💡First, I recognize that the diagonal of a square divides it into two congruent right-angled triangles. Each of these triangles is a 45-45-90 triangle, where the legs are equal in length, and the hypotenuse is ( sqrt{2} ) times the length of each leg.Given that the diagonal (hypotenuse) of the square is ( 14sqrt{2} ) centimeters, I can use the relationship between the side length and the diagonal to find the side length of the square. Let ( s ) represent the side length of the square. According to the properties of a 45-45-90 triangle:[text{Diagonal} = s times sqrt{2}]Substituting the given diagonal:[14sqrt{2} = s times sqrt{2}]To solve for ( s ), I divide both sides of the equation by ( sqrt{2} ):[s = frac{14sqrt{2}}{sqrt{2}} = 14 text{ centimeters}]Now that I have the side length of the square, I can calculate its area. The area ( A ) of a square is given by the formula:[A = s^2]Substituting the value of ( s ):[A = 14^2 = 196 text{ square centimeters}]

💡Okay, so I have this problem where I need to find the minimum length of segment AB. Point A is on one set defined by the equation ( y^2 - 9 + 2xy - 12x - 3x^2 = 0 ), and point B is on another set defined by ( y^2 + 3 - 4x - 2y + x^2 = 0 ). Hmm, that looks a bit complicated, but maybe I can simplify these equations to understand what shapes they represent.Starting with the first equation: ( y^2 - 9 + 2xy - 12x - 3x^2 = 0 ). Let me try to rearrange and see if I can factor it or put it into a more familiar form. Maybe grouping terms will help.Looking at the terms, I see ( y^2 ) and ( 2xy ), which makes me think of completing the square or something similar. Let me try to group the y terms together:( y^2 + 2xy - 3x^2 - 12x - 9 = 0 ).Hmm, maybe I can factor this. Let's see if it can be factored into two binomials. Let me think about how to split the terms. Maybe grouping the first three terms and the last three:( (y^2 + 2xy - 3x^2) + (-12x - 9) = 0 ).Looking at the first group, ( y^2 + 2xy - 3x^2 ), that looks like a quadratic in terms of y. Maybe I can factor it:( y^2 + 2xy - 3x^2 = (y + 3x)(y - x) ). Let me check: ( (y + 3x)(y - x) = y^2 - xy + 3xy - 3x^2 = y^2 + 2xy - 3x^2 ). Yes, that works.So now, the equation becomes:( (y + 3x)(y - x) - 12x - 9 = 0 ).Hmm, maybe I can factor the remaining terms. Let me see:( (y + 3x)(y - x) - 12x - 9 = 0 ).Wait, maybe I can factor out something from the last two terms. Let's see:-12x - 9 = -3(4x + 3). Hmm, not sure if that helps.Alternatively, maybe I can write the entire equation as a product of two binomials. Let me try:( (y + 3x - a)(y - x - b) = 0 ).Expanding this, we get:( y^2 - xy + 3xy - 3x^2 - ay + ax - by + bx + ab = 0 ).Simplify:( y^2 + 2xy - 3x^2 + (-a - b)y + (a + b)x + ab = 0 ).Comparing this with the original equation ( y^2 + 2xy - 3x^2 -12x -9 = 0 ), we can set up equations for the coefficients:- Coefficient of y: -a - b = 0 ⇒ a + b = 0- Coefficient of x: a + b = -12- Constant term: ab = -9Wait, from the first equation, a + b = 0, but from the second, a + b = -12. That's a contradiction. Hmm, maybe my approach is wrong.Alternatively, maybe the equation represents two lines. Let me check if it's a degenerate conic.The general form of a conic is ( Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0 ). For it to be degenerate, the determinant of the matrix should be zero.The matrix is:[begin{bmatrix}A & B/2 & D/2 B/2 & C & E/2 D/2 & E/2 & F end{bmatrix}]Plugging in the coefficients:A = -3, B = 2, C = 1, D = -12, E = 0, F = -9.So the determinant is:[begin{vmatrix}-3 & 1 & -6 1 & 1 & 0 -6 & 0 & -9 end{vmatrix}]Calculating this determinant:-3*(1*(-9) - 0*0) - 1*(1*(-9) - (-6)*0) + (-6)*(1*0 - (-6)*1)= -3*(-9) - 1*(-9) + (-6)*(6)= 27 + 9 - 36= 0.Since the determinant is zero, it's a degenerate conic, meaning it can be factored into two lines.So, going back, I had ( (y + 3x)(y - x) - 12x - 9 = 0 ). Maybe I can factor this as:( (y + 3x - a)(y - x - b) = 0 ).Expanding:( y^2 + 2xy - 3x^2 - (a + b)y + ( -3a + b )x + ab = 0 ).Comparing with original equation:- Coefficient of y: - (a + b) = 0 ⇒ a + b = 0- Coefficient of x: -3a + b = -12- Constant term: ab = -9From a + b = 0, we have b = -a.Substituting into -3a + b = -12:-3a - a = -12 ⇒ -4a = -12 ⇒ a = 3.Then, b = -3.Check ab = 3*(-3) = -9, which matches.So, the equation factors as:( (y + 3x - 3)(y - x + 3) = 0 ).Therefore, the original equation represents two lines:1. ( y + 3x - 3 = 0 ) ⇒ ( y = -3x + 3 )2. ( y - x + 3 = 0 ) ⇒ ( y = x - 3 )Wait, hold on, let me double-check the signs.From ( (y + 3x - 3)(y - x + 3) = 0 ):First factor: ( y + 3x - 3 = 0 ) ⇒ ( y = -3x + 3 )Second factor: ( y - x + 3 = 0 ) ⇒ ( y = x - 3 )Yes, that's correct.So, point A lies on either of these two lines.Now, moving on to the second equation: ( y^2 + 3 - 4x - 2y + x^2 = 0 ).Let me rearrange this:( x^2 - 4x + y^2 - 2y + 3 = 0 ).This looks like a circle equation. Let me complete the squares.For x terms: ( x^2 - 4x ). Take half of -4, which is -2, square it: 4.For y terms: ( y^2 - 2y ). Take half of -2, which is -1, square it: 1.So, adding and subtracting these squares:( (x^2 - 4x + 4) - 4 + (y^2 - 2y + 1) - 1 + 3 = 0 ).Simplify:( (x - 2)^2 + (y - 1)^2 - 4 - 1 + 3 = 0 )⇒ ( (x - 2)^2 + (y - 1)^2 - 2 = 0 )⇒ ( (x - 2)^2 + (y - 1)^2 = 2 ).So, this is a circle centered at (2, 1) with radius √2.Therefore, point B lies on this circle.Now, I need to find the minimum distance between a point A on either of the two lines and a point B on the circle.Since the circle is fixed, the minimum distance from the circle to each line will give me the minimum possible AB.Wait, but actually, the minimum distance between the two sets (the union of the two lines and the circle) is the minimum distance from the circle to either of the lines.But I need to confirm whether the closest point on the lines to the circle is actually on the circle or not.Alternatively, since the circle is a compact set, the minimum distance between the two sets will be the minimum distance from the circle to either of the lines.So, let me compute the distance from the center of the circle (2,1) to each of the lines, and then subtract the radius to find the minimum distance from the circle to the line.Wait, no. The distance from the center to the line is d. If d > r, then the minimum distance from the circle to the line is d - r. If d < r, then the line intersects the circle, and the minimum distance is zero.But in our case, let's compute the distance from (2,1) to each line.First line: y = -3x + 3.The formula for distance from point (x0, y0) to line ax + by + c = 0 is |ax0 + by0 + c| / sqrt(a^2 + b^2).Let me write the line in standard form:y = -3x + 3 ⇒ 3x + y - 3 = 0.So, a = 3, b = 1, c = -3.Distance from (2,1):|3*2 + 1*1 - 3| / sqrt(9 + 1) = |6 + 1 - 3| / sqrt(10) = |4| / sqrt(10) = 4 / sqrt(10) ≈ 1.264.Radius of the circle is sqrt(2) ≈ 1.414.Since 4 / sqrt(10) ≈ 1.264 < sqrt(2) ≈ 1.414, the line intersects the circle. Therefore, the minimum distance from the circle to the line is zero. But wait, that would mean the minimum AB is zero, but that can't be right because the lines and the circle might not intersect.Wait, let me check if the line actually intersects the circle.Substitute y = -3x + 3 into the circle equation:(x - 2)^2 + (y - 1)^2 = 2⇒ (x - 2)^2 + (-3x + 3 - 1)^2 = 2⇒ (x - 2)^2 + (-3x + 2)^2 = 2Expand:(x^2 - 4x + 4) + (9x^2 - 12x + 4) = 2Combine like terms:x^2 - 4x + 4 + 9x^2 - 12x + 4 = 210x^2 - 16x + 8 = 210x^2 - 16x + 6 = 0Divide by 2:5x^2 - 8x + 3 = 0Discriminant: 64 - 60 = 4Solutions: x = [8 ± 2]/10 ⇒ x = (8 + 2)/10 = 1, x = (8 - 2)/10 = 0.6So, x = 1 and x = 0.6.Then y = -3x + 3:For x = 1: y = -3 + 3 = 0 ⇒ (1, 0)For x = 0.6: y = -1.8 + 3 = 1.2 ⇒ (0.6, 1.2)So, the line intersects the circle at two points: (1, 0) and (0.6, 1.2). Therefore, the minimum distance from the circle to the line is zero, but since we are looking for points A and B on the respective sets, the minimum AB would be the distance between these intersection points and points on the other line.Wait, no. Since point A is on either of the two lines, and point B is on the circle. If the circle intersects one of the lines, then the minimum AB is zero because A and B can coincide at the intersection points. But wait, point A is on the lines, and point B is on the circle. If the circle intersects the line, then at the intersection points, A and B can be the same point, making AB zero. But that seems too straightforward. Maybe I'm misunderstanding the problem.Wait, let me read the problem again: "Find the minimum length of segment AB, if point A belongs to the set defined by the equation... and point B belongs to the set defined by the equation...".So, if the two sets intersect, then the minimum distance is zero. But in this case, the circle intersects one of the lines, so yes, the minimum AB is zero. But that seems too simple, and the problem is likely expecting a positive minimum distance. Maybe I made a mistake.Wait, let me check the other line.Second line: y = x - 3.Let me compute the distance from the center (2,1) to this line.First, write the line in standard form: y = x - 3 ⇒ x - y - 3 = 0.So, a = 1, b = -1, c = -3.Distance from (2,1):|1*2 + (-1)*1 - 3| / sqrt(1 + 1) = |2 - 1 - 3| / sqrt(2) = |-2| / sqrt(2) = 2 / sqrt(2) = sqrt(2).Radius of the circle is sqrt(2). So, the distance from the center to the line is equal to the radius. Therefore, the line is tangent to the circle.So, the line y = x - 3 is tangent to the circle at one point.Therefore, the minimum distance from the circle to this line is zero (at the tangent point), but since the line is tangent, the minimum AB is zero again.Wait, but if both lines intersect or are tangent to the circle, then the minimum AB is zero. But maybe the problem expects the minimum distance between the two sets when they don't intersect. Wait, but in this case, they do intersect.Wait, let me double-check the equations.First set: two lines y = -3x + 3 and y = x - 3.Second set: circle centered at (2,1) with radius sqrt(2).We found that y = -3x + 3 intersects the circle at two points, and y = x - 3 is tangent to the circle.Therefore, the two sets do intersect, so the minimum AB is zero.But maybe the problem is expecting the minimum distance between the two sets when they don't intersect, but in this case, they do. So, perhaps the answer is zero.But let me think again. Maybe I misapplied something.Wait, the problem says "point A belongs to the set defined by the equation..." and "point B belongs to the set defined by the equation...". So, if the sets intersect, then A and B can be the same point, making AB zero. Therefore, the minimum length is zero.But maybe the problem is intended to have a positive minimum distance, so perhaps I made a mistake in interpreting the sets.Wait, let me check the first equation again.Original first equation: ( y^2 - 9 + 2xy - 12x - 3x^2 = 0 ).I factored it as (y + 3x - 3)(y - x + 3) = 0, giving y = -3x + 3 and y = x - 3.But let me verify this factoring.Multiply (y + 3x - 3)(y - x + 3):= y(y - x + 3) + 3x(y - x + 3) - 3(y - x + 3)= y^2 - xy + 3y + 3xy - 3x^2 + 9x - 3y + 3x - 9Combine like terms:y^2 + ( -xy + 3xy ) + (3y - 3y) + (-3x^2) + (9x + 3x) + (-9)= y^2 + 2xy - 3x^2 + 12x - 9.Wait, but the original equation is y^2 - 9 + 2xy - 12x - 3x^2 = 0, which is y^2 + 2xy - 3x^2 -12x -9 = 0.But when I expanded (y + 3x - 3)(y - x + 3), I got y^2 + 2xy - 3x^2 + 12x -9 = 0.Comparing to original: y^2 + 2xy - 3x^2 -12x -9 = 0.So, the x terms have opposite signs. Therefore, my factoring was incorrect.I must have made a mistake in the factoring process.Let me try again.We have the equation: y^2 + 2xy - 3x^2 -12x -9 = 0.I tried to factor it as (y + 3x - a)(y - x - b) = 0, but that didn't work because of the sign issue.Alternatively, maybe I need to factor it differently.Let me try to factor it as (y + ax + b)(y + cx + d) = 0.Expanding:y^2 + (a + c)xy + (ac)x^2 + (b + d)y + (ad + bc)x + bd = 0.Comparing with original equation:y^2 + 2xy -3x^2 -12x -9 = 0.So, coefficients:- Coefficient of y^2: 1 = 1*1, so both factors have y term with coefficient 1.- Coefficient of xy: a + c = 2.- Coefficient of x^2: ac = -3.- Coefficient of y: b + d = 0 (since there is no y term).- Coefficient of x: ad + bc = -12.- Constant term: bd = -9.So, we have:1. a + c = 22. ac = -33. b + d = 0 ⇒ d = -b4. ad + bc = -125. bd = -9From 3: d = -b.From 5: b*(-b) = -9 ⇒ -b^2 = -9 ⇒ b^2 = 9 ⇒ b = 3 or b = -3.Case 1: b = 3 ⇒ d = -3.From 4: a*(-3) + c*(3) = -12 ⇒ -3a + 3c = -12 ⇒ -a + c = -4.From 1: a + c = 2.So, we have:-a + c = -4a + c = 2Adding both equations:2c = -2 ⇒ c = -1.Then, from a + c = 2 ⇒ a = 3.Check ac = 3*(-1) = -3, which matches.So, the factors are:(y + 3x + 3)(y - x - 3) = 0.Let me expand this:= y^2 - xy - 3y + 3xy - 3x^2 -9x + 3y - 3x -9Simplify:y^2 + ( -xy + 3xy ) + ( -3y + 3y ) + (-3x^2) + (-9x -3x) + (-9)= y^2 + 2xy -3x^2 -12x -9.Yes, that matches the original equation.Therefore, the correct factoring is (y + 3x + 3)(y - x - 3) = 0.So, the two lines are:1. y + 3x + 3 = 0 ⇒ y = -3x - 32. y - x - 3 = 0 ⇒ y = x + 3Okay, so earlier I had the signs wrong. The lines are y = -3x -3 and y = x + 3.Now, let's check if these lines intersect the circle.First, check y = x + 3.Substitute into the circle equation:(x - 2)^2 + (y - 1)^2 = 2⇒ (x - 2)^2 + (x + 3 - 1)^2 = 2⇒ (x - 2)^2 + (x + 2)^2 = 2Expand:(x^2 -4x +4) + (x^2 +4x +4) = 22x^2 + 8 = 22x^2 = -6 ⇒ x^2 = -3.No real solutions. So, the line y = x + 3 does not intersect the circle.Now, check y = -3x -3.Substitute into the circle equation:(x - 2)^2 + (y - 1)^2 = 2⇒ (x - 2)^2 + (-3x -3 -1)^2 = 2⇒ (x - 2)^2 + (-3x -4)^2 = 2Expand:(x^2 -4x +4) + (9x^2 +24x +16) = 2Combine like terms:10x^2 +20x +20 = 210x^2 +20x +18 = 0Divide by 2:5x^2 +10x +9 = 0Discriminant: 100 - 180 = -80 < 0.No real solutions. So, the line y = -3x -3 does not intersect the circle either.Wait, that's different from earlier. So, both lines do not intersect the circle. Therefore, the minimum distance AB is the minimum distance between the two sets, which are two lines and a circle.Since the lines don't intersect the circle, the minimum distance AB is the minimum distance between the circle and either of the lines.So, I need to compute the distance from the center of the circle (2,1) to each of the lines, subtract the radius, and take the minimum.First, for line y = x + 3.Write in standard form: x - y + 3 = 0.Distance from (2,1):|1*2 + (-1)*1 + 3| / sqrt(1 + 1) = |2 -1 +3| / sqrt(2) = |4| / sqrt(2) = 4 / sqrt(2) = 2*sqrt(2).Radius is sqrt(2). So, the minimum distance from the circle to this line is 2*sqrt(2) - sqrt(2) = sqrt(2).Second, for line y = -3x -3.Write in standard form: 3x + y + 3 = 0.Distance from (2,1):|3*2 +1*1 +3| / sqrt(9 +1) = |6 +1 +3| / sqrt(10) = |10| / sqrt(10) = sqrt(10).Radius is sqrt(2). So, the minimum distance from the circle to this line is sqrt(10) - sqrt(2).Now, compare sqrt(2) and sqrt(10) - sqrt(2).Compute sqrt(10) ≈ 3.162, sqrt(2) ≈ 1.414.So, sqrt(10) - sqrt(2) ≈ 3.162 - 1.414 ≈ 1.748.Compare with sqrt(2) ≈ 1.414.So, sqrt(2) is smaller. Therefore, the minimum distance AB is sqrt(2).But wait, let me confirm.The distance from the center to the first line is 2*sqrt(2), subtract radius sqrt(2), gives sqrt(2).The distance from the center to the second line is sqrt(10), subtract radius sqrt(2), gives sqrt(10) - sqrt(2) ≈ 1.748.So, sqrt(2) ≈ 1.414 is smaller. Therefore, the minimum AB is sqrt(2).But let me think again. The minimum distance between the circle and the line y = x + 3 is sqrt(2). But is that the actual minimum distance between any point A on the lines and point B on the circle?Yes, because the minimum distance from the circle to the line is achieved along the perpendicular from the center to the line, and since the line does not intersect the circle, the closest point on the line to the circle is along this perpendicular, at a distance of sqrt(2).Similarly, for the other line, the minimum distance is sqrt(10) - sqrt(2), which is larger.Therefore, the minimum AB is sqrt(2).But wait, let me visualize. The circle is at (2,1), and the line y = x + 3 is above and to the left of the circle. The distance from the center to this line is 2*sqrt(2), which is about 2.828, and the radius is sqrt(2) ≈ 1.414. So, the closest point on the line to the circle is 2*sqrt(2) - sqrt(2) = sqrt(2).Similarly, for the other line y = -3x -3, which is below and to the left of the circle, the distance from the center is sqrt(10) ≈ 3.162, so the closest distance is sqrt(10) - sqrt(2) ≈ 1.748.Therefore, the minimum AB is sqrt(2).But let me confirm by parametrizing.Alternatively, I can parametrize a general point on the circle and a general point on the lines, then minimize the distance.But that might be more complicated.Alternatively, since the minimum distance from the circle to the line y = x + 3 is sqrt(2), and this is achieved at the point on the line closest to the circle, which is along the perpendicular from the center.Similarly, the minimum distance from the circle to the line y = -3x -3 is sqrt(10) - sqrt(2).Therefore, the overall minimum AB is sqrt(2).So, the minimum length of segment AB is sqrt(2).But let me write it in boxed form.**Final Answer**The minimum length of segment ( AB ) is boxed{sqrt{2}}.

💡Okay, so I have this problem where I need to find the smallest positive value of x in degrees for which the function f(x) = sin(x/3) + sin(x/11) reaches its maximum value. Hmm, let me think about how to approach this.First, I know that the sine function reaches its maximum value of 1 at 90 degrees plus any multiple of 360 degrees. So, for each sine term in the function, I can set up equations where each sine term equals 1.So, for sin(x/3) to be 1, x/3 must be equal to 90 degrees plus some multiple of 360 degrees. Similarly, for sin(x/11) to be 1, x/11 must be equal to 90 degrees plus some multiple of 360 degrees.Let me write that down:1. x/3 = 90 + 360a2. x/11 = 90 + 360bWhere a and b are integers representing the number of full cycles plus the quarter cycle to reach the peak.Now, solving these equations for x:From equation 1:x = 3*(90 + 360a) = 270 + 1080aFrom equation 2:x = 11*(90 + 360b) = 990 + 3960bSo, I have two expressions for x:x = 270 + 1080ax = 990 + 3960bSince both expressions equal x, I can set them equal to each other:270 + 1080a = 990 + 3960bLet me simplify this equation. Subtract 270 from both sides:1080a = 720 + 3960bNow, I can divide both sides by 360 to make the numbers smaller:3a = 2 + 11bSo, 3a - 11b = 2Now, I need to find integer solutions for a and b that satisfy this equation. This is a linear Diophantine equation. Let me try to find the smallest non-negative integers a and b that satisfy this.Let me rearrange the equation:3a = 11b + 2So, 3a must be equal to 11b + 2. I need to find the smallest b such that 11b + 2 is divisible by 3.Let me try b = 0: 11*0 + 2 = 2, which is not divisible by 3.b = 1: 11 + 2 = 13, not divisible by 3.b = 2: 22 + 2 = 24, which is divisible by 3. So, b = 2.Then, 3a = 24, so a = 8.So, a = 8 and b = 2.Now, plugging back into the expressions for x:From x = 270 + 1080a:x = 270 + 1080*8 = 270 + 8640 = 8910 degrees.From x = 990 + 3960b:x = 990 + 3960*2 = 990 + 7920 = 8910 degrees.So, both give x = 8910 degrees.Is this the smallest positive x? Let me check if there are smaller positive solutions.Since we found the smallest b that satisfies the equation is b = 2, which gives a = 8, and plugging back gives x = 8910 degrees. If I try smaller b, like b = 1, then 3a = 13, which is not an integer. Similarly, b = 0 gives 3a = 2, which is not an integer. So, 8910 degrees is indeed the smallest positive x where both sine terms reach their maximum simultaneously.Wait, but just to be thorough, is there a smaller x where both sine functions reach their maximum? Let me think about the periods of the two sine functions.The period of sin(x/3) is 360*3 = 1080 degrees.The period of sin(x/11) is 360*11 = 3960 degrees.So, the function f(x) = sin(x/3) + sin(x/11) will have a period equal to the least common multiple (LCM) of 1080 and 3960.Let me compute LCM(1080, 3960).First, factor both numbers:1080 = 2^3 * 3^3 * 53960 = 2^3 * 3^2 * 5 * 11So, LCM is the product of the highest powers of all primes present:2^3 * 3^3 * 5 * 11 = 8 * 27 * 5 * 11 = 8 * 27 = 216; 216 * 5 = 1080; 1080 * 11 = 11880 degrees.So, the function f(x) repeats every 11880 degrees. Therefore, the maximum value should occur at x = 8910 degrees, which is less than 11880, so it's within the first period. Therefore, 8910 degrees is indeed the smallest positive x where the function reaches its maximum.Wait, but just to make sure, is there a way to get a smaller x by considering negative multiples? But since we're looking for the smallest positive x, negative multiples would give negative x, which we don't want. So, 8910 degrees is the answer.I think that's it. So, the smallest positive x is 8910 degrees.

💡First, I need to understand the dimensions of the wooden block and how it will be divided into smaller cubes. The block is 5 inches long, 5 inches wide, and 1 inch high. When it is cut into 1-inch cubes, there will be a total of 5 x 5 x 1 = 25 small cubes.Next, I should consider the painting process. The entire outer surface of the block is painted blue on all six sides. This means that the cubes on the surfaces of the block will have blue faces, while the cubes inside the block will not have any blue faces.Since the block is only 1 inch high, it essentially forms a single layer of cubes. This simplifies the analysis because there are no cubes stacked vertically, which would have different numbers of exposed faces.Now, I'll classify the small cubes based on their positions within the block:1. **Corner Cubes**: There are four corner cubes in the 5x5 grid. Each corner cube has three exposed faces (one on each of the three dimensions). Since all three faces are painted blue, each corner cube has three blue faces.2. **Edge Cubes (excluding corners)**: Along each edge of the block, excluding the corners, there are 5 - 2 = 3 cubes. With four edges in the 5x5 grid, there are 4 x 3 = 12 edge cubes. Each edge cube has two exposed faces, both of which are painted blue.3. **Center Cubes**: The remaining cubes in the center of the block are not on any edge or corner. There are 5 x 5 - 4 (corners) - 12 (edges) = 5 center cubes. Each center cube has only one exposed face, which is painted blue.Finally, I need to determine how many of these small cubes have an odd number of blue faces. The corner cubes have three blue faces (an odd number), and the center cubes have one blue face (also an odd number). The edge cubes have two blue faces (an even number), so they are not included.Adding the number of corner and center cubes gives the total number of small cubes with an odd number of blue faces: 4 (corner cubes) + 5 (center cubes) = 9 cubes.

💡Okay, so I have this equation to solve: (frac{1}{x + 8} + frac{1}{x + 5} = frac{1}{x + 11} + frac{1}{x + 4}). Hmm, it's a bit intimidating with all those fractions, but I think I can handle it step by step.First, I remember that when dealing with equations involving fractions, a good strategy is to combine the fractions on each side so that I can simplify the equation. Let me try that.On the left side, I have (frac{1}{x + 8} + frac{1}{x + 5}). To combine these, I need a common denominator, which would be ((x + 8)(x + 5)). So, I'll rewrite each fraction with this common denominator:[frac{1}{x + 8} = frac{x + 5}{(x + 8)(x + 5)}]and[frac{1}{x + 5} = frac{x + 8}{(x + 8)(x + 5)}]Adding these together gives:[frac{x + 5 + x + 8}{(x + 8)(x + 5)} = frac{2x + 13}{(x + 8)(x + 5)}]Okay, so the left side simplifies to (frac{2x + 13}{(x + 8)(x + 5)}). Now, let me do the same for the right side: (frac{1}{x + 11} + frac{1}{x + 4}).The common denominator here is ((x + 11)(x + 4)). So, rewriting each fraction:[frac{1}{x + 11} = frac{x + 4}{(x + 11)(x + 4)}]and[frac{1}{x + 4} = frac{x + 11}{(x + 11)(x + 4)}]Adding these together:[frac{x + 4 + x + 11}{(x + 11)(x + 4)} = frac{2x + 15}{(x + 11)(x + 4)}]So, the right side simplifies to (frac{2x + 15}{(x + 11)(x + 4)}).Now, the equation looks like this:[frac{2x + 13}{(x + 8)(x + 5)} = frac{2x + 15}{(x + 11)(x + 4)}]Hmm, okay. To solve this, I think I can cross-multiply to eliminate the denominators. That should make the equation easier to handle. So, cross-multiplying gives:[(2x + 13)(x + 11)(x + 4) = (2x + 15)(x + 8)(x + 5)]Wow, that looks a bit complicated, but let's take it step by step. I'll need to expand both sides.First, let's expand the left side: ((2x + 13)(x + 11)(x + 4)).I'll start by expanding ((x + 11)(x + 4)):[(x + 11)(x + 4) = x^2 + 4x + 11x + 44 = x^2 + 15x + 44]Now, multiply this by ((2x + 13)):[(2x + 13)(x^2 + 15x + 44)]Let me distribute each term:First, (2x times x^2 = 2x^3),(2x times 15x = 30x^2),(2x times 44 = 88x),Then, (13 times x^2 = 13x^2),(13 times 15x = 195x),(13 times 44 = 572).Adding all these together:[2x^3 + 30x^2 + 88x + 13x^2 + 195x + 572]Combine like terms:- (2x^3)- (30x^2 + 13x^2 = 43x^2)- (88x + 195x = 283x)- (572)So, the left side expands to:[2x^3 + 43x^2 + 283x + 572]Now, let's work on the right side: ((2x + 15)(x + 8)(x + 5)).Again, I'll start by expanding ((x + 8)(x + 5)):[(x + 8)(x + 5) = x^2 + 5x + 8x + 40 = x^2 + 13x + 40]Now, multiply this by ((2x + 15)):[(2x + 15)(x^2 + 13x + 40)]Distribute each term:First, (2x times x^2 = 2x^3),(2x times 13x = 26x^2),(2x times 40 = 80x),Then, (15 times x^2 = 15x^2),(15 times 13x = 195x),(15 times 40 = 600).Adding all these together:[2x^3 + 26x^2 + 80x + 15x^2 + 195x + 600]Combine like terms:- (2x^3)- (26x^2 + 15x^2 = 41x^2)- (80x + 195x = 275x)- (600)So, the right side expands to:[2x^3 + 41x^2 + 275x + 600]Now, let's set both expanded sides equal to each other:[2x^3 + 43x^2 + 283x + 572 = 2x^3 + 41x^2 + 275x + 600]Hmm, okay. Let's subtract (2x^3) from both sides to eliminate the cubic terms:[43x^2 + 283x + 572 = 41x^2 + 275x + 600]Now, subtract (41x^2 + 275x + 600) from both sides to bring all terms to one side:[43x^2 - 41x^2 + 283x - 275x + 572 - 600 = 0]Simplify each term:- (43x^2 - 41x^2 = 2x^2)- (283x - 275x = 8x)- (572 - 600 = -28)So, the equation simplifies to:[2x^2 + 8x - 28 = 0]Hmm, I can factor out a 2 from all terms to simplify further:[2(x^2 + 4x - 14) = 0]Divide both sides by 2:[x^2 + 4x - 14 = 0]Wait, that doesn't seem right. Let me double-check my expansions because I think I might have made a mistake somewhere.Going back to the left side expansion:[(2x + 13)(x^2 + 15x + 44) = 2x^3 + 30x^2 + 88x + 13x^2 + 195x + 572]Which combines to:[2x^3 + 43x^2 + 283x + 572]That seems correct.Right side expansion:[(2x + 15)(x^2 + 13x + 40) = 2x^3 + 26x^2 + 80x + 15x^2 + 195x + 600]Which combines to:[2x^3 + 41x^2 + 275x + 600]That also seems correct.Subtracting right side from left side:[(2x^3 + 43x^2 + 283x + 572) - (2x^3 + 41x^2 + 275x + 600) = 0]Which simplifies to:[2x^2 + 8x - 28 = 0]Yes, that's correct. So, I can factor out a 2:[2(x^2 + 4x - 14) = 0]Divide both sides by 2:[x^2 + 4x - 14 = 0]Wait, but in the initial problem, I think I might have made a mistake in the cross-multiplication step. Let me check that again.Original equation after combining fractions:[frac{2x + 13}{(x + 8)(x + 5)} = frac{2x + 15}{(x + 11)(x + 4)}]Cross-multiplying:[(2x + 13)(x + 11)(x + 4) = (2x + 15)(x + 8)(x + 5)]Yes, that's correct. So, the expansions seem right, leading to the quadratic equation (x^2 + 4x - 14 = 0).Wait, but in the initial problem, the user's solution ended up with (x^2 + 3x - 7 = 0). Did I make a mistake in the expansion?Let me check the expansions again.Left side:[(2x + 13)(x + 11)(x + 4)]First, ((x + 11)(x + 4) = x^2 + 15x + 44). Correct.Then, multiplying by (2x + 13):[2x(x^2 + 15x + 44) = 2x^3 + 30x^2 + 88x][13(x^2 + 15x + 44) = 13x^2 + 195x + 572]Adding together: (2x^3 + 43x^2 + 283x + 572). Correct.Right side:[(2x + 15)(x + 8)(x + 5)]First, ((x + 8)(x + 5) = x^2 + 13x + 40). Correct.Then, multiplying by (2x + 15):[2x(x^2 + 13x + 40) = 2x^3 + 26x^2 + 80x][15(x^2 + 13x + 40) = 15x^2 + 195x + 600]Adding together: (2x^3 + 41x^2 + 275x + 600). Correct.Subtracting right side from left side:[(2x^3 + 43x^2 + 283x + 572) - (2x^3 + 41x^2 + 275x + 600) = 0]Simplifies to:[2x^2 + 8x - 28 = 0]Divide by 2:[x^2 + 4x - 14 = 0]Hmm, so my result is different from the initial solution. That means I must have made a mistake somewhere. Let me check the initial steps again.Wait, in the initial problem, the user combined the fractions on each side correctly:Left side: (frac{2x + 13}{(x + 8)(x + 5)})Right side: (frac{2x + 15}{(x + 11)(x + 4)})Cross-multiplying gives:[(2x + 13)(x + 11)(x + 4) = (2x + 15)(x + 8)(x + 5)]Yes, that's correct.But when I expanded, I got different coefficients. Let me try expanding again, perhaps I made an arithmetic error.Left side:[(2x + 13)(x + 11)(x + 4)]First, ((x + 11)(x + 4) = x^2 + 15x + 44). Correct.Multiply by (2x + 13):[2x(x^2 + 15x + 44) = 2x^3 + 30x^2 + 88x][13(x^2 + 15x + 44) = 13x^2 + 195x + 572]Adding together: (2x^3 + 43x^2 + 283x + 572). Correct.Right side:[(2x + 15)(x + 8)(x + 5)]First, ((x + 8)(x + 5) = x^2 + 13x + 40). Correct.Multiply by (2x + 15):[2x(x^2 + 13x + 40) = 2x^3 + 26x^2 + 80x][15(x^2 + 13x + 40) = 15x^2 + 195x + 600]Adding together: (2x^3 + 41x^2 + 275x + 600). Correct.Subtracting right side from left side:[(2x^3 + 43x^2 + 283x + 572) - (2x^3 + 41x^2 + 275x + 600) = 0]Simplifies to:[2x^2 + 8x - 28 = 0]Divide by 2:[x^2 + 4x - 14 = 0]Wait, but the initial solution had (x^2 + 3x - 7 = 0). So, there's a discrepancy. Let me check if I made a mistake in the cross-multiplication step.Wait, perhaps I should have subtracted the other way around. Let me try subtracting left side minus right side:[(2x^3 + 43x^2 + 283x + 572) - (2x^3 + 41x^2 + 275x + 600) = 0]Which is:[2x^2 + 8x - 28 = 0]Which simplifies to (x^2 + 4x - 14 = 0).Alternatively, if I subtract right side minus left side:[(2x^3 + 41x^2 + 275x + 600) - (2x^3 + 43x^2 + 283x + 572) = 0]Which would be:[-2x^2 -8x +28 = 0]Or (2x^2 + 8x -28 = 0), same as before.So, regardless, the quadratic is (x^2 + 4x -14 = 0).Wait, but the initial solution had (x^2 + 3x -7 = 0). So, perhaps there was a mistake in the initial solution.Alternatively, maybe I made a mistake in the cross-multiplication step.Wait, let me try cross-multiplying again.Original equation after combining fractions:[frac{2x + 13}{(x + 8)(x + 5)} = frac{2x + 15}{(x + 11)(x + 4)}]Cross-multiplying:[(2x + 13)(x + 11)(x + 4) = (2x + 15)(x + 8)(x + 5)]Yes, that's correct.Alternatively, perhaps the initial solution made a mistake in the cross-multiplication step.Wait, let me check the initial solution's steps.They combined the fractions correctly:Left side: (frac{2x +13}{(x+8)(x+5)})Right side: (frac{2x +15}{(x+11)(x+4)})Then cross-multiplied:[(2x +13)(x+11)(x+4) = (2x +15)(x+8)(x+5)]Then, they expanded and got:Left side: (2x^3 +35x^2 +122x +572)Right side: (2x^3 +31x^2 +110x +600)Then, subtracting, they got:(4x^2 +12x -28 =0), which simplifies to (x^2 +3x -7=0).Wait, so in their expansion, the left side was (2x^3 +35x^2 +122x +572), but when I expanded, I got (2x^3 +43x^2 +283x +572). So, their coefficients are different.Similarly, their right side was (2x^3 +31x^2 +110x +600), but I got (2x^3 +41x^2 +275x +600).So, clearly, there's a discrepancy in the expansion steps. Let me check their expansion.Wait, perhaps they made a mistake in expanding ((2x +13)(x +11)(x +4)).Let me try expanding it their way.First, ((x +11)(x +4) = x^2 +15x +44). Then, multiply by (2x +13):[2x(x^2 +15x +44) = 2x^3 +30x^2 +88x][13(x^2 +15x +44) =13x^2 +195x +572]Adding together: (2x^3 +43x^2 +283x +572). So, their expansion must be wrong.Wait, in the initial solution, they have left side as (2x^3 +35x^2 +122x +572). That suggests that perhaps they expanded ((2x +13)(x +11)(x +4)) differently.Wait, maybe they expanded ((2x +13)(x +11)) first, then multiplied by ((x +4)).Let me try that.First, expand ((2x +13)(x +11)):[2x(x) +2x(11) +13(x) +13(11) = 2x^2 +22x +13x +143 = 2x^2 +35x +143]Then, multiply by ((x +4)):[(2x^2 +35x +143)(x +4) = 2x^3 +8x^2 +35x^2 +140x +143x +572]Combine like terms:[2x^3 + (8x^2 +35x^2) + (140x +143x) +572 = 2x^3 +43x^2 +283x +572]So, same result as before. So, the initial solution must have made a mistake in their expansion.Similarly, for the right side, they expanded ((2x +15)(x +8)(x +5)) as (2x^3 +31x^2 +110x +600), but when I expanded, I got (2x^3 +41x^2 +275x +600).Let me check their expansion:First, expand ((2x +15)(x +8)):[2x(x) +2x(8) +15(x) +15(8) = 2x^2 +16x +15x +120 = 2x^2 +31x +120]Then, multiply by ((x +5)):[(2x^2 +31x +120)(x +5) = 2x^3 +10x^2 +31x^2 +155x +120x +600]Combine like terms:[2x^3 + (10x^2 +31x^2) + (155x +120x) +600 = 2x^3 +41x^2 +275x +600]So, the initial solution's expansion was incorrect. They must have made a mistake in their multiplication.Therefore, the correct quadratic equation is (x^2 +4x -14 =0), not (x^2 +3x -7=0).Wait, but let me solve my quadratic equation to see if it makes sense.So, (x^2 +4x -14 =0).Using the quadratic formula:[x = frac{-b pm sqrt{b^2 -4ac}}{2a}]Where (a=1), (b=4), (c=-14).So,[x = frac{-4 pm sqrt{16 +56}}{2} = frac{-4 pm sqrt{72}}{2} = frac{-4 pm 6sqrt{2}}{2} = -2 pm 3sqrt{2}]So, the solutions are (x = -2 + 3sqrt{2}) and (x = -2 - 3sqrt{2}).Wait, but let me check if these solutions are valid by plugging them back into the original equation to ensure they don't make any denominators zero.Original equation denominators: (x +8), (x +5), (x +11), (x +4).For (x = -2 + 3sqrt{2}):Calculate each denominator:- (x +8 = (-2 +3sqrt{2}) +8 =6 +3sqrt{2} neq 0)- (x +5 = (-2 +3sqrt{2}) +5 =3 +3sqrt{2} neq 0)- (x +11 = (-2 +3sqrt{2}) +11 =9 +3sqrt{2} neq 0)- (x +4 = (-2 +3sqrt{2}) +4 =2 +3sqrt{2} neq 0)All denominators are non-zero.For (x = -2 -3sqrt{2}):Calculate each denominator:- (x +8 = (-2 -3sqrt{2}) +8 =6 -3sqrt{2} neq 0)- (x +5 = (-2 -3sqrt{2}) +5 =3 -3sqrt{2} neq 0)- (x +11 = (-2 -3sqrt{2}) +11 =9 -3sqrt{2} neq 0)- (x +4 = (-2 -3sqrt{2}) +4 =2 -3sqrt{2} neq 0)Again, all denominators are non-zero.So, both solutions are valid.Wait, but in the initial solution, they had (x = frac{-3 pm sqrt{37}}{2}). Let me calculate that:[frac{-3 + sqrt{37}}{2} approx frac{-3 +6.082}{2} approx frac{3.082}{2} approx1.541][frac{-3 - sqrt{37}}{2} approx frac{-3 -6.082}{2} approx frac{-9.082}{2} approx-4.541]But my solutions are (x = -2 pm 3sqrt{2}), which are approximately:[-2 +4.242 =2.242][-2 -4.242 =-6.242]So, different solutions. Therefore, the initial solution must have made a mistake in their expansion.Therefore, the correct solutions are (x = -2 pm 3sqrt{2}).Wait, but let me double-check my quadratic equation.From the expanded equation:[2x^2 +8x -28 =0]Divide by 2:[x^2 +4x -14 =0]Yes, that's correct.Alternatively, perhaps the initial solution made a mistake in the cross-multiplication step.Wait, let me try solving the equation again, but this time, perhaps using a different approach to see if I get the same result.Alternative approach:Instead of cross-multiplying immediately, perhaps I can subtract one side from the other and combine fractions.Original equation:[frac{1}{x +8} + frac{1}{x +5} - frac{1}{x +11} - frac{1}{x +4} =0]Let me combine the fractions:Group terms:[left(frac{1}{x +8} - frac{1}{x +11}right) + left(frac{1}{x +5} - frac{1}{x +4}right) =0]Compute each difference:First difference: (frac{1}{x +8} - frac{1}{x +11})Common denominator: ((x +8)(x +11))[frac{(x +11) - (x +8)}{(x +8)(x +11)} = frac{3}{(x +8)(x +11)}]Second difference: (frac{1}{x +5} - frac{1}{x +4})Common denominator: ((x +5)(x +4))[frac{(x +4) - (x +5)}{(x +5)(x +4)} = frac{-1}{(x +5)(x +4)}]So, the equation becomes:[frac{3}{(x +8)(x +11)} - frac{1}{(x +5)(x +4)} =0]Move the second term to the other side:[frac{3}{(x +8)(x +11)} = frac{1}{(x +5)(x +4)}]Now, cross-multiply:[3(x +5)(x +4) = (x +8)(x +11)]Expand both sides:Left side:[3(x +5)(x +4) =3(x^2 +9x +20) =3x^2 +27x +60]Right side:[(x +8)(x +11) =x^2 +19x +88]Set equal:[3x^2 +27x +60 =x^2 +19x +88]Subtract (x^2 +19x +88) from both sides:[2x^2 +8x -28 =0]Which simplifies to:[x^2 +4x -14 =0]Same result as before. So, this confirms that the correct quadratic equation is (x^2 +4x -14 =0), leading to solutions (x = -2 pm 3sqrt{2}).Therefore, the initial solution must have made a mistake in their expansion steps, leading to an incorrect quadratic equation.So, the correct solutions are (x = -2 + 3sqrt{2}) and (x = -2 - 3sqrt{2}).

💡Okay, so I have this problem here about functions and their symmetry. Let me try to figure it out step by step. First, the function given is f(x) = x² + 2x. They mention that the graph of function g(x) is symmetric to the graph of f(x) about the origin. Hmm, symmetry about the origin usually means that if you rotate the graph 180 degrees around the origin, it looks the same. So, for every point (x, y) on f(x), there should be a point (-x, -y) on g(x). That makes sense.So, for part (1), I need to find the explicit expression for g(x). Let me think. If (x, y) is on f(x), then (-x, -y) is on g(x). So, if y = f(x) = x² + 2x, then for g(x), -y = f(-x). Wait, is that right? Let me check.If I take a point (x, y) on f(x), then the symmetric point about the origin would be (-x, -y). So, if I plug -x into f(x), I should get -y. So, f(-x) = (-x)² + 2*(-x) = x² - 2x. But since this equals -y, then -y = x² - 2x, so y = -x² + 2x. Therefore, g(x) = -x² + 2x. That seems straightforward. Let me write that down.For part (2), I need to solve the inequality g(x) ≥ f(x) - |x| - 1. So, substituting the expressions we have:g(x) = -x² + 2xf(x) = x² + 2xSo, the inequality becomes:- x² + 2x ≥ (x² + 2x) - |x| - 1Let me simplify this step by step. First, expand the right-hand side:(x² + 2x) - |x| - 1 = x² + 2x - |x| - 1So, the inequality is:- x² + 2x ≥ x² + 2x - |x| - 1Now, let me bring all terms to one side to make it easier to handle:- x² + 2x - (x² + 2x - |x| - 1) ≥ 0Simplify the left-hand side:- x² + 2x - x² - 2x + |x| + 1 ≥ 0Combine like terms:(-x² - x²) + (2x - 2x) + |x| + 1 ≥ 0Which simplifies to:-2x² + |x| + 1 ≥ 0Hmm, okay, so we have -2x² + |x| + 1 ≥ 0. Let me rewrite this as:2x² - |x| - 1 ≤ 0Because multiplying both sides by -1 reverses the inequality. So now, we have 2x² - |x| - 1 ≤ 0.This looks like a quadratic in terms of |x|. Let me set t = |x|, so t ≥ 0. Then the inequality becomes:2t² - t - 1 ≤ 0Now, let's solve this quadratic inequality. First, find the roots of 2t² - t - 1 = 0.Using the quadratic formula:t = [1 ± √(1 + 8)] / 4 = [1 ± 3] / 4So, t = (1 + 3)/4 = 1 or t = (1 - 3)/4 = -0.5Since t = |x| ≥ 0, we discard the negative root. So, t = 1 is the critical point.Now, the quadratic 2t² - t - 1 opens upwards (since the coefficient of t² is positive), so it is ≤ 0 between its roots. But since one root is negative and we only consider t ≥ 0, the inequality 2t² - t - 1 ≤ 0 holds for t between -0.5 and 1. But since t ≥ 0, this reduces to 0 ≤ t ≤ 1.Therefore, |x| ≤ 1, which implies that x is between -1 and 1.So, the solution to the inequality is -1 ≤ x ≤ 1.Wait, let me double-check my steps to make sure I didn't make a mistake.Starting from the inequality:-2x² + |x| + 1 ≥ 0Which I rewrote as:2x² - |x| - 1 ≤ 0Then set t = |x|, so 2t² - t - 1 ≤ 0Solving 2t² - t - 1 = 0 gives t = 1 and t = -0.5Since t ≥ 0, the inequality holds for t between 0 and 1, so |x| ≤ 1, hence x ∈ [-1, 1].Yes, that seems correct. So, the solution is x between -1 and 1.**Final Answer**(1) boxed{g(x) = -x^2 + 2x}(2) boxed{[-1, 1]}