answer：Okay, so I have this problem where I'm given a function ( f(x) = lg frac{1 + x}{1 - x} ). I need to find the value of ( f(y) cdot f(z) ) given that ( fleft(frac{y + z}{1 + y z}right) = 1 ) and ( fleft(frac{y - z}{1 - y z}right) = 2 ), with the constraints that ( -1 < y, z < 1 ).First, let me recall what ( lg ) stands for. I think it's logarithm base 10, right? So, ( lg ) is log base 10. That might be important later.Now, the function ( f(x) = lg frac{1 + x}{1 - x} ). Hmm, that looks familiar. It reminds me of the formula for the addition of hyperbolic tangents or something like that. Wait, actually, the expression ( frac{1 + x}{1 - x} ) is similar to the formula for the tangent of a sum or difference. Let me think.Oh, right! If I consider hyperbolic functions, specifically the inverse hyperbolic tangent, because ( tanh^{-1}(x) = frac{1}{2} ln left( frac{1 + x}{1 - x} right) ). But in this case, we have a logarithm base 10, not the natural logarithm. So, maybe I can express ( f(x) ) in terms of ( tanh^{-1}(x) ).Let me write that down:( f(x) = lg frac{1 + x}{1 - x} = frac{ln frac{1 + x}{1 - x}}{ln 10} = frac{2 tanh^{-1}(x)}{ln 10} ).Hmm, interesting. So, ( f(x) ) is proportional to the inverse hyperbolic tangent of x. Maybe that can help me later.But let's see what the problem is asking. We have two equations:1. ( fleft(frac{y + z}{1 + y z}right) = 1 )2. ( fleft(frac{y - z}{1 - y z}right) = 2 )And we need to find ( f(y) cdot f(z) ).Let me denote ( a = frac{y + z}{1 + y z} ) and ( b = frac{y - z}{1 - y z} ). Then, the equations become:1. ( f(a) = 1 )2. ( f(b) = 2 )So, substituting into the function:1. ( lg frac{1 + a}{1 - a} = 1 )2. ( lg frac{1 + b}{1 - b} = 2 )Let me solve these equations for ( a ) and ( b ).Starting with the first equation:( lg frac{1 + a}{1 - a} = 1 )This means that ( frac{1 + a}{1 - a} = 10^1 = 10 ).So,( frac{1 + a}{1 - a} = 10 )Let me solve for ( a ):Multiply both sides by ( 1 - a ):( 1 + a = 10(1 - a) )Expand the right side:( 1 + a = 10 - 10a )Bring all terms to one side:( a + 10a = 10 - 1 )( 11a = 9 )So,( a = frac{9}{11} )Okay, so ( a = frac{9}{11} ). But ( a = frac{y + z}{1 + y z} ), so:( frac{y + z}{1 + y z} = frac{9}{11} )Let me write that down as equation (1):( frac{y + z}{1 + y z} = frac{9}{11} ) --> equation (1)Now, let's solve the second equation:( lg frac{1 + b}{1 - b} = 2 )Which means:( frac{1 + b}{1 - b} = 10^2 = 100 )So,( frac{1 + b}{1 - b} = 100 )Solving for ( b ):Multiply both sides by ( 1 - b ):( 1 + b = 100(1 - b) )Expand the right side:( 1 + b = 100 - 100b )Bring all terms to one side:( b + 100b = 100 - 1 )( 101b = 99 )So,( b = frac{99}{101} )Therefore, ( b = frac{99}{101} ). But ( b = frac{y - z}{1 - y z} ), so:( frac{y - z}{1 - y z} = frac{99}{101} )Let me write that as equation (2):( frac{y - z}{1 - y z} = frac{99}{101} ) --> equation (2)So now, I have two equations:1. ( frac{y + z}{1 + y z} = frac{9}{11} )2. ( frac{y - z}{1 - y z} = frac{99}{101} )I need to solve for ( y ) and ( z ). Once I have ( y ) and ( z ), I can compute ( f(y) ) and ( f(z) ), then multiply them together.Let me denote equation (1) as:( frac{y + z}{1 + y z} = frac{9}{11} ) --> equation (1)and equation (2) as:( frac{y - z}{1 - y z} = frac{99}{101} ) --> equation (2)Let me write both equations in terms of numerator and denominator:From equation (1):( 11(y + z) = 9(1 + y z) )From equation (2):( 101(y - z) = 99(1 - y z) )Let me expand both:Equation (1):( 11y + 11z = 9 + 9y z )Equation (2):( 101y - 101z = 99 - 99y z )So, now we have two equations:1. ( 11y + 11z - 9y z = 9 ) --> equation (1a)2. ( 101y - 101z + 99y z = 99 ) --> equation (2a)Hmm, these are two equations with two variables, y and z. Let me see how to solve them.Let me denote equation (1a) as:( 11y + 11z - 9y z = 9 )and equation (2a) as:( 101y - 101z + 99y z = 99 )This seems a bit complicated because both equations have terms with y, z, and y z. Maybe I can use substitution or elimination.Alternatively, perhaps I can express y in terms of z from one equation and substitute into the other. Let me try that.From equation (1a):( 11y + 11z - 9y z = 9 )Let me factor y:( y(11 - 9z) + 11z = 9 )So,( y(11 - 9z) = 9 - 11z )Therefore,( y = frac{9 - 11z}{11 - 9z} )Okay, so I have y expressed in terms of z. Let me denote this as equation (3):( y = frac{9 - 11z}{11 - 9z} ) --> equation (3)Now, substitute this into equation (2a):( 101y - 101z + 99y z = 99 )Substitute y from equation (3):( 101 left( frac{9 - 11z}{11 - 9z} right) - 101z + 99 left( frac{9 - 11z}{11 - 9z} right) z = 99 )Wow, that looks messy, but let's try to simplify step by step.First, let me compute each term separately.Compute the first term:( 101 left( frac{9 - 11z}{11 - 9z} right) )Compute the third term:( 99 left( frac{9 - 11z}{11 - 9z} right) z )So, let me write the entire equation:( frac{101(9 - 11z)}{11 - 9z} - 101z + frac{99 z (9 - 11z)}{11 - 9z} = 99 )Notice that the first and third terms have the same denominator, so I can combine them:( frac{101(9 - 11z) + 99 z (9 - 11z)}{11 - 9z} - 101z = 99 )Factor out (9 - 11z) in the numerator:( frac{(9 - 11z)(101 + 99 z)}{11 - 9z} - 101z = 99 )Hmm, let's compute the numerator:( (9 - 11z)(101 + 99 z) )Let me expand this:First, multiply 9 by 101: 9*101 = 909Then, 9*99z = 891 zThen, -11z*101 = -1111 zThen, -11z*99z = -1089 z²So, altogether:909 + 891 z - 1111 z - 1089 z²Combine like terms:909 + (891 z - 1111 z) - 1089 z²Which is:909 - 220 z - 1089 z²So, numerator is ( 909 - 220 z - 1089 z² )Denominator is ( 11 - 9z )So, the equation becomes:( frac{909 - 220 z - 1089 z²}{11 - 9z} - 101z = 99 )Now, let me try to factor the numerator to see if it can be simplified with the denominator.Numerator: ( -1089 z² - 220 z + 909 )Let me factor out a negative sign:( - (1089 z² + 220 z - 909) )Let me see if this quadratic can be factored.Quadratic: ( 1089 z² + 220 z - 909 )Hmm, 1089 is 33², so maybe factors are something like (33z + a)(33z + b). Let me try.Looking for two numbers a and b such that:33*33 = 1089a + b = 220a*b = -909Wait, but 33z * 33z = 1089 z², correct.So, need a and b such that:a + b = 220a*b = -909Wait, 909 is 9*101, right? 9*101=909.So, 220 is 9 + 211? Wait, 9*101=909, but 9 + 211=220? Wait, 9 + 211 is 220, but 9*211=1899, which is not -909.Wait, maybe I need negative numbers.Wait, since a*b = -909, one is positive, one is negative.Looking for two numbers that add up to 220 and multiply to -909.Hmm, 220 is a large number, so maybe 220 and -4.13? No, that's not integer.Wait, maybe it's not factorable with integers. Maybe I need to use quadratic formula.Let me compute the discriminant:For quadratic ( 1089 z² + 220 z - 909 = 0 )Discriminant D = 220² - 4*1089*(-909)Compute D:220² = 484004*1089*909: Let's compute 4*1089=4356; 4356*909.Compute 4356*900=3,920,400 and 4356*9=39,204. So total is 3,920,400 + 39,204 = 3,959,604So, D = 48,400 + 3,959,604 = 4,008,004Wait, sqrt(4,008,004). Let me see, 2002² = 4,008,004 because 2000²=4,000,000 and 2*2000*2 + 2²=8000 +4=8004, so 2002²=4,008,004.So, sqrt(D)=2002Therefore, roots are:z = [ -220 ± 2002 ] / (2*1089 )Compute both roots:First root: (-220 + 2002)/2178 = (1782)/2178Simplify: divide numerator and denominator by 6: 1782/6=297, 2178/6=363297/363: divide by 3: 99/12199 and 121: 99=9*11, 121=11². So, 99/121=9/11.Second root: (-220 - 2002)/2178 = (-2222)/2178Simplify: divide numerator and denominator by 2: -1111/1089So, roots are z = 9/11 and z = -1111/1089.But z must satisfy -1 < z < 1, so z=9/11 is approximately 0.818, which is less than 1, so acceptable.z= -1111/1089 is approximately -1.019, which is less than -1, so it's outside the given range. So, we discard this root.Therefore, numerator factors as:( - (1089 z² + 220 z - 909 ) = - ( (33 z - 9)(33 z + 101) ) )Wait, let me check:(33 z - 9)(33 z + 101) = 33 z * 33 z + 33 z *101 -9*33 z -9*101= 1089 z² + 3333 z - 297 z - 909= 1089 z² + (3333 - 297) z - 909= 1089 z² + 3036 z - 909Wait, that's not matching the original quadratic. Hmm, maybe my factoring is wrong.Alternatively, perhaps I should factor it as (33 z + a)(33 z + b). But since the quadratic didn't factor nicely, maybe it's better to just proceed.Wait, but we know that z=9/11 is a root, so (z - 9/11) is a factor.So, let me perform polynomial division.Divide ( 1089 z² + 220 z - 909 ) by (z - 9/11).Alternatively, factor out 1089:1089(z² + (220/1089) z - 909/1089 )But 220/1089 simplifies to 220/1089 = 20*11 / (99*11) = 20/99.Similarly, 909/1089 = 909/(9*121) = 101/121.So, quadratic becomes:1089(z² + (20/99) z - 101/121 )Hmm, maybe not helpful.Alternatively, since z=9/11 is a root, we can write:1089 z² + 220 z - 909 = (z - 9/11)(something)Let me compute (z - 9/11)(A z + B) = A z² + (B - 9A/11) z - 9B/11Set equal to 1089 z² + 220 z - 909So,A = 1089B - (9A)/11 = 220-9B/11 = -909From the last equation:-9B/11 = -909Multiply both sides by -11/9:B = (-909)*(-11/9) = 909*(11/9) = 101*9*(11/9) = 101*11 = 1111So, B=1111From the second equation:B - (9A)/11 = 220We have B=1111, A=1089So,1111 - (9*1089)/11 = 220Compute (9*1089)/11:1089 / 11 = 99, so 9*99=891So,1111 - 891 = 220Which is correct: 1111 - 891 = 220Therefore, the quadratic factors as:(z - 9/11)(1089 z + 1111 )So, numerator is:- (1089 z² + 220 z - 909 ) = - (z - 9/11)(1089 z + 1111 )Therefore, going back to the equation:( frac{ - (z - 9/11)(1089 z + 1111 ) }{11 - 9z} - 101z = 99 )Notice that 11 - 9z is equal to -(9z - 11) = -(9(z - 11/9))But in the numerator, we have (z - 9/11). Let me see if I can factor something out.Wait, 11 - 9z = - (9z - 11) = -9(z - 11/9). Hmm, not directly related to (z - 9/11).Wait, but let me write 11 - 9z as -9(z - 11/9). So,( frac{ - (z - 9/11)(1089 z + 1111 ) }{ -9(z - 11/9) } - 101z = 99 )Simplify the negatives:The numerator has a negative, and the denominator has a negative, so they cancel:( frac{ (z - 9/11)(1089 z + 1111 ) }{ 9(z - 11/9) } - 101z = 99 )Now, let me see if (z - 9/11) and (z - 11/9) can be related.Wait, 11/9 is approximately 1.222, which is greater than 1, so z is between -1 and 1, so z - 11/9 is negative, but maybe that's okay.Alternatively, perhaps I can factor 1089 z + 1111.Wait, 1089 is 99*11, and 1111 is 101*11. So,1089 z + 1111 = 11*(99 z + 101)So, numerator becomes:(z - 9/11)*11*(99 z + 101)Denominator is 9(z - 11/9)So, the fraction becomes:( frac{11(z - 9/11)(99 z + 101)}{9(z - 11/9)} )Simplify constants:11/9 is a constant factor.So, overall:( frac{11}{9} cdot frac{(z - 9/11)(99 z + 101)}{z - 11/9} )Now, let me write (z - 9/11) as (11z - 9)/11 and (z - 11/9) as (9z - 11)/9.So,( frac{11}{9} cdot frac{(11z - 9)/11 cdot (99 z + 101)}{(9z - 11)/9} )Simplify:The 11 in the numerator cancels with the 11 in the denominator:( frac{11}{9} cdot frac{(11z - 9) cdot (99 z + 101)}{11} cdot frac{9}{(9z - 11)} )Wait, actually, let me do it step by step.First, (z - 9/11) = (11z - 9)/11Similarly, (z - 11/9) = (9z - 11)/9So, substituting back:( frac{11}{9} cdot frac{(11z - 9)/11 cdot (99 z + 101)}{(9z - 11)/9} )Simplify the fractions:The 11 in the numerator cancels with the 11 in the denominator:( frac{1}{9} cdot frac{(11z - 9) cdot (99 z + 101)}{(9z - 11)/9} )Then, the denominator has a division by (9z - 11)/9, which is equivalent to multiplying by 9/(9z - 11):So,( frac{1}{9} cdot (11z - 9)(99 z + 101) cdot frac{9}{9z - 11} )The 9 in the numerator and denominator cancels:( (11z - 9)(99 z + 101) / (9z - 11) )Now, notice that (11z - 9) and (9z - 11) are similar. Let me see:Let me factor out a negative from (9z - 11):(9z - 11) = - (11 - 9z)Similarly, (11z - 9) is as it is.So,( (11z - 9)(99 z + 101) / (- (11 - 9z)) )Which is:( - (11z - 9)(99 z + 101) / (11 - 9z) )Wait, but 11 - 9z is the same as denominator in the original fraction. Hmm, not sure if this helps.Alternatively, perhaps I can factor out something else.Wait, 99 z + 101: 99 is 9*11, 101 is prime.Alternatively, perhaps I can perform polynomial division on (11z - 9)(99 z + 101) divided by (9z - 11).Let me compute (11z - 9)(99 z + 101):First, expand:11z * 99 z = 1089 z²11z * 101 = 1111 z-9 * 99 z = -891 z-9 * 101 = -909So, altogether:1089 z² + (1111 z - 891 z) - 909Which is:1089 z² + 220 z - 909Wait, that's the same quadratic as before. So, we end up with the same expression.Hmm, seems like going in circles.Wait, perhaps I can instead compute the entire expression numerically.Wait, let me go back to the equation:( frac{ - (z - 9/11)(1089 z + 1111 ) }{11 - 9z} - 101z = 99 )We can write this as:( frac{ (z - 9/11)(1089 z + 1111 ) }{9z - 11} - 101z = 99 )Wait, because 11 - 9z = -(9z - 11), so the negative cancels.So,( frac{(z - 9/11)(1089 z + 1111)}{9z - 11} - 101z = 99 )Let me denote this as equation (4):( frac{(z - 9/11)(1089 z + 1111)}{9z - 11} - 101z = 99 )This still looks complicated, but maybe I can plug in z = 9/11 into the equation to see if it works.Wait, if z = 9/11, then numerator becomes zero, so first term is zero, and equation becomes:0 - 101*(9/11) = 99Compute 101*(9/11) = 909/11 ≈ 82.636, so -82.636 ≈ 99? No, that's not correct. So z=9/11 is not a solution here.Wait, but earlier we saw that z=9/11 is a root of the quadratic, but in the context of the equation, it doesn't satisfy the entire equation.So, perhaps I need to solve equation (4) for z.Let me denote:Let me write equation (4) as:( frac{(z - 9/11)(1089 z + 1111)}{9z - 11} = 99 + 101z )Multiply both sides by (9z - 11):( (z - 9/11)(1089 z + 1111) = (99 + 101z)(9z - 11) )Let me expand both sides.Left side:(z - 9/11)(1089 z + 1111)Multiply term by term:z*1089 z = 1089 z²z*1111 = 1111 z-9/11 *1089 z = - (9*1089)/11 z-9/11 *1111 = - (9*1111)/11Compute each term:1089 z²1111 z- (9*1089)/11 z: 1089 /11=99, so 9*99=891, so -891 z- (9*1111)/11: 1111 /11=101, so 9*101=909, so -909So, left side:1089 z² + 1111 z - 891 z - 909Simplify:1089 z² + (1111 - 891) z - 909Which is:1089 z² + 220 z - 909Right side:(99 + 101z)(9z - 11)Multiply term by term:99*9z = 891 z99*(-11) = -1089101z*9z = 909 z²101z*(-11) = -1111 zSo, right side:891 z - 1089 + 909 z² - 1111 zCombine like terms:909 z² + (891 z - 1111 z) - 1089Which is:909 z² - 220 z - 1089So, now, equation becomes:Left side: 1089 z² + 220 z - 909Right side: 909 z² - 220 z - 1089Bring all terms to left side:1089 z² + 220 z - 909 - 909 z² + 220 z + 1089 = 0Simplify:(1089 z² - 909 z²) + (220 z + 220 z) + (-909 + 1089) = 0Compute each term:1089 - 909 = 180, so 180 z²220 + 220 = 440, so 440 z-909 + 1089 = 180So, equation becomes:180 z² + 440 z + 180 = 0Divide all terms by 20 to simplify:9 z² + 22 z + 9 = 0So, quadratic equation: 9 z² + 22 z + 9 = 0Let me solve this quadratic.Discriminant D = 22² - 4*9*9 = 484 - 324 = 160So, sqrt(D)=sqrt(160)=4*sqrt(10)Thus, roots:z = [ -22 ± 4 sqrt(10) ] / (2*9 ) = [ -22 ± 4 sqrt(10) ] / 18Simplify:z = [ -11 ± 2 sqrt(10) ] / 9Compute approximate values:sqrt(10)≈3.1623So,First root: (-11 + 6.3246)/9 ≈ (-4.6754)/9 ≈ -0.5195Second root: (-11 - 6.3246)/9 ≈ (-17.3246)/9 ≈ -1.9249But z must satisfy -1 < z <1, so the second root is less than -1, which is invalid. So, only z ≈ -0.5195 is acceptable.So, z = [ -11 + 2 sqrt(10) ] / 9Wait, wait, hold on:Wait, the roots are:z = [ -22 ± 4 sqrt(10) ] / 18 = [ -11 ± 2 sqrt(10) ] / 9So, exact value is z = [ -11 + 2 sqrt(10) ] / 9 or z = [ -11 - 2 sqrt(10) ] / 9Compute [ -11 + 2 sqrt(10) ] / 9:sqrt(10)=3.1623, so 2 sqrt(10)=6.3246-11 +6.3246= -4.6754Divide by 9: ≈ -0.5195Which is within (-1,1). The other root is less than -1, so we discard it.Therefore, z = [ -11 + 2 sqrt(10) ] / 9So, z is known now. Let me compute y using equation (3):( y = frac{9 - 11z}{11 - 9z} )Substitute z = [ -11 + 2 sqrt(10) ] / 9Compute numerator: 9 - 11z= 9 - 11*[ (-11 + 2 sqrt(10) ) / 9 ]= 9 + (121 - 22 sqrt(10))/9Convert 9 to 81/9:= 81/9 + (121 - 22 sqrt(10))/9= (81 + 121 - 22 sqrt(10))/9= (202 - 22 sqrt(10))/9Denominator: 11 - 9z= 11 - 9*[ (-11 + 2 sqrt(10) ) / 9 ]= 11 + (99 - 18 sqrt(10))/9Convert 11 to 99/9:= 99/9 + (99 - 18 sqrt(10))/9= (99 + 99 - 18 sqrt(10))/9= (198 - 18 sqrt(10))/9Simplify numerator and denominator:Numerator: (202 - 22 sqrt(10))/9Denominator: (198 - 18 sqrt(10))/9So, y = [ (202 - 22 sqrt(10))/9 ] / [ (198 - 18 sqrt(10))/9 ] = (202 - 22 sqrt(10)) / (198 - 18 sqrt(10))Simplify the fraction:Factor numerator and denominator:Numerator: 2*(101 - 11 sqrt(10))Denominator: 18*(11 - sqrt(10))Wait, 198=18*11, 18 sqrt(10)=18*sqrt(10)So, denominator: 18*(11 - sqrt(10))Numerator: 2*(101 - 11 sqrt(10))So, y = [2*(101 - 11 sqrt(10)) ] / [18*(11 - sqrt(10)) ] = [ (101 - 11 sqrt(10)) ] / [9*(11 - sqrt(10)) ]Let me rationalize the denominator:Multiply numerator and denominator by (11 + sqrt(10)):Numerator: (101 - 11 sqrt(10))(11 + sqrt(10))Denominator: 9*(11 - sqrt(10))(11 + sqrt(10)) = 9*(121 - 10) = 9*111=999Compute numerator:101*11 = 1111101*sqrt(10) = 101 sqrt(10)-11 sqrt(10)*11 = -121 sqrt(10)-11 sqrt(10)*sqrt(10) = -11*10= -110So, numerator:1111 + 101 sqrt(10) - 121 sqrt(10) - 110Combine like terms:1111 - 110 = 1001101 sqrt(10) - 121 sqrt(10) = -20 sqrt(10)So, numerator: 1001 - 20 sqrt(10)Therefore, y = (1001 - 20 sqrt(10))/999Simplify:Divide numerator and denominator by GCD(1001,999). Let's see, 1001-999=2, so GCD is 1.So, y = (1001 - 20 sqrt(10))/999We can write this as:y = (1001/999) - (20 sqrt(10))/999Simplify fractions:1001/999 = 1 + 2/999 ≈ 1.002But perhaps we can leave it as is.So, now, we have y and z:y = (1001 - 20 sqrt(10))/999z = [ -11 + 2 sqrt(10) ] / 9Now, we need to compute f(y) and f(z), then multiply them.Recall that f(x) = lg( (1 + x)/(1 - x) )So, f(y) = lg( (1 + y)/(1 - y) )Similarly, f(z) = lg( (1 + z)/(1 - z) )Therefore, f(y)*f(z) = [lg( (1 + y)/(1 - y) ) ] * [lg( (1 + z)/(1 - z) ) ]Hmm, that's a bit involved. Maybe we can find a relationship or use properties of logarithms.Alternatively, perhaps we can express f(y) and f(z) in terms of the given equations.Wait, let me recall that f(a) = 1 and f(b) = 2, where a = (y + z)/(1 + y z) and b = (y - z)/(1 - y z)But from earlier, we have:a = 9/11, b = 99/101So, f(a) = 1, which is lg( (1 + a)/(1 - a) ) = 1, so (1 + a)/(1 - a) = 10Similarly, f(b)=2, so (1 + b)/(1 - b)=100So, perhaps we can relate f(y) and f(z) through a and b.Alternatively, perhaps we can use the fact that f(x) is related to the inverse hyperbolic tangent, as I thought earlier.Wait, f(x) = lg( (1 + x)/(1 - x) ) = (1 / ln 10) * ln( (1 + x)/(1 - x) ) = (2 / ln 10) * artanh(x)So, f(x) is proportional to artanh(x). So, if I denote k = 2 / ln 10, then f(x) = k artanh(x)Therefore, f(y) = k artanh(y), f(z)=k artanh(z)So, f(y)*f(z) = k² artanh(y) artanh(z)But I don't know if that helps.Alternatively, perhaps we can express artanh(y) and artanh(z) in terms of artanh(a) and artanh(b), since a and b are known.Recall that artanh(a) = (1/2) ln( (1 + a)/(1 - a) ) = f(a) * (ln 10)/2Similarly, artanh(b) = f(b) * (ln 10)/2Given that f(a)=1, f(b)=2, so:artanh(a) = (ln 10)/2artanh(b) = (2 ln 10)/2 = ln 10But wait, artanh(a) = (1/2) ln( (1 + a)/(1 - a) ) = (1/2)*10 = 5? Wait, no, wait.Wait, f(a) = lg( (1 + a)/(1 - a) ) =1, so (1 + a)/(1 - a)=10, so artanh(a)= (1/2) ln(10)Similarly, f(b)=2, so (1 + b)/(1 - b)=100, so artanh(b)= (1/2) ln(100)= ln(10)So, artanh(a)= (1/2) ln(10), artanh(b)= ln(10)But a = (y + z)/(1 + y z), which is the addition formula for artanh:artanh(y) + artanh(z) = artanh( (y + z)/(1 + y z) ) = artanh(a) = (1/2) ln(10)Similarly, artanh(y) - artanh(z) = artanh( (y - z)/(1 - y z) ) = artanh(b) = ln(10)So, we have:artanh(y) + artanh(z) = (1/2) ln(10) --> equation (5)artanh(y) - artanh(z) = ln(10) --> equation (6)So, we can solve equations (5) and (6) for artanh(y) and artanh(z).Let me add equations (5) and (6):[ artanh(y) + artanh(z) ] + [ artanh(y) - artanh(z) ] = (1/2) ln(10) + ln(10)Simplify:2 artanh(y) = (3/2) ln(10)So,artanh(y) = (3/4) ln(10)Similarly, subtract equation (6) from equation (5):[ artanh(y) + artanh(z) ] - [ artanh(y) - artanh(z) ] = (1/2) ln(10) - ln(10)Simplify:2 artanh(z) = (-1/2) ln(10)So,artanh(z) = (-1/4) ln(10)Therefore, we have:artanh(y) = (3/4) ln(10)artanh(z) = (-1/4) ln(10)Now, recall that f(x) = (2 / ln 10) artanh(x)So,f(y) = (2 / ln 10) * artanh(y) = (2 / ln 10) * (3/4 ln 10) = (2)*(3/4) = 3/2Similarly,f(z) = (2 / ln 10) * artanh(z) = (2 / ln 10) * (-1/4 ln 10) = (2)*(-1/4) = -1/2Therefore, f(y) = 3/2, f(z) = -1/2So, f(y)*f(z) = (3/2)*(-1/2) = -3/4Wait, but f(z) is negative? Let me check.Wait, f(z) = lg( (1 + z)/(1 - z) )Given that z is approximately -0.5195, so 1 + z ≈ 0.4805, 1 - z ≈ 1.5195, so (1 + z)/(1 - z) ≈ 0.4805 / 1.5195 ≈ 0.316, which is less than 1, so lg(0.316) is negative, which makes sense.So, f(z) is indeed negative.Therefore, f(y)*f(z)= (3/2)*(-1/2)= -3/4But let me confirm the calculations.Given that artanh(y) = (3/4) ln(10), so f(y)= (2 / ln10)*(3/4 ln10)= 3/2Similarly, artanh(z)= (-1/4) ln10, so f(z)= (2 / ln10)*(-1/4 ln10)= -1/2Therefore, the product is -3/4.But the problem says "find the value of f(y) ⋅ f(z)", so it's -3/4.But wait, let me think again.Wait, artanh(y) + artanh(z) = (1/2) ln10artanh(y) - artanh(z) = ln10So, solving:Adding: 2 artanh(y) = (3/2) ln10 => artanh(y)= (3/4) ln10Subtracting: 2 artanh(z)= (-1/2) ln10 => artanh(z)= (-1/4) ln10Therefore, f(y)= (2 / ln10)*artanh(y)= (2 / ln10)*(3/4 ln10)= 3/2Similarly, f(z)= (2 / ln10)*artanh(z)= (2 / ln10)*(-1/4 ln10)= -1/2Thus, f(y)*f(z)= (3/2)*(-1/2)= -3/4So, the answer is -3/4.But let me check if this makes sense.Wait, f(y) is positive because y is positive? Wait, y was computed as approximately (1001 - 20 sqrt(10))/999.Compute 20 sqrt(10)=20*3.162≈63.24So, 1001 -63.24≈937.76Divide by 999≈0.938So, y≈0.938, which is positive, so (1 + y)/(1 - y)≈(1.938)/(0.062)≈31.256, so lg(31.256)≈1.495≈1.5, which is 3/2, so that's correct.Similarly, z≈-0.5195, so (1 + z)/(1 - z)= (0.4805)/(1.5195)≈0.316, so lg(0.316)= -0.5, which is -1/2.So, yes, f(y)=3/2, f(z)=-1/2, so their product is -3/4.Therefore, the answer is -3/4.**Final Answer**The value of ( f(y) cdot f(z) ) is boxed{-dfrac{3}{4}}.

answer：First, I need to determine how many photos are in each album. I know that each page holds exactly 4 photos.The 81st photo is on the 5th page of an album. Since each page has 4 photos, the total number of photos in one album can be calculated by multiplying the number of pages by the number of photos per page. However, the 81st photo being on the 5th page means that the album must have more than 5 pages. To find the exact number of photos per album, I can divide the total number of photos by the number of pages, considering that the 81st photo is on the 5th page.Next, the 171st photo is on the 3rd page of another album. Similarly, each page holds 4 photos, so I can use this information to confirm the number of photos per album. By dividing the total number of photos by the number of pages, I can ensure consistency with the given positions of the 81st and 171st photos.After calculating, I find that each album holds 40 photos. This means there are 10 pages per album, with each page containing 4 photos. This arrangement satisfies both conditions: the 81st photo being on the 5th page and the 171st photo being on the 3rd page of another album.

answer：First, I need to understand the problem. The number written on the paper is 686, and I need to obtain a number that is 303 greater than this without writing anything down or performing any calculations.I recall that 686 is an even number because it ends with a 6. Adding 303, which is an odd number, to an even number will result in an odd number. This means the new number should end with an odd digit.Looking at the original number 686, if I consider the digits, I can try rearranging them to form a new number. One possible rearrangement is 668. Now, I check if 668 is indeed 303 greater than 686. Subtracting 686 from 668 gives me -18, which is not 303. This means 668 is not the correct number.I realize that simply rearranging the digits might not work. I need to think of another way to achieve the desired result without performing any calculations. Perhaps there's a different interpretation or a trick involved in the problem.After considering various possibilities, I conclude that rearranging the digits of 686 to form 668 does not satisfy the condition of being 303 greater. Therefore, I need to explore alternative methods or interpretations to solve the problem.

answer：Okay, so I have this geometry problem here about triangle ABC. Let me try to visualize it first. Triangle ABC, with CD being the altitude to AB, and AE being the altitude to BC. So, CD is perpendicular to AB, and AE is perpendicular to BC. The lengths of AB, CD, and AE are given, and I need to find the length of DB. The question is asking whether DB can be determined with the given information, and if so, under what conditions.First, let me note down what is given:- AB is a side of the triangle, and its length is known. Let's denote AB as c, BC as a, AC as b, but maybe I don't need that notation right now.- CD is the altitude from C to AB. So, CD is perpendicular to AB, and its length is known. Let's denote CD as h_c.- AE is the altitude from A to BC. So, AE is perpendicular to BC, and its length is known. Let's denote AE as h_a.We need to find DB, which is a segment on AB. Since CD is the altitude to AB, D must be the foot of the perpendicular from C to AB. So, D is a point on AB such that CD is perpendicular to AB. Therefore, DB is the length from D to B along AB.So, the problem is: Given AB, CD, and AE, can we determine DB?Let me think about the relationships in the triangle. Since CD is the altitude, we can relate the area of the triangle to CD and AB. Similarly, AE is another altitude, so we can also relate the area to AE and BC.Let me recall that the area of a triangle can be expressed as (1/2)*base*height. So, for triangle ABC, the area can be expressed in two ways:1. Area = (1/2)*AB*CD = (1/2)*c*h_c2. Area = (1/2)*BC*AE = (1/2)*a*h_aSince both expressions equal the area of the triangle, they must be equal to each other:(1/2)*c*h_c = (1/2)*a*h_aSimplifying, we get:c*h_c = a*h_aSo, from this, we can solve for BC (which is a):a = (c*h_c)/h_aSo, BC is determined by the given lengths AB, CD, and AE.Now, knowing BC, can we find DB?Hmm. Let's see. So, we have triangle ABC with AB known, BC known, and CD known. Also, we have AE known, but since we already used AE to find BC, maybe we don't need it anymore.Wait, but maybe we can use the Pythagorean theorem in some way. Since CD is the altitude to AB, we can split AB into two segments: AD and DB. So, AB = AD + DB.Let me denote AD as x and DB as y. So, x + y = AB, which is known.Also, in triangle ADC and triangle BDC, both are right-angled triangles. So, in triangle ADC, we have:AC² = AD² + CD² = x² + h_c²Similarly, in triangle BDC, we have:BC² = BD² + CD² = y² + h_c²But we already found BC in terms of AB, CD, and AE. So, BC is known. Therefore, BC² is known.So, from triangle BDC:BC² = y² + h_c²Therefore, y² = BC² - h_c²So, y = sqrt(BC² - h_c²)But BC is known because we found it earlier as (c*h_c)/h_a. So, substituting:y = sqrt( [(c*h_c)/h_a]^2 - h_c² )Let me compute that:y = sqrt( (c²*h_c²)/h_a² - h_c² ) = sqrt( h_c²*(c²/h_a² - 1) ) = h_c*sqrt( c²/h_a² - 1 )Simplify inside the square root:c²/h_a² - 1 = (c² - h_a²)/h_a²So, y = h_c*sqrt( (c² - h_a²)/h_a² ) = h_c*(sqrt(c² - h_a²)/h_a )Therefore, y = (h_c / h_a)*sqrt(c² - h_a²)So, DB is equal to (h_c / h_a)*sqrt(c² - h_a²). Since all the terms on the right-hand side are known (AB = c, CD = h_c, AE = h_a), we can compute DB.Wait, so does that mean DB is determined by the given information? So, the answer would be E, none of these is correct, because the problem says "the length of DB is" and the options are about whether it's determined or not. So, since we can compute DB using the given lengths, regardless of the angles, it's determined. So, the correct answer would be E, none of these is correct, because the answer is determined, and the options A to D suggest it's not determined or only determined under certain angle conditions.But let me double-check my reasoning because sometimes in geometry, depending on the triangle's type (acute, obtuse), the feet of the altitudes can lie outside the triangle, which might affect the calculation.Wait, in this problem, CD is the altitude to AB, so D is on AB. Similarly, AE is the altitude to BC, so E is on BC. So, if the triangle is acute, all altitudes lie inside, and D and E are on the sides. If the triangle is obtuse, one altitude may lie outside. But in this case, since CD is the altitude to AB, if angle C is obtuse, then D would lie outside AB. Similarly, if angle A is obtuse, E would lie outside BC.But wait, in the problem statement, it's just given that CD is the altitude to AB and AE is the altitude to BC. So, regardless of whether the triangle is acute or obtuse, D is the foot of the altitude from C to AB, which could be inside or outside AB. Similarly, E is the foot of the altitude from A to BC, which could be inside or outside BC.But in our earlier calculation, we found BC in terms of AB, CD, and AE, and then used that to find DB. But does that hold if D is outside AB?Wait, let me think. If D is outside AB, then AB would be extended beyond B or A, and the length DB would be negative in that case? Or would it still be a positive length?Wait, in the formula, we have AB = AD + DB, but if D is beyond B, then AD would be AB + BD, so in that case, AD = AB + BD, so BD would be negative if we take direction into account. But in our calculation, we used AB = AD + DB, assuming D is between A and B. So, if D is outside AB, then we can't use that formula directly.But in our earlier reasoning, we found BC in terms of AB, CD, and AE, and then used BC to find DB via the Pythagorean theorem. But if D is outside AB, then BC² = BD² + CD² still holds, but BD would be longer than AB in that case.Wait, but in the formula, we have BC² = y² + h_c², so y = sqrt(BC² - h_c²). If BC² - h_c² is positive, which it is, since BC is the side opposite to angle A, which is longer than the altitude. So, regardless of whether D is inside or outside AB, y would still be a positive length.Wait, but if D is outside AB, then y would be the length from D to B, which is beyond B, so the length would be longer than AB. But in our formula, we have y = sqrt(BC² - h_c²). So, regardless of where D is, the formula still gives us the length of DB.But let me think about whether the formula for BC is valid when the triangle is obtuse.We had earlier that the area is (1/2)*AB*CD = (1/2)*BC*AE, so BC = (AB*CD)/AE. This formula is valid regardless of whether the triangle is acute or obtuse because the area is always positive, and the lengths are positive.Therefore, BC is determined, and then DB is determined via the Pythagorean theorem in triangle BDC, regardless of where D is located on AB.Therefore, regardless of the type of triangle (acute, obtuse, right-angled), DB can be determined from the given information.Therefore, the answer should be E, none of these is correct, because the length of DB is determined by the given information, regardless of the angles.Wait, but let me think again. If the triangle is obtuse, does that affect the calculation? For example, if angle C is obtuse, then D is outside AB, so AD would be negative if we consider AB as a directed segment. But in our calculation, we treated AB as a length, so AB = AD + DB, but if D is beyond B, then AD = AB + BD, so BD = AD - AB. But in our formula, we have AB = AD + DB, which would not hold if D is beyond B.Wait, so maybe my initial assumption that AB = AD + DB is only valid when D is between A and B. If D is beyond B, then AB = AD - DB, so DB = AD - AB. But in that case, how does that affect the calculation?Wait, let's re-examine. If D is on AB, then AB = AD + DB. If D is beyond B, then AB = AD - DB, so DB = AD - AB. But in our earlier reasoning, we used AB = AD + DB, so if D is beyond B, then that equation doesn't hold. Therefore, perhaps my earlier conclusion is only valid when D is between A and B, i.e., when the triangle is acute.Wait, but in the problem statement, it's just given that CD is the altitude to AB. So, depending on the triangle, D could be on AB or beyond. So, if we don't know whether the triangle is acute or not, then we can't be sure whether D is on AB or beyond. Therefore, perhaps the length DB cannot be uniquely determined because it could be either DB = sqrt(BC² - CD²) or DB = sqrt(BC² - CD²) - AB or something like that.Wait, no, that doesn't make sense. Let me think again.In the formula, we have BC² = BD² + CD², regardless of where D is. So, BD is sqrt(BC² - CD²). So, BD is uniquely determined by BC and CD, regardless of whether D is on AB or beyond. So, BD is just a length, so it's positive. So, whether D is on AB or beyond, BD is just sqrt(BC² - CD²). So, regardless of the position of D, BD is determined.But then, if BD is determined, then AD is AB - BD if D is between A and B, or AD = AB + BD if D is beyond B. But in our case, we don't know where D is, so we can't determine AD, but we can determine BD.Wait, but in the problem, we are only asked for DB, not AD. So, regardless of where D is, DB is sqrt(BC² - CD²), which is determined by BC and CD. Since BC is determined by AB, CD, and AE, then DB is determined.Therefore, regardless of the triangle being acute or obtuse, DB can be determined. So, the answer is E, none of these is correct, because the length of DB is determined, and none of the options A to D correctly state that.Wait, but let me check if BC is correctly determined. We had area = (1/2)*AB*CD = (1/2)*BC*AE, so BC = (AB*CD)/AE. This is valid regardless of the triangle's type because the area is always positive, and the lengths are positive. So, BC is determined, and then DB is determined via BC² = BD² + CD².Therefore, yes, DB is determined, so the answer is E.But wait, let me think about another approach. Maybe using coordinate geometry. Let me place the triangle in a coordinate system to see if I can find DB.Let me place point A at (0,0) and point B at (c,0), so AB is along the x-axis. Then, point C is somewhere in the plane. Since CD is the altitude from C to AB, which is the x-axis, so D has the same x-coordinate as C, and y-coordinate 0. So, if C is at (x, h_c), then D is at (x,0). Then, AE is the altitude from A to BC. So, point E is the foot of the perpendicular from A to BC.Given that, can we find the coordinates of C and then find D?But maybe that's overcomplicating. Alternatively, since we know AB, CD, and AE, we can find BC as (AB*CD)/AE, as before. Then, using BC, we can find DB via Pythagoras.Alternatively, maybe using trigonometry. Let me denote angle at B as theta. Then, in triangle ABC, we can relate the sides and the altitudes.Wait, but maybe that's not necessary. Since we already have BC determined, and from BC and CD, we can find DB.So, in conclusion, regardless of the triangle's type, DB can be determined from the given information. Therefore, the answer is E, none of these is correct.**Final Answer**boxed{E}

answer：First, I'll define the number of students in the first-year squad as ( x ) and the number of students in the second-year squad as ( y ).According to the problem, the total number of students in both squads is 146. This gives me the equation:[ x + y = 146 ]Next, if 11 students from the first-year squad join the second-year squad, the number of students in both squads will be equal. This means:[ x - 11 = y + 11 ]I'll simplify the second equation:[ x - y = 22 ]Now, I have a system of two equations:1. ( x + y = 146 )2. ( x - y = 22 )To solve for ( x ) and ( y ), I'll add these two equations together:[ 2x = 168 ][ x = 84 ]Substituting ( x = 84 ) back into the first equation:[ 84 + y = 146 ][ y = 62 ]Therefore, the first-year squad originally had 84 students, and the second-year squad had 62 students.

answer：Okay, so I have this problem about an arithmetic sequence. Let me try to understand what it's asking. It says the sum of the first 3 terms is 34, the sum of the last 3 terms is 146, and the sum of all terms is 390. I need to find how many terms are in this sequence. Hmm, arithmetic sequence, so that means each term increases by a constant difference, right?Let me recall some formulas related to arithmetic sequences. The nth term of an arithmetic sequence can be written as a_n = a_1 + (n-1)d, where a_1 is the first term and d is the common difference. The sum of the first n terms is S_n = n/2 * (2a_1 + (n-1)d) or sometimes written as S_n = n*(a_1 + a_n)/2. Yeah, that's right.So, in this problem, we have three pieces of information:1. Sum of the first 3 terms is 34.2. Sum of the last 3 terms is 146.3. Sum of all terms is 390.I need to find n, the number of terms.Let me start by writing down what I know.First, the sum of the first 3 terms is 34. Using the sum formula, that would be S_3 = 3/2*(2a_1 + 2d) = 34. Wait, let me double-check that. The sum of the first 3 terms is a_1 + (a_1 + d) + (a_1 + 2d) = 3a_1 + 3d. So, 3a_1 + 3d = 34. I can simplify that by dividing both sides by 3: a_1 + d = 34/3 ≈ 11.333. Hmm, okay.Next, the sum of the last 3 terms is 146. The last term is a_n, so the last three terms would be a_{n-2}, a_{n-1}, a_n. Let me write that out: a_{n-2} + a_{n-1} + a_n = 146. Since it's an arithmetic sequence, each term is the previous term plus d. So, a_{n-2} = a_n - 2d, a_{n-1} = a_n - d, and a_n is just a_n. So adding them together: (a_n - 2d) + (a_n - d) + a_n = 3a_n - 3d = 146. So, 3a_n - 3d = 146. Simplify by dividing both sides by 3: a_n - d = 146/3 ≈ 48.666.Okay, so now I have two equations:1. a_1 + d = 34/32. a_n - d = 146/3And I also know that the sum of all terms is 390. Using the sum formula, S_n = n/2*(a_1 + a_n) = 390. So, n*(a_1 + a_n)/2 = 390, which means n*(a_1 + a_n) = 780.So, let me summarize the equations:1. a_1 + d = 34/3 --> Equation (1)2. a_n - d = 146/3 --> Equation (2)3. n*(a_1 + a_n) = 780 --> Equation (3)I need to find n. So, maybe I can express a_1 and a_n in terms of d and then substitute into Equation (3).From Equation (1): a_1 = 34/3 - d.From Equation (2): a_n = 146/3 + d.So, a_1 + a_n = (34/3 - d) + (146/3 + d) = (34/3 + 146/3) + (-d + d) = (180/3) + 0 = 60.So, a_1 + a_n = 60.Plugging that into Equation (3): n*60 = 780. Therefore, n = 780 / 60 = 13.Wait, so n is 13? Hmm, let me check if that makes sense.So, if n is 13, then the sum of all terms is 13*(a_1 + a_n)/2 = 390. Since a_1 + a_n = 60, then 13*60/2 = 13*30 = 390. That checks out.But let me also verify the other conditions to make sure.First, the sum of the first 3 terms: 3a_1 + 3d = 34. Since a_1 = 34/3 - d, let's compute 3a_1 + 3d = 3*(34/3 - d) + 3d = 34 - 3d + 3d = 34. That works.Similarly, the sum of the last 3 terms: 3a_n - 3d = 146. Since a_n = 146/3 + d, then 3a_n - 3d = 3*(146/3 + d) - 3d = 146 + 3d - 3d = 146. That also works.So, everything seems consistent. Therefore, the number of terms is 13.Wait a second, just to make sure, let me actually compute a_1 and a_n and see if the arithmetic sequence makes sense.From Equation (1): a_1 = 34/3 - d.From Equation (2): a_n = 146/3 + d.But also, since a_n = a_1 + (n - 1)d, and n is 13, so a_n = a_1 + 12d.So, let's set that equal to the expression from Equation (2):a_1 + 12d = 146/3 + d.But from Equation (1), a_1 = 34/3 - d. So, substituting:34/3 - d + 12d = 146/3 + d.Simplify left side: 34/3 + 11d.Right side: 146/3 + d.So, 34/3 + 11d = 146/3 + d.Subtract 34/3 from both sides: 11d = (146 - 34)/3 + d = 112/3 + d.Subtract d from both sides: 10d = 112/3.Therefore, d = (112/3)/10 = 112/30 = 56/15 ≈ 3.733.So, d is 56/15.Then, a_1 = 34/3 - d = 34/3 - 56/15. Let's compute that.Convert 34/3 to fifteenths: 34/3 = 170/15.So, 170/15 - 56/15 = 114/15 = 38/5 = 7.6.So, a_1 is 38/5, which is 7.6, and d is 56/15, which is approximately 3.733.Let me compute a few terms to see if the sequence makes sense.a_1 = 38/5 = 7.6a_2 = a_1 + d = 7.6 + 56/15 ≈ 7.6 + 3.733 ≈ 11.333a_3 = a_2 + d ≈ 11.333 + 3.733 ≈ 15.066So, sum of first 3 terms: 7.6 + 11.333 + 15.066 ≈ 34, which matches.Now, a_n is a_13 = a_1 + 12d = 38/5 + 12*(56/15).Compute 12*(56/15) = (12*56)/15 = 672/15 = 44.8.So, a_13 = 38/5 + 44.8 = 7.6 + 44.8 = 52.4.So, a_13 is 52.4.Then, the last three terms would be a_11, a_12, a_13.Compute a_11 = a_1 + 10d = 38/5 + 10*(56/15) = 7.6 + 37.333 ≈ 44.933a_12 = a_1 + 11d ≈ 7.6 + 41.066 ≈ 48.666a_13 = 52.4Sum of last three terms: 44.933 + 48.666 + 52.4 ≈ 146, which is correct.So, all the conditions are satisfied. Therefore, the number of terms is 13.**Final Answer**The sequence has boxed{13} terms.