Since ( theta = frac{pi}{4} + 2kpi ), let's take ( theta = frac{pi}{4} ) for simplicity.Substituting ( theta = frac{pi}{4} ) into the parametric equations of ( C_2 ):[x = sqrt{2} cos left( frac{pi}{4} right ) = sqrt{2} cdot frac{sqrt{2}}{2} = 1][y = 2 sin left( frac{pi}{4} right ) = 2 cdot frac{sqrt{2}}{2} = sqrt{2}]So, the point ( P ) is ( (1, sqrt{2}) ).Let me double-check my calculations. The parametric equations for ( C_2 ) are ( x = sqrt{2} cos theta ) and ( y = 2 sin theta ). Plugging in ( theta = frac{pi}{4} ), we get ( x = sqrt{2} cdot frac{sqrt{2}}{2} = 1 ) and ( y = 2 cdot frac{sqrt{2}}{2} = sqrt{2} ). That seems correct.Also, checking the distance from ( P(1, sqrt{2}) ) to the line ( sqrt{2}x + y - 4 = 0 ):[d = frac{|sqrt{2}(1) + sqrt{2} - 4|}{sqrt{(sqrt{2})^2 + 1^2}} = frac{|sqrt{2} + sqrt{2} - 4|}{sqrt{3}} = frac{|2sqrt{2} - 4|}{sqrt{3}} = frac{4 - 2sqrt{2}}{sqrt{3}} = frac{4sqrt{3} - 2sqrt{6}}{3}]Which matches what I found earlier. So, that seems consistent.Wait, hold on a second. When I calculated ( sqrt{2}x + y ) for ( P(1, sqrt{2}) ), I got ( sqrt{2}(1) + sqrt{2} = 2sqrt{2} ), which is approximately 2.828, and since the line is ( sqrt{2}x + y = 4 ), which is approximately 4, the distance is the difference divided by the norm, which is ( sqrt{3} ). So, yes, the distance is ( (4 - 2sqrt{2}) / sqrt{3} ), which is approximately (4 - 2.828)/1.732 ≈ 1.172/1.732 ≈ 0.676. That seems reasonable.I think I've covered all the steps and verified my calculations. So, the point ( P ) is ( (1, sqrt{2}) ) and the minimum distance is ( frac{4sqrt{3} - 2sqrt{6}}{3} ).**Final Answer**(1) The polar equation of ( C_1 ) is ( rho = 1 ) and the parametric equations of ( C_2 ) are ( begin{cases} x = sqrt{2} cos theta y = 2 sin theta end{cases} ).(2) The point ( P ) on ( C_2 ) with the minimum distance to line ( l ) is ( (1, sqrt{2}) ) and the minimum distance is ( boxed{dfrac{4sqrt{3} - 2sqrt{6}}{3}} ).

answer：First, I identify the snowfall amounts for both the morning and afternoon.Morning snowfall is 0.125 inch, and afternoon snowfall is 0.5 inch.To find the total snowfall, I add these two amounts together.Adding 0.125 inch and 0.5 inch gives a total of 0.625 inch.Therefore, the total snowfall in Yardley was 0.625 inch.

answer：Okay, so I have this problem about a geometric sequence. Let me try to figure it out step by step. The problem says that in the geometric sequence {a_n}, a₁ + a₅ = 82, and a₂ · a₄ = 81. I need to find the value of a₃.First, I remember that in a geometric sequence, each term is the previous term multiplied by a common ratio, r. So, the nth term can be written as a₁ * r^(n-1). That means:a₁ = a₁ a₂ = a₁ * r a₃ = a₁ * r² a₄ = a₁ * r³ a₅ = a₁ * r⁴ Okay, so I can express all the terms in terms of a₁ and r. Let me write down the given equations using this.The first equation is a₁ + a₅ = 82. Substituting a₅, that becomes:a₁ + a₁ * r⁴ = 82 Which simplifies to: a₁(1 + r⁴) = 82 ...(1)The second equation is a₂ · a₄ = 81. Substituting a₂ and a₄, that becomes:(a₁ * r) * (a₁ * r³) = 81 Simplifying that: a₁² * r⁴ = 81 ...(2)So now I have two equations:1. a₁(1 + r⁴) = 82 2. a₁² * r⁴ = 81 Hmm, I need to solve for a₁ and r. Maybe I can express a₁ from equation (1) and substitute into equation (2). Let me try that.From equation (1): a₁ = 82 / (1 + r⁴)Substitute this into equation (2):(82 / (1 + r⁴))² * r⁴ = 81Let me write that out:(82² * r⁴) / (1 + r⁴)² = 81Hmm, that looks a bit complicated, but maybe I can let x = r⁴ to simplify the equation. Let me set x = r⁴. Then the equation becomes:(82² * x) / (1 + x)² = 81Let me compute 82². 82 * 82 is 6724. So:6724 * x / (1 + x)² = 81Let me write that as:6724x = 81(1 + x)²Expanding the right side:6724x = 81(1 + 2x + x²) 6724x = 81 + 162x + 81x²Let me bring all terms to one side:81x² + 162x + 81 - 6724x = 0 81x² - 6562x + 81 = 0Wait, that seems a bit messy. Let me check my calculations again.Wait, 82² is 6724, correct. Then, 6724x = 81(1 + x)².Expanding 81(1 + x)²: 81*(1 + 2x + x²) = 81 + 162x + 81x².So, moving everything to the left:6724x - 81 - 162x - 81x² = 0 Which simplifies to:-81x² + (6724x - 162x) - 81 = 0 -81x² + 6562x - 81 = 0Multiplying both sides by -1 to make it positive:81x² - 6562x + 81 = 0Hmm, that's a quadratic in terms of x. Let me write it as:81x² - 6562x + 81 = 0This seems a bit complicated. Maybe I made a mistake earlier. Let me check.Wait, another approach: Maybe instead of substituting a₁ from equation (1) into equation (2), I can find a relationship between a₁ and r without substituting.Looking back at equation (2): a₁² * r⁴ = 81. So, a₁ * r² = sqrt(81) = 9. Because (a₁ * r²)^2 = a₁² * r⁴ = 81, so a₁ * r² = 9 or -9. But since we're dealing with a geometric sequence, unless specified otherwise, I think we can assume the terms are positive, so a₁ * r² = 9.Wait, that's interesting. So, a₁ * r² = 9. But a₃ is a₁ * r², so a₃ = 9. Is that possible? Wait, that seems too straightforward. Let me verify.If a₁ * r² = 9, then a₃ = 9. But let me check if this holds with the other equation.From equation (1): a₁(1 + r⁴) = 82.But since a₁ * r² = 9, then a₁ = 9 / r².Substituting into equation (1):(9 / r²)(1 + r⁴) = 82 9(1 + r⁴) / r² = 82 9(1/r² + r²) = 82 Let me set y = r², then 1/r² = 1/y.So, 9(1/y + y) = 82 Which is 9(y + 1/y) = 82 Divide both sides by 9: y + 1/y = 82/9 ≈ 9.111...Multiply both sides by y: y² + 1 = (82/9)y Which is y² - (82/9)y + 1 = 0Multiply both sides by 9 to eliminate the fraction:9y² - 82y + 9 = 0Now, let's solve for y using the quadratic formula:y = [82 ± sqrt(82² - 4*9*9)] / (2*9) Compute discriminant: 82² - 4*9*9 = 6724 - 324 = 6400 sqrt(6400) = 80So, y = [82 ± 80] / 18So, two solutions:1. y = (82 + 80)/18 = 162/18 = 9 2. y = (82 - 80)/18 = 2/18 = 1/9So, y = 9 or y = 1/9.Since y = r², then:If y = 9, then r² = 9, so r = 3 or r = -3 If y = 1/9, then r² = 1/9, so r = 1/3 or r = -1/3Now, let's find a₁ for each case.Case 1: y = 9, so r² = 9, r = 3 or r = -3From a₁ * r² = 9, so a₁ = 9 / r² = 9 / 9 = 1So, a₁ = 1Then, a₅ = a₁ * r⁴ = 1 * (r²)² = 1 * 9² = 81So, a₁ + a₅ = 1 + 81 = 82, which matches equation (1). Good.Case 2: y = 1/9, so r² = 1/9, r = 1/3 or r = -1/3From a₁ * r² = 9, so a₁ = 9 / (1/9) = 81So, a₁ = 81Then, a₅ = a₁ * r⁴ = 81 * (r²)² = 81 * (1/9)² = 81 * 1/81 = 1So, a₁ + a₅ = 81 + 1 = 82, which also matches equation (1). Good.So, we have two possible cases:1. a₁ = 1, r = 3 or r = -3 2. a₁ = 81, r = 1/3 or r = -1/3Now, let's find a₃ in each case.Case 1: a₁ = 1, r = 3 a₃ = a₁ * r² = 1 * 9 = 9Case 1: a₁ = 1, r = -3 a₃ = 1 * (-3)² = 1 * 9 = 9Case 2: a₁ = 81, r = 1/3 a₃ = 81 * (1/3)² = 81 * 1/9 = 9Case 2: a₁ = 81, r = -1/3 a₃ = 81 * (-1/3)² = 81 * 1/9 = 9So, in all cases, a₃ = 9.Wait, so regardless of the common ratio being positive or negative, a₃ is positive 9. That makes sense because a₃ is a₁ * r², and r² is always positive, so a₃ is positive.Therefore, the value of a₃ is 9.

answer：First, I recognize that ( f(x) ) is an odd function, which means ( f(-x) = -f(x) ).Given that ( f(x) = x(x - 1) ) for ( x > 0 ), I can find ( f(3) ) by substituting ( x = 3 ) into the equation:[f(3) = 3(3 - 1) = 3 times 2 = 6]Since ( f(x) ) is odd, ( f(-3) = -f(3) ). Therefore:[f(-3) = -6]

answer：Okay, so I have this problem about a sequence of numbers where the sum of any five consecutive terms is negative, and the sum of any nine consecutive terms is positive. I need to figure out the maximum possible value of n, which is the number of terms in the sequence. Hmm, that sounds a bit tricky, but let me try to break it down.First, let me restate the problem to make sure I understand it. We have a sequence a₁, a₂, ..., aₙ. For any five consecutive terms, say aᵢ, aᵢ₊₁, aᵢ₊₂, aᵢ₊₃, aᵢ₊₄, their sum is negative. Similarly, for any nine consecutive terms, like aⱼ, aⱼ₊₁, ..., aⱼ₊₈, their sum is positive. I need to find the largest possible n where this can happen.Alright, so maybe I can start by considering smaller values of n and see how the conditions apply. Let's see.If n is 5, then the sum of all five terms is negative, which is straightforward. But since n is 5, there's no way to have nine consecutive terms, so the second condition doesn't apply. So n=5 is possible, but we need to go higher.If n is 9, then the sum of all nine terms is positive. But also, every set of five consecutive terms must be negative. So, for example, the sum of a₁ to a₅ is negative, a₂ to a₆ is negative, and so on, up to a₅ to a₉. But the total sum from a₁ to a₉ is positive. Hmm, that seems possible if the later terms are positive enough to make the total sum positive, but each group of five is still negative. Maybe.Wait, let's think about the total sum. If the sum of a₁ to a₉ is positive, and the sum of a₁ to a₅ is negative, then the sum of a₆ to a₉ must be positive enough to make the total positive. Similarly, the sum of a₂ to a₆ is negative, so the sum of a₇ to a₉ must be positive enough to offset that. This seems a bit conflicting, but maybe it's possible.But let's not get bogged down with n=9. The problem is asking for the maximum n, so I need to see how high n can go before these conditions contradict each other.I remember that in similar problems, overlapping sums can lead to contradictions when n is too large. Maybe I can use that idea here. Let me try to think about how the sums overlap.Suppose n is 13. Then, I can have multiple overlapping sets of nine consecutive terms. For example:1. a₁ to a₉2. a₂ to a₁₀3. a₃ to a₁₁4. a₄ to a₁₂5. a₅ to a₁₃Each of these sets must have a positive sum. Now, if I add up all these five sums, what do I get?Let me write that out:Sum1 = a₁ + a₂ + a₃ + a₄ + a₅ + a₆ + a₇ + a₈ + a₉ > 0Sum2 = a₂ + a₃ + a₄ + a₅ + a₆ + a₇ + a₈ + a₉ + a₁₀ > 0Sum3 = a₃ + a₄ + a₅ + a₆ + a₇ + a₈ + a₉ + a₁₀ + a₁₁ > 0Sum4 = a₄ + a₅ + a₆ + a₇ + a₈ + a₉ + a₁₀ + a₁₁ + a₁₂ > 0Sum5 = a₅ + a₆ + a₇ + a₈ + a₉ + a₁₀ + a₁₁ + a₁₂ + a₁₃ > 0If I add all these sums together, how many times does each term appear?- a₁ appears once in Sum1- a₂ appears in Sum1 and Sum2, so twice- a₃ appears in Sum1, Sum2, Sum3, so three times- Similarly, a₄ appears four times- a₅ appears five times- a₆ appears five times- a₇ appears five times- a₈ appears five times- a₉ appears five times- a₁₀ appears four times- a₁₁ appears three times- a₁₂ appears twice- a₁₃ appears onceSo the total sum of all five sums is:a₁ + 2a₂ + 3a₃ + 4a₄ + 5a₅ + 5a₆ + 5a₇ + 5a₈ + 5a₉ + 4a₁₀ + 3a₁₁ + 2a₁₂ + a₁₃ > 0But also, each of the five consecutive terms must be negative. Let's consider the sum of each five-term block:SumA = a₁ + a₂ + a₃ + a₄ + a₅ < 0SumB = a₂ + a₃ + a₄ + a₅ + a₆ < 0SumC = a₃ + a₄ + a₅ + a₆ + a₇ < 0SumD = a₄ + a₅ + a₆ + a₇ + a₈ < 0SumE = a₅ + a₆ + a₇ + a₈ + a₉ < 0SumF = a₆ + a₇ + a₈ + a₉ + a₁₀ < 0SumG = a₇ + a₈ + a₉ + a₁₀ + a₁₁ < 0SumH = a₈ + a₉ + a₁₀ + a₁₁ + a₁₂ < 0SumI = a₉ + a₁₀ + a₁₁ + a₁₂ + a₁₃ < 0So, there are nine such sums, each negative. If I add all these nine sums together, what do I get?Let me write that out:SumA + SumB + SumC + SumD + SumE + SumF + SumG + SumH + SumI < 0Each term a₁ to a₁₃ appears how many times?- a₁ appears once in SumA- a₂ appears in SumA and SumB, so twice- a₃ appears in SumA, SumB, SumC, so three times- Similarly, a₄ appears four times- a₅ appears five times- a₆ appears five times- a₇ appears five times- a₈ appears five times- a₉ appears five times- a₁₀ appears four times- a₁₁ appears three times- a₁₂ appears twice- a₁₃ appears onceSo the total sum is:a₁ + 2a₂ + 3a₃ + 4a₄ + 5a₅ + 5a₆ + 5a₇ + 5a₈ + 5a₉ + 4a₁₀ + 3a₁₁ + 2a₁₂ + a₁₃ < 0Wait a minute, that's the same expression as the total sum of the five nine-term sums! So, from the nine-term sums, we have that expression greater than 0, and from the five-term sums, we have the same expression less than 0. That's a contradiction because the same expression can't be both greater than 0 and less than 0 at the same time.Therefore, n cannot be 13 because it leads to a contradiction. So the maximum n must be less than 13. Let's check if n=12 is possible.For n=12, let's try to construct such a sequence. Maybe I can set up the sequence in a way that the sums of five terms are negative, but the sums of nine terms are positive.Let me think about how to arrange the numbers. Maybe have some negative numbers and some positive numbers in a pattern that satisfies the conditions.Suppose I have a repeating pattern where every fifth term is a large positive number, and the others are negative. For example, let's say a₁=-4, a₂=-4, a₃=-4, a₄=15, a₅=-4, and then repeat this pattern.Wait, let me test this. If I have a₁=-4, a₂=-4, a₃=-4, a₄=15, a₅=-4, a₆=-4, a₇=-4, a₈=15, a₉=-4, a₁₀=-4, a₁₁=-4, a₁₂=15.Now, let's check the sum of any five consecutive terms. For example, a₁ to a₅: -4 -4 -4 +15 -4 = -1, which is negative. Similarly, a₂ to a₆: -4 -4 +15 -4 -4 = -1, also negative. Continuing this way, each five-term sum is -1, which is negative.Now, let's check the sum of any nine consecutive terms. For example, a₁ to a₉: -4 -4 -4 +15 -4 -4 -4 +15 -4 = 2, which is positive. Similarly, a₂ to a₁₀: -4 -4 +15 -4 -4 -4 +15 -4 -4 = 2, positive. And so on. Each nine-term sum is 2, which is positive.So, this sequence with n=12 satisfies both conditions. Therefore, n=12 is possible. Since n=13 leads to a contradiction, the maximum n must be 12.I think that makes sense. By carefully choosing the sequence with a pattern that balances the negative and positive sums appropriately, we can satisfy both conditions for n=12, but not for n=13.