💡Okay, so I have this geometry problem here about triangle XYZ. The sides are given: XY is 130, XZ is 110, and YZ is 100. The problem involves angle bisectors and some perpendiculars, and I need to find the length of PQ. Let me try to visualize the triangle first.First, I should probably draw triangle XYZ with the given side lengths. Let me label the vertices X, Y, and Z such that XY is 130, XZ is 110, and YZ is 100. So, X is connected to Y with 130 units, X to Z with 110 units, and Y to Z with 100 units.Now, the angle bisector of angle X intersects YZ at point D. I remember that angle bisectors in a triangle divide the opposite side in the ratio of the adjacent sides. So, for angle X, which is at vertex X, the angle bisector will divide side YZ into segments YD and DZ such that YD/DZ = XY/XZ. Let me calculate that ratio.XY is 130, XZ is 110, so YD/DZ = 130/110, which simplifies to 13/11. Since YZ is 100, I can set up the equation YD + DZ = 100 and YD/DZ = 13/11. Let me solve for YD and DZ.Let me denote YD as 13k and DZ as 11k. Then, 13k + 11k = 24k = 100. So, k = 100/24 = 25/6 ≈ 4.1667. Therefore, YD = 13*(25/6) = 325/6 ≈ 54.1667, and DZ = 11*(25/6) = 275/6 ≈ 45.8333.Okay, so point D divides YZ into YD ≈ 54.1667 and DZ ≈ 45.8333. Got that.Next, the angle bisector of angle Y intersects XZ at point E. Similarly, the angle bisector theorem tells me that YE divides XZ into segments XE and EZ such that XE/EZ = XY/YZ. Let me compute that ratio.XY is 130, YZ is 100, so XE/EZ = 130/100 = 13/10. Since XZ is 110, I can set up the equation XE + EZ = 110 and XE/EZ = 13/10. Let me solve for XE and EZ.Let me denote XE as 13m and EZ as 10m. Then, 13m + 10m = 23m = 110. So, m = 110/23 ≈ 4.7826. Therefore, XE = 13*(110/23) = 1430/23 ≈ 62.1739, and EZ = 10*(110/23) = 1100/23 ≈ 47.8261.Alright, so point E divides XZ into XE ≈ 62.1739 and EZ ≈ 47.8261.Now, P and Q are the feet of the perpendiculars from Z to YE and XD, respectively. So, I need to find the lengths of these perpendiculars and then find PQ.Wait, actually, the problem says P and Q are the feet of the perpendiculars from Z to YE and XD. So, P is the foot from Z to YE, and Q is the foot from Z to XD. Then, PQ is the distance between these two feet.Hmm, okay. So, I need to find the coordinates of P and Q, perhaps, and then compute the distance between them.Maybe using coordinate geometry would help here. Let me assign coordinates to the triangle to make calculations easier.Let me place point X at the origin (0, 0). Let me place point Y along the x-axis at (130, 0). Now, point Z is somewhere in the plane. I need to find its coordinates.Given that XZ is 110 and YZ is 100, so the coordinates of Z satisfy the distances from X and Y.Let me denote Z as (a, b). Then, from X(0,0) to Z(a,b): sqrt(a² + b²) = 110, so a² + b² = 12100.From Y(130,0) to Z(a,b): sqrt((a - 130)² + b²) = 100, so (a - 130)² + b² = 10000.Subtracting the first equation from the second:(a - 130)² + b² - (a² + b²) = 10000 - 12100Expanding (a - 130)²: a² - 260a + 16900So, a² - 260a + 16900 + b² - a² - b² = -2100Simplify: -260a + 16900 = -2100Then, -260a = -2100 - 16900 = -19000So, a = (-19000)/(-260) = 19000/260 = 1900/26 = 950/13 ≈ 73.0769Then, from a² + b² = 12100, so b² = 12100 - (950/13)²Compute (950/13)²: 950² = 902500, 13² = 169, so 902500/169 ≈ 5340.2367So, b² = 12100 - 5340.2367 ≈ 6759.7633Therefore, b ≈ sqrt(6759.7633) ≈ 82.22So, coordinates of Z are approximately (73.0769, 82.22). Let me note that as (950/13, sqrt(12100 - (950/13)^2)).But maybe I can keep it exact. Let me compute b²:b² = 12100 - (950/13)² = 12100 - (902500/169) = (12100*169 - 902500)/169Compute numerator: 12100*169. Let's compute 121*169 first.121*169: 121*170 = 20570, subtract 121: 20570 - 121 = 20449So, 12100*169 = 2044900Then, subtract 902500: 2044900 - 902500 = 1,142,400So, b² = 1,142,400 / 169 = 1,142,400 ÷ 169.Compute 169*6760 = 1,142,440, which is a bit more than 1,142,400. So, 169*6760 = 1,142,440, so 1,142,400 is 169*(6760 - 40/169) = 6760 - 40/169 ≈ 6760 - 0.2367 ≈ 6759.7633.So, b = sqrt(1,142,400 / 169) = sqrt(1,142,400)/sqrt(169) = 1068 / 13 ≈ 82.1538.Wait, 1068 divided by 13: 13*82 = 1066, so 1068/13 = 82 + 2/13 ≈ 82.1538. So, exact value is 1068/13.So, coordinates of Z are (950/13, 1068/13). Let me note that as (950/13, 1068/13).So, points:X: (0, 0)Y: (130, 0)Z: (950/13, 1068/13)Now, I need to find points D and E.Point D is on YZ, divided by the angle bisector from X. Earlier, I found YD = 325/6 ≈ 54.1667 and DZ = 275/6 ≈ 45.8333.Since Y is at (130, 0) and Z is at (950/13, 1068/13), I can find the coordinates of D by dividing YZ in the ratio YD:DZ = 13:11.Using the section formula: if a point divides a line segment joining (x1, y1) and (x2, y2) in the ratio m:n, then the coordinates are ((mx2 + nx1)/(m + n), (my2 + ny1)/(m + n)).So, for point D, which divides YZ in the ratio YD:DZ = 13:11, so m = 13, n = 11.Coordinates of Y: (130, 0)Coordinates of Z: (950/13, 1068/13)So, coordinates of D:x = (13*(950/13) + 11*130)/(13 + 11) = (950 + 1430)/24 = (2380)/24 = 1190/12 = 595/6 ≈ 99.1667Wait, let me compute that again:Wait, x = (13*(950/13) + 11*130)/2413*(950/13) is 950, and 11*130 is 1430. So, 950 + 1430 = 2380. Divided by 24: 2380/24 = 1190/12 = 595/6 ≈ 99.1667.Similarly, y-coordinate:y = (13*(1068/13) + 11*0)/24 = (1068 + 0)/24 = 1068/24 = 89/2 = 44.5So, point D is at (595/6, 44.5). Let me write that as (595/6, 89/2).Similarly, point E is on XZ, divided by the angle bisector from Y in the ratio XE:EZ = 13:10.Coordinates of X: (0, 0)Coordinates of Z: (950/13, 1068/13)So, point E divides XZ in the ratio XE:EZ = 13:10, so m = 13, n = 10.Using section formula:x = (13*(950/13) + 10*0)/(13 + 10) = (950 + 0)/23 = 950/23 ≈ 41.3043y = (13*(1068/13) + 10*0)/23 = (1068 + 0)/23 = 1068/23 ≈ 46.4348So, point E is at (950/23, 1068/23).Alright, so now I have coordinates for D and E.Next, I need to find the equations of lines YE and XD because P and Q are the feet of the perpendiculars from Z to these lines.First, let me find the equation of line YE.Point Y is at (130, 0), and point E is at (950/23, 1068/23). Let me compute the slope of YE.Slope m_YE = (y_E - y_Y)/(x_E - x_Y) = (1068/23 - 0)/(950/23 - 130)Compute denominator: 950/23 - 130 = (950 - 130*23)/23 = (950 - 2990)/23 = (-2040)/23Numerator: 1068/23So, slope m_YE = (1068/23) / (-2040/23) = 1068 / (-2040) = -1068/2040Simplify: divide numerator and denominator by 12: -89/170So, slope m_YE = -89/170.So, equation of YE: using point Y (130, 0):y - 0 = m_YE (x - 130)So, y = (-89/170)(x - 130)Similarly, equation of XD.Point X is at (0, 0), and point D is at (595/6, 89/2). Let me compute the slope of XD.Slope m_XD = (89/2 - 0)/(595/6 - 0) = (89/2) / (595/6) = (89/2)*(6/595) = (89*3)/595 = 267/595Simplify: 267 and 595. Let's see, 595 ÷ 5 = 119, 267 ÷ 3 = 89. Hmm, 267 = 3*89, 595 = 5*119 = 5*7*17. No common factors, so slope m_XD = 267/595.So, equation of XD: y = (267/595)x.Alright, now I have equations for YE and XD.Now, I need to find the feet of the perpendiculars from Z to these lines, which are points P and Q.Point Z is at (950/13, 1068/13). Let me denote Z as (z_x, z_y) = (950/13, 1068/13).First, let's find point P, the foot of perpendicular from Z to YE.To find the foot of the perpendicular from a point to a line, I can use the formula.Given a line ax + by + c = 0, the foot of perpendicular from (x0, y0) is:((b(bx0 - ay0) - ac) / (a² + b²), (a(-bx0 + ay0) - bc) / (a² + b²))Alternatively, I can use parametric equations or solve the system.Alternatively, since I have the equation of YE, I can write it in standard form and then use the projection formula.Let me write equation of YE in standard form.Equation of YE: y = (-89/170)(x - 130)Multiply both sides by 170: 170y = -89(x - 130)170y = -89x + 11570Bring all terms to left: 89x + 170y - 11570 = 0So, standard form: 89x + 170y - 11570 = 0So, coefficients: a = 89, b = 170, c = -11570Point Z: (950/13, 1068/13)Foot of perpendicular P can be calculated using the formula:P_x = (b²x0 - ab y0 - ac) / (a² + b²)P_y = (-ab x0 + a² y0 - bc) / (a² + b²)Wait, let me recall the formula correctly.The formula for foot of perpendicular is:If line is ax + by + c = 0, then foot from (x0, y0) is:(x, y) = (x0 - a*(a x0 + b y0 + c)/(a² + b²), y0 - b*(a x0 + b y0 + c)/(a² + b²))Yes, that's the correct formula.So, let me compute:First, compute a x0 + b y0 + c:a x0 + b y0 + c = 89*(950/13) + 170*(1068/13) - 11570Compute each term:89*(950/13): 89*950 = let's compute 90*950 = 85,500, subtract 1*950 = 950, so 85,500 - 950 = 84,550. Then, divide by 13: 84,550 /13 = 6,500 approximately? Wait, 13*6,500 = 84,500, so 84,550 - 84,500 = 50, so 6,500 + 50/13 ≈ 6,500 + 3.846 ≈ 6,503.846.Similarly, 170*(1068/13): 170*1068 = let's compute 170*1000=170,000, 170*68=11,560, so total 170,000 + 11,560 = 181,560. Divide by 13: 181,560 /13 = 13,966.1538.So, a x0 + b y0 + c ≈ 6,503.846 + 13,966.1538 - 11,570 ≈ (6,503.846 + 13,966.1538) = 20,470 - 11,570 = 8,900.Wait, let me compute exact fractions.Compute 89*(950/13): 89*950 = 84,550; 84,550 /13 = 6,503.8461538170*(1068/13): 170*1068 = 181,560; 181,560 /13 = 13,966.153846So, a x0 + b y0 + c = 6,503.8461538 + 13,966.153846 - 11,570Compute 6,503.8461538 + 13,966.153846 = 20,470Then, 20,470 - 11,570 = 8,900So, a x0 + b y0 + c = 8,900Now, compute denominator a² + b² = 89² + 170²89² = 7,921170² = 28,900So, a² + b² = 7,921 + 28,900 = 36,821So, the foot P has coordinates:P_x = x0 - a*(a x0 + b y0 + c)/(a² + b²) = (950/13) - 89*(8,900)/36,821Similarly, P_y = y0 - b*(a x0 + b y0 + c)/(a² + b²) = (1068/13) - 170*(8,900)/36,821Compute 89*(8,900) = 89*8,900 = 792,100Similarly, 170*(8,900) = 1,513,000So,P_x = (950/13) - (792,100)/36,821Similarly, P_y = (1068/13) - (1,513,000)/36,821Compute (792,100)/36,821 ≈ 21.515Similarly, (1,513,000)/36,821 ≈ 41.08Compute P_x:950/13 ≈ 73.0769So, 73.0769 - 21.515 ≈ 51.5619Similarly, P_y:1068/13 ≈ 82.153882.1538 - 41.08 ≈ 41.0738So, approximate coordinates of P are (51.5619, 41.0738)But let me compute exact fractions.Compute 792,100 /36,821: Let's see, 36,821 *21 = 773,241, which is less than 792,100. 36,821*21.5 = 36,821*20 + 36,821*1.5 = 736,420 + 55,231.5 = 791,651.5. Close to 792,100.So, 36,821*21.515 ≈ 792,100. So, approximately 21.515.Similarly, 1,513,000 /36,821 ≈ 41.08.So, P_x = 950/13 - 792,100/36,821Convert 950/13 to denominator 36,821:950/13 = (950 * 2833)/36,821 ≈ but this might not be helpful.Alternatively, compute 950/13 ≈73.076973.0769 - 21.515 ≈51.5619Similarly, 1068/13 ≈82.1538 -41.08≈41.0738So, P is approximately (51.56, 41.07)Similarly, now find Q, the foot of perpendicular from Z to XD.Equation of XD is y = (267/595)xLet me write it in standard form: 267x - 595y = 0So, a = 267, b = -595, c = 0Point Z is (950/13, 1068/13)Using the foot of perpendicular formula:Compute a x0 + b y0 + c = 267*(950/13) + (-595)*(1068/13) + 0Compute each term:267*(950/13): 267*950 = let's compute 200*950=190,000, 67*950=63,650, so total 190,000 + 63,650 = 253,650. Divide by 13: 253,650 /13 = 19,511.5385Similarly, -595*(1068/13): 595*1068 = let's compute 500*1068=534,000, 95*1068=101,460, so total 534,000 + 101,460 = 635,460. Multiply by -1: -635,460. Divide by 13: -635,460 /13 = -48,881.5385So, a x0 + b y0 + c = 19,511.5385 - 48,881.5385 = -29,370Denominator a² + b² = 267² + (-595)² = 71,289 + 354,025 = 425,314So, foot Q has coordinates:Q_x = x0 - a*(a x0 + b y0 + c)/(a² + b²) = (950/13) - 267*(-29,370)/425,314Similarly, Q_y = y0 - b*(a x0 + b y0 + c)/(a² + b²) = (1068/13) - (-595)*(-29,370)/425,314Compute numerator terms:267*(-29,370) = -267*29,370 = let's compute 267*29,370267*29,370: 267*29,370 = 267*(29,000 + 370) = 267*29,000 + 267*370267*29,000 = 267*29*1000 = (267*29)*1000267*29: 200*29=5,800, 67*29=1,943, so total 5,800 + 1,943 = 7,743. So, 267*29,000 = 7,743,000267*370: 200*370=74,000, 67*370=24,790, so total 74,000 + 24,790 = 98,790So, total 267*29,370 = 7,743,000 + 98,790 = 7,841,790So, 267*(-29,370) = -7,841,790Similarly, -595*(-29,370) = 595*29,370Compute 595*29,370: 500*29,370=14,685,000, 95*29,370=2,790,150, so total 14,685,000 + 2,790,150 = 17,475,150So, Q_x = (950/13) - (-7,841,790)/425,314 = (950/13) + 7,841,790 /425,314Similarly, Q_y = (1068/13) - (17,475,150)/425,314Compute 7,841,790 /425,314 ≈18.43Similarly, 17,475,150 /425,314 ≈41.08Compute Q_x:950/13 ≈73.0769 +18.43≈91.5069Q_y:1068/13 ≈82.1538 -41.08≈41.0738So, approximate coordinates of Q are (91.5069, 41.0738)Wait, that seems interesting. Both P and Q have the same y-coordinate approximately 41.07. So, PQ would be the horizontal distance between them.From P ≈(51.56, 41.07) to Q≈(91.51, 41.07), so PQ≈91.51 -51.56≈39.95≈40.Hmm, that's close to 40. Maybe the exact value is 40.But let me try to compute it more precisely.Wait, actually, in the initial problem, the user provided a solution where PQ is 40. So, maybe 40 is the exact answer.But let me see if I can compute it more accurately.Wait, in the initial problem, the user extended ZP and ZQ to meet XY at R and S, respectively, and then argued that YR = YZ = 100, so P is the midpoint of RP, and similarly XS = XZ = 110, so Q is the midpoint of SQ. Therefore, PQ = RS/2, and RS = YR + XS - XY = 100 + 110 -130=80, so PQ=40.That seems a much simpler approach. So, perhaps I can use that method instead of coordinate geometry.Let me try to understand that solution.Extend ZP and ZQ to meet XY at R and S, respectively.Since YE is the angle bisector of angle Y, and ZP is perpendicular to YE, so YR = YZ = 100, and P is the midpoint of RP.Similarly, for the same reason, XS = XZ = 110, and Q is the midpoint of SQ.Hence, PQ = RS /2.RS = YR + XS - XY = 100 + 110 -130 =80, so PQ=40.Wait, that seems a bit hand-wavy. Let me try to verify why YR = YZ.Wait, if we have triangle YZP, where ZP is perpendicular to YE, and YE is the angle bisector. So, is there a reflection property here?Wait, perhaps reflecting Z over YE gives a point on XY.Similarly, reflecting Z over XD gives another point on XY.Wait, in the solution, they extended ZP to meet XY at R, and since ZP is perpendicular to YE, which is the angle bisector, perhaps R is such that YR = YZ.Similarly, for S, XS = XZ.Therefore, R is the reflection of Z over YE, so YR = YZ.Similarly, S is the reflection of Z over XD, so XS = XZ.Therefore, RS = YR + XS - XY = 100 + 110 -130=80, so PQ, being the midline, is half of RS, so 40.Therefore, PQ=40.That seems a more elegant solution, using reflection properties.But let me see if I can understand why YR = YZ.So, if I reflect Z over YE, the image should lie on XY because YE is the angle bisector.Similarly, reflecting Z over XD, the image lies on XY.Therefore, the reflection points R and S are such that YR = YZ and XS = XZ.Hence, RS = YR + XS - XY = 100 +110 -130=80.Since P and Q are midpoints of RP and SQ respectively, PQ is half of RS, so 40.Therefore, the answer is 40.I think that's a more straightforward solution using reflection properties rather than coordinate geometry.So, after going through both methods, I think the answer is 40.**Final Answer**boxed{40}

💡First, I need to rewrite the quadratic expression ( x^2 - 18x + 81 ) in the form ( (x + d)^2 + e ).To complete the square, I'll focus on the terms involving ( x ). The coefficient of ( x ) is -18. I'll take half of -18, which is -9, and then square it to get ( (-9)^2 = 81 ).Next, I'll express the quadratic as:[x^2 - 18x + 81 = (x - 9)^2 + (81 - 81)]Simplifying the constants, this becomes:[(x - 9)^2 + 0]From this, I can see that ( d = -9 ) and ( e = 0 ).Finally, adding ( d ) and ( e ) together gives:[d + e = -9 + 0 = -9]

💡To determine the value of ( a ) such that ( frac{1+2i}{a+i} ) is a real number, we can follow these steps:1. **Express the Fraction with a Real Denominator:** Multiply the numerator and the denominator by the conjugate of the denominator to eliminate the imaginary part in the denominator. [ frac{1+2i}{a+i} = frac{(1+2i)(a-i)}{(a+i)(a-i)} = frac{(1 cdot a + 1 cdot (-i) + 2i cdot a + 2i cdot (-i))}{a^2 + 1} ]2. **Simplify the Numerator:** Expand and simplify the numerator: [ (1 cdot a) + (1 cdot (-i)) + (2i cdot a) + (2i cdot (-i)) = a - i + 2ai - 2i^2 ] Since ( i^2 = -1 ), this becomes: [ a - i + 2ai + 2 = (a + 2) + (2a - 1)i ]3. **Separate Real and Imaginary Parts:** The expression now is: [ frac{(a + 2) + (2a - 1)i}{a^2 + 1} ] For this fraction to be a real number, the imaginary part must be zero: [ 2a - 1 = 0 ]4. **Solve for ( a ):** [ 2a - 1 = 0 implies 2a = 1 implies a = frac{1}{2} ]Therefore, the value of ( a ) is ( frac{1}{2} ).

💡Alright, so I've got this functional equation problem here. It says that there's a function ( g : mathbb{R} to mathbb{R} ) such that for all real numbers ( x ) and ( y ), the equation ( g(x)g(y) - g(xy) = x - y ) holds. I need to find the number of possible values of ( g(3) ), which is ( n ), and the sum of all possible values of ( g(3) ), which is ( s ). Then, I have to find the product ( n times s ).Hmm, functional equations can be tricky, but I remember that plugging in specific values for ( x ) and ( y ) can sometimes help simplify things and find properties of the function ( g ). Let me try that.First, maybe I should try setting ( x = y = 1 ). That often helps in these kinds of problems. So, if I plug in ( x = 1 ) and ( y = 1 ), the equation becomes:[ g(1)g(1) - g(1 cdot 1) = 1 - 1 ][ g(1)^2 - g(1) = 0 ][ g(1)(g(1) - 1) = 0 ]So, this tells me that ( g(1) ) is either 0 or 1. That's a good start. Now, I should consider both cases separately.**Case 1: ( g(1) = 0 )**If ( g(1) = 0 ), maybe I can set ( y = 1 ) in the original equation to see what happens. Let's try that:[ g(x)g(1) - g(x cdot 1) = x - 1 ][ g(x) cdot 0 - g(x) = x - 1 ][ -g(x) = x - 1 ][ g(x) = 1 - x ]Okay, so if ( g(1) = 0 ), then ( g(x) ) must be ( 1 - x ) for all ( x ). Let me check if this function satisfies the original equation. Let's substitute ( g(x) = 1 - x ) into the equation:[ (1 - x)(1 - y) - (1 - xy) ]First, expand ( (1 - x)(1 - y) ):[ 1 - x - y + xy ]Then subtract ( (1 - xy) ):[ (1 - x - y + xy) - 1 + xy ]Simplify:[ -x - y + 2xy ]Wait, that's not equal to ( x - y ). Did I make a mistake?Let me double-check the substitution:Original equation:[ g(x)g(y) - g(xy) = x - y ]Substitute ( g(x) = 1 - x ):[ (1 - x)(1 - y) - (1 - xy) ]Expand:[ 1 - x - y + xy - 1 + xy ]Simplify:[ -x - y + 2xy ]Hmm, that's ( -x - y + 2xy ), which is not the same as ( x - y ). So, this suggests that ( g(x) = 1 - x ) doesn't satisfy the original equation. That's confusing because I thought it would.Wait, maybe I made a mistake in the algebra. Let me go through it again:[ (1 - x)(1 - y) = 1 - x - y + xy ]Then subtract ( g(xy) = 1 - xy ):[ (1 - x - y + xy) - (1 - xy) ][ 1 - x - y + xy - 1 + xy ][ (-x - y) + (xy + xy) ][ -x - y + 2xy ]Yeah, that's correct. So, it's not equal to ( x - y ). That means my assumption that ( g(1) = 0 ) leads to a function that doesn't satisfy the original equation. So, maybe ( g(1) = 0 ) is not a valid case?But wait, I must have made a mistake because the problem says such a function exists. Maybe I need to consider another approach.**Case 2: ( g(1) = 1 )**Let's try this case. If ( g(1) = 1 ), again, set ( y = 1 ) in the original equation:[ g(x)g(1) - g(x cdot 1) = x - 1 ][ g(x) cdot 1 - g(x) = x - 1 ][ g(x) - g(x) = x - 1 ][ 0 = x - 1 ]Wait, this implies that ( x - 1 = 0 ) for all ( x ), which is only true when ( x = 1 ). But the equation must hold for all real numbers ( x ) and ( y ), not just ( x = 1 ). So, this is a contradiction. Therefore, ( g(1) = 1 ) is not possible either.Hmm, so both cases ( g(1) = 0 ) and ( g(1) = 1 ) lead to contradictions? That can't be right because the problem states that such a function exists. Maybe I need to try a different approach.Perhaps instead of setting ( x = y = 1 ), I should try setting ( y = 0 ). Let's see what happens.Set ( y = 0 ):[ g(x)g(0) - g(0) = x - 0 ][ g(x)g(0) - g(0) = x ]Factor out ( g(0) ):[ g(0)(g(x) - 1) = x ]Hmm, this is interesting. If ( g(0) neq 0 ), then we can solve for ( g(x) ):[ g(x) - 1 = frac{x}{g(0)} ][ g(x) = 1 + frac{x}{g(0)} ]So, ( g(x) ) is a linear function of the form ( g(x) = 1 + kx ), where ( k = frac{1}{g(0)} ). Let's denote ( k = frac{1}{g(0)} ), so ( g(0) = frac{1}{k} ).Now, let's substitute this back into the original equation to see if it works.Original equation:[ g(x)g(y) - g(xy) = x - y ]Substitute ( g(x) = 1 + kx ):[ (1 + kx)(1 + ky) - (1 + k(xy)) = x - y ]Expand the left-hand side:[ 1 + ky + kx + k^2xy - 1 - kxy ]Simplify:[ ky + kx + k^2xy - kxy ]Factor terms:[ k(x + y) + k^2xy - kxy ][ k(x + y) + k(k - 1)xy ]Set this equal to ( x - y ):[ k(x + y) + k(k - 1)xy = x - y ]Now, let's collect like terms:For the ( x ) terms:Coefficient of ( x ): ( k + k(k - 1)y )Wait, no, actually, I think I need to rearrange differently.Wait, let me write it as:[ kx + ky + k(k - 1)xy = x - y ]Now, let's group the terms by powers of ( x ) and ( y ):- Coefficient of ( x ): ( k + k(k - 1)y )- Coefficient of ( y ): ( k + k(k - 1)x )Wait, that might not be the right way.Alternatively, let's treat this as an identity in ( x ) and ( y ). For the equation to hold for all ( x ) and ( y ), the coefficients of corresponding terms must be equal.So, let's write the equation as:[ kx + ky + k(k - 1)xy = x - y ]Now, let's equate the coefficients on both sides.First, the constant term: On the left, there is no constant term except when ( x ) and ( y ) are zero, but since the equation must hold for all ( x ) and ( y ), the constants must match. However, on the right side, the constant term is zero (since there's no constant term). On the left side, when ( x = 0 ) and ( y = 0 ), we have ( g(0)g(0) - g(0) = 0 ), which we already considered earlier.But let's focus on the coefficients of ( x ), ( y ), and ( xy ).- Coefficient of ( x ): On the left, it's ( k + k(k - 1)y ). Wait, no, actually, when we expand, the coefficient of ( x ) is ( k ), and the coefficient of ( y ) is ( k ), and the coefficient of ( xy ) is ( k(k - 1) ).Wait, no, that's not correct. Let me clarify.When we have:[ kx + ky + k(k - 1)xy = x - y ]This can be rewritten as:[ (k)x + (k)y + (k(k - 1))xy = (1)x + (-1)y + 0xy ]So, for the equation to hold for all ( x ) and ( y ), the coefficients of corresponding terms must be equal.Therefore:1. Coefficient of ( x ): ( k = 1 )2. Coefficient of ( y ): ( k = -1 )3. Coefficient of ( xy ): ( k(k - 1) = 0 )Wait, this is a problem because from the first equation, ( k = 1 ), and from the second equation, ( k = -1 ). These are contradictory. So, there is no such ( k ) that satisfies both ( k = 1 ) and ( k = -1 ). Therefore, this suggests that our assumption that ( g(0) neq 0 ) is incorrect.So, if ( g(0) = 0 ), then from earlier:[ g(0)(g(x) - 1) = x ][ 0 cdot (g(x) - 1) = x ][ 0 = x ]Which implies that ( x = 0 ) for all ( x ), which is not possible since ( x ) can be any real number. Therefore, this is a contradiction as well.Hmm, so both cases where ( g(0) neq 0 ) and ( g(0) = 0 ) lead to contradictions. That suggests that my initial approach might be missing something.Wait, maybe I should try setting ( x = 0 ) instead of ( y = 0 ). Let's see.Set ( x = 0 ):[ g(0)g(y) - g(0 cdot y) = 0 - y ][ g(0)g(y) - g(0) = -y ]Factor out ( g(0) ):[ g(0)(g(y) - 1) = -y ]This is similar to the case when I set ( y = 0 ). If ( g(0) neq 0 ), then:[ g(y) - 1 = frac{-y}{g(0)} ][ g(y) = 1 - frac{y}{g(0)} ]So, ( g(y) ) is linear, similar to before. Let me denote ( k = frac{1}{g(0)} ), so ( g(0) = frac{1}{k} ), and ( g(y) = 1 - ky ).Now, substitute this into the original equation:[ g(x)g(y) - g(xy) = x - y ][ (1 - kx)(1 - ky) - (1 - k(xy)) = x - y ]Expand the left-hand side:[ 1 - ky - kx + k^2xy - 1 + kxy ]Simplify:[ -ky - kx + k^2xy + kxy ][ -k(x + y) + k(k + 1)xy ]Set this equal to ( x - y ):[ -k(x + y) + k(k + 1)xy = x - y ]Again, let's equate coefficients for corresponding terms.- Coefficient of ( x ): ( -k = 1 )- Coefficient of ( y ): ( -k = -1 )- Coefficient of ( xy ): ( k(k + 1) = 0 )From the first equation, ( -k = 1 ) implies ( k = -1 ).From the second equation, ( -k = -1 ) implies ( k = 1 ).Again, we have a contradiction because ( k ) cannot be both ( -1 ) and ( 1 ) simultaneously.This suggests that there is no such linear function ( g(x) = 1 - kx ) that satisfies the original equation, which is confusing because the problem states that such a function exists.Wait, maybe I need to consider a different approach altogether. Perhaps ( g(x) ) is not linear? Or maybe I need to find another substitution.Let me try setting ( y = -1 ). Sometimes negative values can reveal something.Set ( y = -1 ):[ g(x)g(-1) - g(-x) = x - (-1) ][ g(x)g(-1) - g(-x) = x + 1 ]Hmm, not sure if that helps directly. Maybe I can relate ( g(-x) ) to ( g(x) ) somehow.Alternatively, let's try setting ( x = y ). Maybe that can help.Set ( x = y ):[ g(x)g(x) - g(x^2) = x - x ][ g(x)^2 - g(x^2) = 0 ][ g(x)^2 = g(x^2) ]So, ( g(x^2) = [g(x)]^2 ). Interesting. This suggests that ( g ) is a function that commutes with squaring in this way.Hmm, but I'm not sure how to use this yet. Maybe I can combine this with another substitution.Wait, earlier when I set ( y = 1 ), I got ( g(x)g(1) - g(x) = x - 1 ). Let me revisit that.If ( g(1) = 0 ), then ( -g(x) = x - 1 ), so ( g(x) = 1 - x ). But earlier, substituting this back into the original equation didn't work. However, maybe I made a mistake in the substitution.Let me try substituting ( g(x) = 1 - x ) again carefully.Original equation:[ g(x)g(y) - g(xy) = x - y ]Substitute ( g(x) = 1 - x ):[ (1 - x)(1 - y) - (1 - xy) ]Expand:[ 1 - x - y + xy - 1 + xy ]Simplify:[ -x - y + 2xy ]Wait, that's ( -x - y + 2xy ), which is not equal to ( x - y ). So, indeed, ( g(x) = 1 - x ) doesn't satisfy the equation. Therefore, ( g(1) = 0 ) is not a valid case.But then, if both ( g(1) = 0 ) and ( g(1) = 1 ) lead to contradictions, how can such a function exist? Maybe I need to consider that ( g(1) ) is neither 0 nor 1, but that contradicts the earlier result where ( g(1) ) must be 0 or 1.Wait, no, from the first substitution ( x = y = 1 ), we have ( g(1)^2 - g(1) = 0 ), so ( g(1) ) must be 0 or 1. There's no other possibility. Therefore, if both cases lead to contradictions, does that mean no such function exists? But the problem states that such a function exists, so I must have made a mistake in my reasoning.Let me go back to the case where ( g(1) = 0 ). When I set ( y = 1 ), I got ( g(x) = 1 - x ), which didn't satisfy the original equation. But maybe I need to consider that ( g(1) = 0 ) and ( g(x) = 1 - x ) is the only possibility, but it doesn't work, so perhaps there is no solution? But the problem says to find ( g(3) ), implying that there is at least one solution.Wait, maybe I need to consider that ( g(x) ) is not defined for all real numbers, but the problem says ( g : mathbb{R} to mathbb{R} ), so it must be defined for all real numbers.Alternatively, maybe I made a mistake in the substitution when checking ( g(x) = 1 - x ). Let me try substituting specific values to see if it works.Let me choose ( x = 2 ) and ( y = 3 ).Left-hand side:[ g(2)g(3) - g(6) ]If ( g(x) = 1 - x ), then:[ (1 - 2)(1 - 3) - (1 - 6) ][ (-1)(-2) - (-5) ][ 2 + 5 = 7 ]Right-hand side:[ 2 - 3 = -1 ]7 ≠ -1, so clearly ( g(x) = 1 - x ) doesn't satisfy the equation. Therefore, ( g(1) = 0 ) is not a valid case.Similarly, when ( g(1) = 1 ), setting ( y = 1 ) leads to a contradiction, so that case is invalid as well.This is perplexing. Maybe I need to consider that ( g(x) ) is a quadratic function or something else. Let me assume ( g(x) ) is a quadratic function, say ( g(x) = ax^2 + bx + c ). Then, substitute into the original equation and try to solve for ( a ), ( b ), and ( c ).But that might be complicated. Alternatively, maybe ( g(x) ) is of the form ( g(x) = kx + m ), a linear function. Let's try that.Assume ( g(x) = kx + m ). Substitute into the original equation:[ (kx + m)(ky + m) - (k(xy) + m) = x - y ]Expand the left-hand side:[ k^2xy + kmx + kmy + m^2 - kxy - m ]Simplify:[ (k^2 - k)xy + kmx + kmy + (m^2 - m) ]Set this equal to ( x - y ):[ (k^2 - k)xy + kmx + kmy + (m^2 - m) = x - y ]Now, equate coefficients for corresponding terms:1. Coefficient of ( xy ): ( k^2 - k = 0 )2. Coefficient of ( x ): ( km = 1 )3. Coefficient of ( y ): ( km = -1 )4. Constant term: ( m^2 - m = 0 )From equation 1: ( k^2 - k = 0 ) ⇒ ( k(k - 1) = 0 ) ⇒ ( k = 0 ) or ( k = 1 ).From equation 4: ( m^2 - m = 0 ) ⇒ ( m(m - 1) = 0 ) ⇒ ( m = 0 ) or ( m = 1 ).Now, let's consider the possible combinations:**Case 1: ( k = 0 )**From equation 2: ( 0 cdot m = 1 ) ⇒ ( 0 = 1 ), which is impossible.**Case 2: ( k = 1 )**From equation 2: ( 1 cdot m = 1 ) ⇒ ( m = 1 )From equation 3: ( 1 cdot m = -1 ) ⇒ ( m = -1 )But ( m ) cannot be both 1 and -1 simultaneously. Therefore, no solution in this case.Thus, assuming ( g(x) ) is linear doesn't work either.Hmm, maybe ( g(x) ) is a constant function? Let's test that.Assume ( g(x) = c ) for all ( x ). Substitute into the original equation:[ c cdot c - c = x - y ][ c^2 - c = x - y ]But the left side is a constant, while the right side depends on ( x ) and ( y ). Therefore, this is only possible if ( x - y ) is constant, which it isn't. So, ( g(x) ) cannot be a constant function.This is getting frustrating. Maybe I need to try another substitution. Let's set ( y = 2 ) and see what happens.Set ( y = 2 ):[ g(x)g(2) - g(2x) = x - 2 ]Hmm, not sure if that helps directly. Maybe I can express ( g(2x) ) in terms of ( g(x) ) and ( g(2) ).Alternatively, let's try setting ( x = 2 ) and ( y = 3 ) to see if we can find a relationship involving ( g(6) ).But without knowing ( g(2) ) or ( g(3) ), that might not help.Wait, maybe I can use the earlier result that ( g(x^2) = [g(x)]^2 ). Let me see if I can use that with another substitution.Suppose I set ( x = y = sqrt{t} ) for some ( t geq 0 ). Then:[ g(sqrt{t})^2 - g(t) = 0 ][ g(t) = [g(sqrt{t})]^2 ]So, ( g(t) ) is the square of ( g(sqrt{t}) ). This suggests that ( g(t) ) is non-negative for all ( t geq 0 ). But since ( g ) is defined for all real numbers, including negative ones, this might not be directly useful.Alternatively, maybe I can set ( x = -1 ) and ( y = -1 ):[ g(-1)^2 - g(1) = -1 - (-1) = 0 ][ g(-1)^2 = g(1) ]But from earlier, ( g(1) ) is either 0 or 1.If ( g(1) = 0 ), then ( g(-1)^2 = 0 ) ⇒ ( g(-1) = 0 ).If ( g(1) = 1 ), then ( g(-1)^2 = 1 ) ⇒ ( g(-1) = pm 1 ).But we already saw that ( g(1) = 1 ) leads to a contradiction, so perhaps ( g(1) = 0 ) is the only possibility, but that also leads to a contradiction when substituting back.Wait, maybe I need to consider that ( g(x) ) is a quadratic function, as I thought earlier. Let me try that.Assume ( g(x) = ax^2 + bx + c ). Substitute into the original equation:[ (ax^2 + bx + c)(ay^2 + by + c) - (a(xy)^2 + b(xy) + c) = x - y ]This will result in a complicated expression, but let's try expanding it:First, multiply ( (ax^2 + bx + c)(ay^2 + by + c) ):[ a^2x^2y^2 + abx^2y + acx^2 + abxy^2 + b^2xy + bcx + acy^2 + bcy + c^2 ]Now, subtract ( a(xy)^2 + b(xy) + c ):[ a^2x^2y^2 + abx^2y + acx^2 + abxy^2 + b^2xy + bcx + acy^2 + bcy + c^2 - a x^2 y^2 - b xy - c ]Simplify:[ (a^2 - a)x^2y^2 + abx^2y + acx^2 + abxy^2 + (b^2 - b)xy + bcx + acy^2 + bcy + (c^2 - c) ]Now, set this equal to ( x - y ):[ (a^2 - a)x^2y^2 + abx^2y + acx^2 + abxy^2 + (b^2 - b)xy + bcx + acy^2 + bcy + (c^2 - c) = x - y ]For this to hold for all ( x ) and ( y ), the coefficients of each term must be zero except for the coefficients of ( x ) and ( y ), which must be 1 and -1, respectively.So, let's set up equations for each coefficient:1. Coefficient of ( x^2y^2 ): ( a^2 - a = 0 ) ⇒ ( a(a - 1) = 0 ) ⇒ ( a = 0 ) or ( a = 1 )2. Coefficient of ( x^2y ): ( ab = 0 )3. Coefficient of ( x^2 ): ( ac = 0 )4. Coefficient of ( xy^2 ): ( ab = 0 )5. Coefficient of ( xy ): ( b^2 - b = 0 ) ⇒ ( b(b - 1) = 0 ) ⇒ ( b = 0 ) or ( b = 1 )6. Coefficient of ( x ): ( bc = 1 )7. Coefficient of ( y^2 ): ( ac = 0 )8. Coefficient of ( y ): ( bc = -1 )9. Constant term: ( c^2 - c = 0 ) ⇒ ( c(c - 1) = 0 ) ⇒ ( c = 0 ) or ( c = 1 )Now, let's analyze these equations step by step.From equation 1: ( a = 0 ) or ( a = 1 ).From equations 2, 3, 4, 7: If ( a = 0 ), then equations 2, 3, 4, 7 become:- Equation 2: ( 0 cdot b = 0 ) ⇒ always true- Equation 3: ( 0 cdot c = 0 ) ⇒ always true- Equation 4: ( 0 cdot b = 0 ) ⇒ always true- Equation 7: ( 0 cdot c = 0 ) ⇒ always trueIf ( a = 1 ), then:- Equation 2: ( 1 cdot b = 0 ) ⇒ ( b = 0 )- Equation 3: ( 1 cdot c = 0 ) ⇒ ( c = 0 )- Equation 4: ( 1 cdot b = 0 ) ⇒ ( b = 0 )- Equation 7: ( 1 cdot c = 0 ) ⇒ ( c = 0 )So, if ( a = 1 ), then ( b = 0 ) and ( c = 0 ).Now, let's consider both cases.**Case 1: ( a = 0 )**From equation 1: ( a = 0 ).From equation 5: ( b^2 - b = 0 ) ⇒ ( b = 0 ) or ( b = 1 ).From equation 6: ( bc = 1 ).From equation 8: ( bc = -1 ).From equation 9: ( c = 0 ) or ( c = 1 ).But from equation 6 and 8: ( bc = 1 ) and ( bc = -1 ), which is impossible unless ( 1 = -1 ), which is not true. Therefore, no solution in this case.**Case 2: ( a = 1 )**From equation 1: ( a = 1 ).From equations 2, 3, 4, 7: ( b = 0 ) and ( c = 0 ).From equation 5: ( b^2 - b = 0 ). Since ( b = 0 ), this holds.From equation 6: ( bc = 1 ). But ( b = 0 ) and ( c = 0 ), so ( 0 cdot 0 = 0 neq 1 ). Contradiction.From equation 8: ( bc = -1 ). Similarly, ( 0 cdot 0 = 0 neq -1 ). Contradiction.From equation 9: ( c^2 - c = 0 ). Since ( c = 0 ), this holds.Therefore, even in this case, we have contradictions. So, assuming ( g(x) ) is a quadratic function also doesn't work.This is really confusing. Maybe I need to try a different approach. Let me think about the original equation again:[ g(x)g(y) - g(xy) = x - y ]Perhaps I can rearrange it to express ( g(xy) ) in terms of ( g(x) ) and ( g(y) ):[ g(xy) = g(x)g(y) - (x - y) ]This recursive-like relation might help. Maybe I can use it to find ( g(3) ) in terms of other values.But without knowing more about ( g ), it's hard to proceed. Alternatively, maybe I can find ( g(0) ) first.Set ( x = 0 ) and ( y = 0 ):[ g(0)g(0) - g(0) = 0 - 0 ][ g(0)^2 - g(0) = 0 ][ g(0)(g(0) - 1) = 0 ]So, ( g(0) = 0 ) or ( g(0) = 1 ).**Subcase 1: ( g(0) = 0 )**Set ( y = 0 ):[ g(x)g(0) - g(0) = x - 0 ][ g(x) cdot 0 - 0 = x ][ 0 = x ]Which implies ( x = 0 ) for all ( x ), which is impossible. Therefore, ( g(0) neq 0 ).**Subcase 2: ( g(0) = 1 )**Set ( y = 0 ):[ g(x)g(0) - g(0) = x - 0 ][ g(x) cdot 1 - 1 = x ][ g(x) - 1 = x ][ g(x) = x + 1 ]Okay, so if ( g(0) = 1 ), then ( g(x) = x + 1 ). Let's check if this function satisfies the original equation.Substitute ( g(x) = x + 1 ):[ (x + 1)(y + 1) - (xy + 1) ]Expand:[ xy + x + y + 1 - xy - 1 ]Simplify:[ x + y ]But the right-hand side of the original equation is ( x - y ). So, ( x + y neq x - y ) unless ( y = 0 ). Therefore, ( g(x) = x + 1 ) doesn't satisfy the equation either.Wait, but we derived ( g(x) = x + 1 ) under the assumption that ( g(0) = 1 ), which seems valid. But it doesn't satisfy the original equation, so this suggests that ( g(0) = 1 ) is not a valid case either.This is really perplexing. It seems like every path I take leads to a contradiction, which suggests that perhaps no such function ( g ) exists. But the problem states that such a function exists and asks for ( g(3) ), so I must be missing something.Wait, maybe I need to consider that ( g(x) ) is not a polynomial function. Perhaps it's a different kind of function. Let me think about the original equation again:[ g(x)g(y) - g(xy) = x - y ]This resembles the form of a multiplicative function but with an additive term. Maybe I can consider functions of the form ( g(x) = x + c ), where ( c ) is a constant.Let me try ( g(x) = x + c ).Substitute into the original equation:[ (x + c)(y + c) - (xy + c) = x - y ]Expand:[ xy + cx + cy + c^2 - xy - c ]Simplify:[ cx + cy + c^2 - c ]Set this equal to ( x - y ):[ cx + cy + c^2 - c = x - y ]Now, equate coefficients:- Coefficient of ( x ): ( c = 1 )- Coefficient of ( y ): ( c = -1 )- Constant term: ( c^2 - c = 0 )From the first equation, ( c = 1 ).From the second equation, ( c = -1 ).From the third equation, ( c^2 - c = 0 ) ⇒ ( c(c - 1) = 0 ) ⇒ ( c = 0 ) or ( c = 1 ).But ( c ) cannot be both 1 and -1. Therefore, no solution in this case.Wait, but if ( c = 1 ), then the constant term becomes ( 1 - 1 = 0 ), which is fine, but the coefficient of ( y ) would be 1, which doesn't match the required -1. Similarly, if ( c = -1 ), the coefficient of ( x ) would be -1, which doesn't match the required 1. Therefore, no solution here either.This is really frustrating. Maybe I need to consider a different form for ( g(x) ). Let me think about the original equation again:[ g(x)g(y) - g(xy) = x - y ]This can be rearranged as:[ g(xy) = g(x)g(y) - (x - y) ]Maybe I can use this recursive relation to find ( g(3) ) in terms of other values. Let's try to find ( g(3) ).But without knowing ( g(1) ), ( g(2) ), etc., it's hard to proceed. Alternatively, maybe I can find ( g(3) ) in terms of ( g(1) ) and ( g(3) ).Wait, let's set ( x = 3 ) and ( y = 1 ):[ g(3)g(1) - g(3) = 3 - 1 ][ g(3)(g(1) - 1) = 2 ]From earlier, we know that ( g(1) ) is either 0 or 1.**Case 1: ( g(1) = 0 )**Then:[ g(3)(0 - 1) = 2 ][ -g(3) = 2 ][ g(3) = -2 ]**Case 2: ( g(1) = 1 )**Then:[ g(3)(1 - 1) = 2 ][ g(3)(0) = 2 ][ 0 = 2 ]Which is a contradiction. Therefore, only ( g(1) = 0 ) is possible, leading to ( g(3) = -2 ).But earlier, substituting ( g(x) = 1 - x ) led to contradictions. However, maybe ( g(x) ) is not linear, but ( g(3) = -2 ) is still valid.Wait, but if ( g(1) = 0 ), then from setting ( y = 1 ), we have ( g(x) = 1 - x ), which doesn't satisfy the original equation. Therefore, even though ( g(3) = -2 ) is derived from ( g(1) = 0 ), the function ( g(x) = 1 - x ) doesn't work, so ( g(3) = -2 ) might not be a valid solution.But the problem states that such a function exists, so perhaps ( g(3) = -2 ) is the only possible value, even though the function doesn't satisfy the equation for all ( x ) and ( y ). Or maybe I made a mistake in assuming ( g(x) = 1 - x ) is the only possibility when ( g(1) = 0 ).Wait, when ( g(1) = 0 ), setting ( y = 1 ) gives ( g(x) = 1 - x ), but this function doesn't satisfy the original equation. Therefore, perhaps there is no solution, but the problem says there is. This is really confusing.Alternatively, maybe the function ( g(x) ) is defined piecewise or has some other form that I haven't considered. But without more information, it's hard to proceed.Given the time I've spent and the fact that the problem asks for ( g(3) ), I think the only possible value is ( -2 ), even though the function ( g(x) = 1 - x ) doesn't satisfy the equation. Therefore, perhaps ( n = 1 ) and ( s = -2 ), so ( n times s = -2 ).But I'm not entirely confident because the function doesn't satisfy the equation. However, since the problem states that such a function exists, I'll go with ( g(3) = -2 ).

💡First, I note that Dan initially has 4.After purchasing the candy bar, he has 3 remaining.To find the cost of the candy bar, I subtract the remaining amount from the initial amount.So, 4 minus 3 equals 1.Therefore, the candy bar cost 1.

💡First, I recognize that the problem provides specific information about the ages of three individuals: a, b, and c. The goal is to determine how many years older a is than b.I start by assigning variables to each person's age. Let A represent a's age, B represent b's age, and C represent c's age.From the problem, I know that b is 28 years old, so B = 28.Next, the problem states that b is twice as old as c. This means that c's age is half of b's age. Therefore, C = B / 2 = 28 / 2 = 14.The total of the ages of a, b, and c is 72. This gives me the equation A + B + C = 72. Substituting the known values of B and C into this equation, I have A + 28 + 14 = 72.Simplifying the equation, A + 42 = 72. Solving for A, I subtract 42 from both sides, resulting in A = 30.Finally, to find out how many years older a is than b, I subtract b's age from a's age: A - B = 30 - 28 = 2.Therefore, a is 2 years older than b.