💡Okay, so I have this trigonometric function to simplify and then evaluate given a specific condition. Let me try to tackle part (1) first.The function is:[ f(α) = frac{sin left(α - frac{π}{2}right) cdot cos left(frac{3π}{2} + αright) cdot tan (π - α)}{tan (-α - π) cdot sin (-α - π)} ]Hmm, that looks a bit complicated, but I remember there are some trigonometric identities that can help simplify each of these terms. Let me recall them.Starting with the numerator:1. **Simplify (sin left(α - frac{π}{2}right)):**I remember that (sin (θ - frac{π}{2}) = -cos θ). So, applying that identity:[ sin left(α - frac{π}{2}right) = -cos α ]2. **Simplify (cos left(frac{3π}{2} + αright)):**There's an identity for cosine of an angle plus π/2 or 3π/2. Specifically, (cos left(frac{3π}{2} + αright) = sin α). Wait, let me verify that.Actually, (cos left(frac{3π}{2} + αright)) can be rewritten using the cosine addition formula:[ cos left(frac{3π}{2} + αright) = cos frac{3π}{2} cos α - sin frac{3π}{2} sin α ]We know that (cos frac{3π}{2} = 0) and (sin frac{3π}{2} = -1), so:[ 0 cdot cos α - (-1) cdot sin α = sin α ]So, yes, (cos left(frac{3π}{2} + αright) = sin α).3. **Simplify (tan (π - α)):**The tangent of π minus α is equal to (-tan α). Because tangent is periodic with period π and is odd function. So:[ tan (π - α) = -tan α ]So, putting it all together, the numerator becomes:[ (-cos α) cdot sin α cdot (-tan α) ]Multiplying the negatives: negative times negative is positive, so:[ cos α cdot sin α cdot tan α ]But wait, let me compute the signs step by step to avoid mistakes.First term: (-cos α)Second term: (sin α)Third term: (-tan α)So, multiplying them together:[ (-cos α) cdot sin α cdot (-tan α) = (-1) cdot cos α cdot sin α cdot (-1) cdot tan α ]The two negatives multiply to positive, so:[ cos α cdot sin α cdot tan α ]Now, let's move to the denominator:1. **Simplify (tan (-α - π)):**First, note that (tan (-θ) = -tan θ), so:[ tan (-α - π) = tan (-(α + π)) = -tan (α + π) ]But (tan (α + π) = tan α) because tangent has a period of π. So:[ tan (-α - π) = -tan α ]2. **Simplify (sin (-α - π)):**Similarly, sine is an odd function, so (sin (-θ) = -sin θ). Also, (sin (θ + π) = -sin θ). Let me apply that step by step.First, (sin (-α - π) = sin (-(α + π)) = -sin (α + π)).Then, (sin (α + π) = -sin α), so:[ -sin (α + π) = -(-sin α) = sin α ]Wait, let me verify that again.Starting with (sin (-α - π)):[ sin (-α - π) = sin (-(α + π)) = -sin (α + π) ]And (sin (α + π) = -sin α), so:[ -sin (α + π) = -(-sin α) = sin α ]Yes, that's correct. So, (sin (-α - π) = sin α).So, the denominator becomes:[ (-tan α) cdot sin α ]Putting it all together, the function (f(α)) is:[ f(α) = frac{cos α cdot sin α cdot tan α}{(-tan α) cdot sin α} ]Wait, hold on. Let me write the numerator and denominator clearly.Numerator: (cos α cdot sin α cdot tan α)Denominator: (-tan α cdot sin α)So, let's write it as:[ f(α) = frac{cos α cdot sin α cdot tan α}{-tan α cdot sin α} ]Now, I can cancel out common terms in numerator and denominator.First, (sin α) cancels out, assuming (sin α neq 0), which it isn't in the third quadrant because sine is negative there.Similarly, (tan α) cancels out, assuming (tan α neq 0), which it isn't in the third quadrant because tangent is positive there (since both sine and cosine are negative, their ratio is positive).So, after cancellation:[ f(α) = frac{cos α}{-1} = -cos α ]So, that simplifies nicely to (-cos α).Wait, let me double-check my steps to make sure I didn't make a mistake.Starting with the numerator:- (sin (α - π/2) = -cos α)- (cos (3π/2 + α) = sin α)- (tan (π - α) = -tan α)Multiplying these together:[ (-cos α) cdot sin α cdot (-tan α) = (-1) cdot cos α cdot sin α cdot (-1) cdot tan α = cos α cdot sin α cdot tan α ]Denominator:- (tan (-α - π) = -tan α)- (sin (-α - π) = sin α)Multiplying these together:[ (-tan α) cdot sin α ]So, putting it all together:[ frac{cos α cdot sin α cdot tan α}{(-tan α) cdot sin α} ]Canceling (sin α) and (tan α):[ frac{cos α}{-1} = -cos α ]Yes, that seems correct.So, part (1) simplifies to (f(α) = -cos α).Now, moving on to part (2):Given that (cos left(α - frac{3π}{2}right) = frac{1}{5}), find the value of (f(α)).We already know from part (1) that (f(α) = -cos α), so we just need to find (cos α) given (cos left(α - frac{3π}{2}right) = frac{1}{5}).Let me simplify (cos left(α - frac{3π}{2}right)).Using the cosine of a difference identity:[ cos (θ - φ) = cos θ cos φ + sin θ sin φ ]So, applying that:[ cos left(α - frac{3π}{2}right) = cos α cos frac{3π}{2} + sin α sin frac{3π}{2} ]We know that:- (cos frac{3π}{2} = 0)- (sin frac{3π}{2} = -1)So, substituting these values:[ cos left(α - frac{3π}{2}right) = cos α cdot 0 + sin α cdot (-1) = -sin α ]Therefore, the given equation becomes:[ -sin α = frac{1}{5} ]Which implies:[ sin α = -frac{1}{5} ]Since α is in the third quadrant, both sine and cosine are negative. So, (sin α = -frac{1}{5}) is consistent with α being in the third quadrant.Now, we need to find (cos α). We can use the Pythagorean identity:[ sin^2 α + cos^2 α = 1 ]Substituting (sin α = -frac{1}{5}):[ left(-frac{1}{5}right)^2 + cos^2 α = 1 ][ frac{1}{25} + cos^2 α = 1 ][ cos^2 α = 1 - frac{1}{25} ][ cos^2 α = frac{25}{25} - frac{1}{25} ][ cos^2 α = frac{24}{25} ][ cos α = pm sqrt{frac{24}{25}} ][ cos α = pm frac{2sqrt{6}}{5} ]But since α is in the third quadrant, cosine is negative. Therefore:[ cos α = -frac{2sqrt{6}}{5} ]Now, recall that (f(α) = -cos α), so substituting the value of (cos α):[ f(α) = -left(-frac{2sqrt{6}}{5}right) = frac{2sqrt{6}}{5} ]So, the value of (f(α)) is (frac{2sqrt{6}}{5}).Let me just recap to make sure I didn't make any mistakes.1. Simplified each trigonometric function using identities.2. Canceled out common terms in numerator and denominator to get (f(α) = -cos α).3. Used the given condition to find (sin α = -frac{1}{5}).4. Applied the Pythagorean identity to find (cos α = -frac{2sqrt{6}}{5}).5. Substituted back into (f(α)) to get the final result.Everything seems to check out. I think I'm confident with this solution.

💡Okay, so I have this problem where someone converted a proper fraction into a decimal, and during the calculation, the digits 1, 6, 7 appeared consecutively. The denominator of the fraction is less than 100. I need to prove that an error was made. Hmm, interesting.First, let me recall what a proper fraction is. A proper fraction is one where the numerator is less than the denominator. So, we have a fraction like (frac{p}{q}) where (p < q) and (q < 100). When converting this to a decimal, we perform long division of (p) by (q). The digits 1, 6, 7 appearing consecutively means that at some point in the decimal expansion, we have ...167...I need to show that this is impossible if the denominator is less than 100. So, maybe I can approach this by considering the properties of decimal expansions of fractions with denominators less than 100.Let me think about how decimal expansions work. When you divide a number by another, the decimal expansion either terminates or repeats. Since the denominator is less than 100, the decimal expansion will either terminate or become periodic with a period that divides the totient of the denominator or something like that.But in this case, the digits 1, 6, 7 appear consecutively. So, perhaps I can model the long division process and see if such a sequence can occur.Let me try to model the long division process step by step. Suppose we have a fraction (frac{p}{q}), and during the division, at some point, we get a remainder (r). Then, we multiply (r) by 10 to get the next digit. So, if we have the digits 1, 6, 7 appearing consecutively, that means:1. After some steps, we have a remainder (r_1) such that (10r_1 = q times 1 + r_2), so the digit 1 is obtained, and the new remainder is (r_2).2. Then, (10r_2 = q times 6 + r_3), so the digit 6 is obtained, and the new remainder is (r_3).3. Then, (10r_3 = q times 7 + r_4), so the digit 7 is obtained, and the new remainder is (r_4).So, we have the following equations:1. (10r_1 = q + r_2)2. (10r_2 = 6q + r_3)3. (10r_3 = 7q + r_4)Since all remainders must be less than (q), we have (0 leq r_1, r_2, r_3, r_4 < q).From the first equation, (10r_1 = q + r_2). Since (r_2 < q), we have (10r_1 < q + q = 2q), so (r_1 < frac{2q}{10} = frac{q}{5}).From the second equation, (10r_2 = 6q + r_3). Since (r_3 < q), we have (10r_2 < 6q + q = 7q), so (r_2 < frac{7q}{10}).From the third equation, (10r_3 = 7q + r_4). Since (r_4 < q), we have (10r_3 < 7q + q = 8q), so (r_3 < frac{8q}{10} = frac{4q}{5}).Now, let's try to express (r_1) in terms of (q). From the first equation, (r_2 = 10r_1 - q). Plugging this into the second equation:(10r_2 = 6q + r_3) (10(10r_1 - q) = 6q + r_3) (100r_1 - 10q = 6q + r_3) (100r_1 = 16q + r_3)From the third equation, (r_4 = 10r_3 - 7q). But (r_4 < q), so (10r_3 - 7q < q) (10r_3 < 8q) (r_3 < frac{8q}{10} = frac{4q}{5}), which we already have.From (100r_1 = 16q + r_3), since (r_3 < frac{4q}{5}), we have:(100r_1 < 16q + frac{4q}{5} = frac{80q + 4q}{5} = frac{84q}{5}) So, (r_1 < frac{84q}{500} = frac{21q}{125})But earlier, we had (r_1 < frac{q}{5}). Since (frac{21q}{125} = 0.168q) and (frac{q}{5} = 0.2q), the more restrictive condition is (r_1 < 0.168q).So, (r_1 < frac{21q}{125}).Now, let's see if we can find (q) such that these conditions hold.From (100r_1 = 16q + r_3), and (r_3 < frac{4q}{5}), we have:(100r_1 < 16q + frac{4q}{5} = frac{84q}{5}) So, (r_1 < frac{84q}{500} = frac{21q}{125}), as before.But also, from (10r_1 = q + r_2), and (r_2 < frac{7q}{10}), we have:(10r_1 < q + frac{7q}{10} = frac{17q}{10}) So, (r_1 < frac{17q}{100} = 0.17q)Wait, but earlier we had (r_1 < 0.168q), which is slightly more restrictive.So, combining these, (r_1 < 0.168q).Now, let's see if we can express (q) in terms of (r_1).From (100r_1 = 16q + r_3), and (r_3 < frac{4q}{5}), we can write:(16q < 100r_1 < 16q + frac{4q}{5}) So, (16q < 100r_1 < frac{84q}{5})Wait, that might not be helpful. Maybe I should try to find a relationship between (q) and (r_1).From (10r_1 = q + r_2), and (r_2 < frac{7q}{10}), we have:(10r_1 < q + frac{7q}{10} = frac{17q}{10}) So, (r_1 < frac{17q}{100})But we also have from (100r_1 = 16q + r_3), and (r_3 < frac{4q}{5}):(100r_1 < 16q + frac{4q}{5} = frac{84q}{5}) So, (r_1 < frac{84q}{500} = frac{21q}{125})So, combining these two inequalities:(frac{21q}{125} > r_1 > frac{q}{10}) (since (r_1) must be positive)Wait, no, (r_1) is a remainder, so (r_1) must be less than (q), but more specifically, from (10r_1 = q + r_2), since (r_2 < q), (10r_1 < q + q = 2q), so (r_1 < frac{2q}{10} = frac{q}{5}).So, (r_1 < frac{q}{5}).But from the other inequality, (r_1 < frac{21q}{125}), which is approximately (0.168q), which is less than (frac{q}{5} = 0.2q).So, (r_1 < 0.168q).Now, let's try to find (q) such that (r_1) is an integer, as remainders must be integers.So, (r_1) must satisfy (10r_1 = q + r_2), and (r_2 = 10r_1 - q).Similarly, (r_3 = 10r_2 - 6q = 10(10r_1 - q) - 6q = 100r_1 - 10q - 6q = 100r_1 - 16q).And (r_4 = 10r_3 - 7q = 10(100r_1 - 16q) - 7q = 1000r_1 - 160q - 7q = 1000r_1 - 167q).Since (r_4) must be less than (q), we have:(1000r_1 - 167q < q) (1000r_1 < 168q) (r_1 < frac{168q}{1000} = frac{42q}{250} = frac{21q}{125}), which is consistent with earlier.But also, (r_4) must be non-negative, so:(1000r_1 - 167q geq 0) (1000r_1 geq 167q) (r_1 geq frac{167q}{1000})So, combining with the previous inequality:(frac{167q}{1000} leq r_1 < frac{21q}{125})Let me compute these fractions:(frac{167q}{1000} = 0.167q) (frac{21q}{125} = 0.168q)So, (0.167q leq r_1 < 0.168q)But (r_1) must be an integer. So, (r_1) must be in the interval ([0.167q, 0.168q)). Since (r_1) is an integer, the length of this interval is (0.168q - 0.167q = 0.001q). For (r_1) to be an integer, this interval must contain at least one integer. But since the interval is only (0.001q) long, which is very small, especially since (q < 100), the interval is less than 0.1 in length (since (0.001 * 100 = 0.1)). Therefore, it's unlikely that there's an integer (r_1) in this interval unless (q) is very large, but (q < 100).Wait, but let's test this with actual numbers. Let's pick (q) values and see if such an (r_1) exists.Let me try (q = 97), which is just below 100.Compute (0.167 * 97 = 16.209) Compute (0.168 * 97 = 16.296)So, (r_1) must be between 16.209 and 16.296. Since (r_1) must be an integer, the only possible value is 16. But 16 is less than 16.209, so it doesn't satisfy the lower bound. Therefore, no integer (r_1) exists for (q = 97).Similarly, try (q = 99):(0.167 * 99 = 16.533) (0.168 * 99 = 16.632)So, (r_1) must be between 16.533 and 16.632. The only integer in this range is 17, but 17 is greater than 16.632, so no solution.Wait, actually, 16.533 to 16.632 is between 16 and 17, but 16.533 is greater than 16, so the only integer is 17, but 17 is outside the upper bound.Similarly, for (q = 98):(0.167 * 98 = 16.366) (0.168 * 98 = 16.464)So, (r_1) must be between 16.366 and 16.464. Again, the only integer is 16, but 16 is less than 16.366, so no solution.Wait, maybe I made a mistake here. Let me check (q = 125), but (q < 100), so that's not allowed.Wait, perhaps I need to consider smaller (q). Let's try (q = 50):(0.167 * 50 = 8.35) (0.168 * 50 = 8.4)So, (r_1) must be between 8.35 and 8.4. The only integer is 8, but 8 is less than 8.35, so no solution.Similarly, (q = 60):(0.167 * 60 = 10.02) (0.168 * 60 = 10.08)So, (r_1) must be between 10.02 and 10.08. The only integer is 10, but 10 is less than 10.02, so no solution.Wait, maybe I'm approaching this wrong. Let me think differently.From (1000r_1 - 167q geq 0), we have (1000r_1 geq 167q), so (q leq frac{1000r_1}{167}).But (q) must be an integer, so (q leq lfloor frac{1000r_1}{167} rfloor).Also, from (r_1 < frac{21q}{125}), we have (q > frac{125r_1}{21}).So, combining these:(frac{125r_1}{21} < q leq frac{1000r_1}{167})Let me compute the ratio (frac{1000}{167} / frac{125}{21}):(frac{1000}{167} / frac{125}{21} = frac{1000}{167} * frac{21}{125} = frac{1000 * 21}{167 * 125} = frac{21000}{20875} ≈ 1.006)So, the upper bound is just slightly larger than the lower bound, meaning that for each (r_1), there might be at most one (q) that satisfies the inequality.But since (q < 100), let's see for which (r_1) this is possible.Let me try (r_1 = 1):(frac{125 * 1}{21} ≈ 5.952 < q leq frac{1000 * 1}{167} ≈ 5.988)So, (q) must be between approximately 5.952 and 5.988. The only integer (q) is 6, but 6 is less than 5.988? No, 6 is greater than 5.988, so no solution.Wait, actually, 5.952 < q ≤ 5.988, so q must be 6, but 6 is greater than 5.988, so no solution.Similarly, (r_1 = 2):(frac{125 * 2}{21} ≈ 11.905 < q leq frac{1000 * 2}{167} ≈ 11.976)So, (q) must be between 11.905 and 11.976. The only integer is 12, but 12 is greater than 11.976, so no solution.Wait, this pattern seems to continue. Let me try (r_1 = 3):(frac{125 * 3}{21} ≈ 17.857 < q leq frac{1000 * 3}{167} ≈ 17.964)So, (q) must be between 17.857 and 17.964. The only integer is 18, but 18 is greater than 17.964, so no solution.Similarly, (r_1 = 4):(frac{125 * 4}{21} ≈ 23.81 < q leq frac{1000 * 4}{167} ≈ 23.952)So, (q) must be between 23.81 and 23.952. The only integer is 24, but 24 is greater than 23.952, so no solution.This pattern suggests that for each (r_1), the interval for (q) is just below an integer, but (q) must be an integer, so there's no solution.Wait, maybe I need to check higher (r_1). Let me try (r_1 = 17):(frac{125 * 17}{21} ≈ 100.714 < q leq frac{1000 * 17}{167} ≈ 101.808)But (q < 100), so this is beyond our limit.Wait, so for (r_1 = 17), (q) would have to be around 100, which is not allowed.So, it seems that for all (r_1), the required (q) is just below an integer, but (q) must be an integer, so there's no solution where (q < 100).Therefore, it's impossible to have the digits 1, 6, 7 appearing consecutively in the decimal expansion of a proper fraction with denominator less than 100. Hence, an error was made.

💡Alright, so I have this problem about a regular hexagon named ABCDEF. The midpoints of sides AB, CD, and EF are labeled as J, K, and L respectively. It's given that the area of triangle JKL is 100, and I need to find the area of the entire hexagon ABCDEF. Hmm, okay, let's try to break this down step by step.First, I remember that a regular hexagon can be divided into six equilateral triangles, all congruent, with each triangle having a vertex at the center of the hexagon. So, if I can find the area of one of these equilateral triangles, I can multiply it by six to get the total area of the hexagon. But how does triangle JKL fit into this?Well, J, K, and L are midpoints of their respective sides. So, triangle JKL is formed by connecting these midpoints. I think this triangle is also an equilateral triangle because the hexagon is regular, meaning all sides and angles are equal. So, triangle JKL should be equilateral as well.Now, to find the relationship between the area of triangle JKL and the area of the hexagon. Maybe I can figure out how big triangle JKL is compared to the hexagon. Let me visualize the hexagon and the triangle inside it.If I draw the regular hexagon and mark the midpoints J, K, and L, connecting them should form a smaller equilateral triangle inside the hexagon. I wonder how many smaller triangles like JKL fit into the hexagon. Maybe if I can find the ratio of their areas, I can use that to find the total area.Wait, perhaps it's better to assign a variable to the side length of the hexagon to make things more concrete. Let's say the side length of the hexagon is 's'. Since it's regular, all sides are equal, and all internal angles are 120 degrees. The area of a regular hexagon can be calculated using the formula:Area = (3√3 / 2) * s²So, if I can find 's', I can plug it into this formula to get the area. But how does triangle JKL help me find 's'?Well, triangle JKL is an equilateral triangle formed by connecting midpoints. So, each side of triangle JKL is equal to half the length of the side of the hexagon. Wait, is that right? Let me think.If J is the midpoint of AB, K is the midpoint of CD, and L is the midpoint of EF, then the distance between J and K should be equal to half the distance between A and D, right? Because J is halfway along AB, and K is halfway along CD. Similarly, the distance between K and L would be half the distance between C and F, and so on.But wait, in a regular hexagon, the distance between opposite sides is twice the side length times (√3 / 2), which is the height of each equilateral triangle. So, maybe the side length of triangle JKL is related to the side length of the hexagon in a specific ratio.Alternatively, maybe I can use coordinate geometry to find the coordinates of J, K, and L, and then calculate the area of triangle JKL in terms of 's'. That might be a more precise approach.Let's place the regular hexagon on a coordinate system with its center at the origin (0,0). Let's assume the hexagon is oriented such that one of its vertices is at (s, 0). Then, the coordinates of the vertices can be determined using trigonometric functions.The coordinates of the six vertices of the hexagon can be given by:A: (s, 0)B: (s/2, (s√3)/2)C: (-s/2, (s√3)/2)D: (-s, 0)E: (-s/2, -(s√3)/2)F: (s/2, -(s√3)/2)Okay, so now I can find the coordinates of the midpoints J, K, and L.Point J is the midpoint of AB. So, the coordinates of J would be the average of the coordinates of A and B.J_x = (s + s/2) / 2 = (3s/2) / 2 = 3s/4J_y = (0 + (s√3)/2) / 2 = (s√3)/4Wait, hold on. That doesn't seem right. Let me recalculate.Actually, midpoint formula is ( (x1 + x2)/2, (y1 + y2)/2 )So, for point J, midpoint of AB:A is (s, 0), B is (s/2, (s√3)/2)So,J_x = (s + s/2)/2 = (3s/2)/2 = 3s/4J_y = (0 + (s√3)/2)/2 = (s√3)/4Okay, that seems correct.Similarly, point K is the midpoint of CD.C is (-s/2, (s√3)/2), D is (-s, 0)So,K_x = (-s/2 + (-s))/2 = (-3s/2)/2 = -3s/4K_y = ((s√3)/2 + 0)/2 = (s√3)/4And point L is the midpoint of EF.E is (-s/2, -(s√3)/2), F is (s/2, -(s√3)/2)So,L_x = (-s/2 + s/2)/2 = 0/2 = 0L_y = (-(s√3)/2 + -(s√3)/2)/2 = (-s√3)/2Wait, that doesn't seem right. Let me check.E is (-s/2, -(s√3)/2), F is (s/2, -(s√3)/2)So,L_x = (-s/2 + s/2)/2 = 0/2 = 0L_y = (-(s√3)/2 + -(s√3)/2)/2 = (-s√3)/2Wait, that's correct. So, point L is at (0, -(s√3)/2)So, now we have the coordinates of J, K, and L:J: (3s/4, s√3/4)K: (-3s/4, s√3/4)L: (0, -s√3/2)Now, let's plot these points or at least visualize them. Points J and K are on the upper half of the hexagon, symmetric about the y-axis, and point L is at the bottom center.So, triangle JKL is formed by these three points. To find its area, we can use the shoelace formula or calculate the lengths of the sides and then use the formula for the area of an equilateral triangle.Wait, is triangle JKL equilateral? Let me check the distances between J, K, and L.First, distance between J and K.J: (3s/4, s√3/4)K: (-3s/4, s√3/4)Distance JK:√[( (3s/4 - (-3s/4))² + (s√3/4 - s√3/4)² )]Simplify:√[( (6s/4)² + 0 )] = √[( (3s/2)² )] = 3s/2Okay, so distance JK is 3s/2.Now, distance between K and L.K: (-3s/4, s√3/4)L: (0, -s√3/2)Distance KL:√[( (-3s/4 - 0)² + (s√3/4 - (-s√3/2))² )]Simplify:√[( (9s²/16) + ( (s√3/4 + s√3/2 )² ) ]First, compute the y-component:s√3/4 + s√3/2 = s√3/4 + 2s√3/4 = 3s√3/4So, the distance KL:√[ (9s²/16) + ( (3s√3/4)² ) ] = √[ 9s²/16 + 27s²/16 ) ] = √[36s²/16] = √[9s²/4] = 3s/2Okay, so distance KL is also 3s/2.Now, distance between L and J.L: (0, -s√3/2)J: (3s/4, s√3/4)Distance LJ:√[( (3s/4 - 0)² + (s√3/4 - (-s√3/2))² )]Simplify:√[ (9s²/16) + ( (s√3/4 + s√3/2 )² ) ]Again, compute the y-component:s√3/4 + s√3/2 = 3s√3/4So, distance LJ:√[ 9s²/16 + (27s²/16) ] = √[36s²/16] = √[9s²/4] = 3s/2So, all sides of triangle JKL are equal to 3s/2, which confirms that triangle JKL is indeed equilateral.Now, the area of an equilateral triangle is given by:Area = (√3 / 4) * (side length)²So, the area of triangle JKL is:(√3 / 4) * (3s/2)² = (√3 / 4) * (9s²/4) = (9√3 s²) / 16According to the problem, this area is 100. So,(9√3 s²) / 16 = 100We can solve for s²:s² = (100 * 16) / (9√3) = (1600) / (9√3)But we need to find the area of the hexagon, which is:Area_hexagon = (3√3 / 2) * s²Substituting s² from above:Area_hexagon = (3√3 / 2) * (1600 / (9√3)) = (3√3 / 2) * (1600 / (9√3))Simplify:The √3 in the numerator and denominator cancels out:= (3 / 2) * (1600 / 9) = (3 * 1600) / (2 * 9) = (4800) / (18) = 266.666...Wait, 4800 divided by 18 is equal to 266.666..., which is 800/3.So, the area of the hexagon is 800/3.But let me double-check my calculations to make sure I didn't make a mistake.First, area of triangle JKL:(√3 / 4) * (3s/2)² = (√3 / 4) * (9s²/4) = (9√3 s²)/16 = 100So, s² = (100 * 16) / (9√3) = 1600 / (9√3)Then, area of hexagon:(3√3 / 2) * s² = (3√3 / 2) * (1600 / (9√3)) = (3√3 * 1600) / (2 * 9√3) = (4800√3) / (18√3) = 4800 / 18 = 266.666...Yes, that's 800/3.So, the area of the hexagon is 800/3.I think that makes sense. Triangle JKL has an area of 100, and through the calculations, we found that the hexagon's area is 800/3, which is approximately 266.67.Just to make sure, let's think about the ratio of areas. The triangle JKL is formed by connecting midpoints, so it's similar to the hexagon but scaled down. The ratio of areas is (800/3)/100 = 8/3, which is approximately 2.666...But since triangle JKL is inside the hexagon, it's smaller, so the hexagon is larger than the triangle. The ratio makes sense because the hexagon is made up of multiple smaller triangles, one of which is JKL.Alternatively, another way to think about it is that the area of the hexagon is 6 times the area of one of the equilateral triangles that make it up. Each of those has an area of (√3 / 4) * s², so 6 times that is (3√3 / 2) * s², which matches the formula I used earlier.Given that, and the calculations leading to 800/3, I feel confident that the area of the hexagon is indeed 800/3.**Final Answer**The area of the regular hexagon (ABCDEF) is (boxed{dfrac{800}{3}}).

💡First, I need to convert the base 8 number (543_8) to base 10. To do this, I'll break down each digit and multiply it by (8) raised to the power of its position, starting from 0 on the right.The number (543_8) has three digits:- The rightmost digit is 3, which is in the (8^0) position.- The middle digit is 4, which is in the (8^1) position.- The leftmost digit is 5, which is in the (8^2) position.Calculating each term:- (3 times 8^0 = 3 times 1 = 3)- (4 times 8^1 = 4 times 8 = 32)- (5 times 8^2 = 5 times 64 = 320)Adding these values together gives the base 10 equivalent:(3 + 32 + 320 = 355)Therefore, the bird discovered 355 different ways to build a nest in base 10.

💡Alright, so I have this problem here: "The sum of a positive number and its square is 210. What is the number?" Hmm, okay, let's see how to approach this.First, I need to understand what the problem is asking. It says that if I take a positive number and add its square to it, the result is 210. So, I need to find that positive number. Let's denote this number as ( x ). So, according to the problem, ( x + x^2 = 210 ).Now, I can write this as an equation: ( x^2 + x = 210 ). To solve for ( x ), I should rearrange this equation into a standard quadratic form. That means moving all terms to one side so that the equation equals zero. So, subtracting 210 from both sides gives me ( x^2 + x - 210 = 0 ).Okay, now I have a quadratic equation: ( x^2 + x - 210 = 0 ). Quadratic equations are of the form ( ax^2 + bx + c = 0 ), and I can solve them using various methods like factoring, completing the square, or the quadratic formula. Let's see if this equation can be factored easily.To factor ( x^2 + x - 210 ), I need two numbers that multiply to -210 and add up to 1 (the coefficient of the ( x ) term). Hmm, let's think about the factors of 210. 210 is a product of 2, 3, 5, and 7. So, the positive factors are 1, 2, 3, 5, 6, 7, 10, 14, 15, 21, 30, 35, 42, 70, 105, and 210.Since the product is negative, one of the numbers must be positive and the other negative. I'm looking for two numbers whose difference is 1 because their sum is 1. Let's see:- 15 and 14: 15 * 14 = 210, and 15 - 14 = 1. But since the product is negative, one should be positive and the other negative. So, if I take 15 and -14, their product is -210, and their sum is 1. Perfect!So, I can factor the quadratic as ( (x + 15)(x - 14) = 0 ). Setting each factor equal to zero gives the potential solutions:1. ( x + 15 = 0 ) → ( x = -15 )2. ( x - 14 = 0 ) → ( x = 14 )But the problem specifies that the number is positive, so ( x = -15 ) is not a valid solution. Therefore, the number must be 14.Wait a second, let me double-check my factoring to make sure I didn't make a mistake. If I expand ( (x + 15)(x - 14) ), I should get ( x^2 -14x +15x -210 ), which simplifies to ( x^2 +x -210 ). Yes, that matches the original equation, so my factoring is correct.Just to be thorough, I'll plug ( x = 14 ) back into the original equation to verify:( 14 + (14)^2 = 14 + 196 = 210 ). Yep, that works out perfectly.So, after going through the steps of setting up the equation, factoring, solving for ( x ), and verifying the solution, I'm confident that the positive number we're looking for is 14.

💡Alright, so I've got this problem about 1000 toy bears in a row, each colored white, brown, or black. The key condition is that among any three consecutive bears, there's at least one of each color. Iskander made five guesses about specific bears' colors, and exactly one of them is wrong. I need to figure out which guess is incorrect.First, let me list out the guesses:1. The 2nd bear is white.2. The 20th bear is brown.3. The 400th bear is black.4. The 600th bear is brown.5. The 800th bear is white.So, exactly one of these is wrong. My task is to determine which one could be the incorrect guess.Given the condition that among any three consecutive bears, there's at least one of each color, this suggests a repeating pattern or some kind of periodicity in the colors. Maybe every three bears repeat the same color sequence? Or perhaps there's a cycle of colors that repeats every few bears.Let me think about the positions of the bears and their colors. If I can figure out a pattern or a rule that the colors must follow, I can check which of Iskander's guesses violate that rule.Since the condition applies to any three consecutive bears, it implies that no three consecutive bears can have the same color or even two of the same color without the third being different. So, for example, you can't have two whites in a row without a different color in between, and similarly for the other colors.This makes me think that the colors must follow a specific repeating sequence to satisfy the condition. Maybe something like white, brown, black, white, brown, black, and so on. Or perhaps a different permutation of the three colors.If that's the case, then the color of each bear depends on its position modulo 3. That is, the color repeats every three bears. So, if I can figure out the color for position 1, 2, and 3, I can determine the colors for all positions based on their modulo 3 value.Let's test this idea. Suppose the colors repeat every three bears. Then, the color of the nth bear depends on n modulo 3. For example:- If n ≡ 1 mod 3, then the color is white.- If n ≡ 2 mod 3, then the color is brown.- If n ≡ 0 mod 3, then the color is black.Of course, the actual colors could be in a different order, but the key is that the color depends on the position modulo 3.Given this, let's calculate the modulo 3 for each of the guessed positions:1. 2nd bear: 2 mod 3 = 22. 20th bear: 20 mod 3 = 23. 400th bear: 400 mod 3 = 14. 600th bear: 600 mod 3 = 05. 800th bear: 800 mod 3 = 2So, positions 2, 20, and 800 all have the same modulo 3 value of 2. According to the repeating pattern idea, these should all have the same color. However, Iskander guessed that the 2nd and 800th bears are white, while the 20th bear is brown. This is a contradiction because if positions 2, 20, and 800 have the same modulo 3 value, they should all be the same color.Therefore, at least one of these three guesses must be wrong. But we know that exactly one of all five guesses is wrong. So, if positions 2, 20, and 800 are supposed to be the same color, and two of them are guessed as white and one as brown, then the one that's brown must be the incorrect guess.So, the 20th bear being brown is likely the incorrect guess. Let me verify this.If I assume that the 20th bear is actually white, then all three positions (2, 20, 800) would be white, which fits the repeating pattern. Then, the other guesses can be checked for consistency.Looking at the 400th bear, which is position 400. 400 mod 3 is 1. If the color for modulo 1 is, say, black, then the 400th bear being black is correct. Similarly, the 600th bear is position 600, which is 0 mod 3. If the color for modulo 0 is brown, then the 600th bear being brown is correct.So, if I adjust the 20th bear to be white instead of brown, all the other guesses fit into a consistent repeating pattern where:- Modulo 1: Black- Modulo 2: White- Modulo 0: BrownThis satisfies the condition that among any three consecutive bears, there is at least one of each color. For example, positions 1 (black), 2 (white), 3 (brown) would have all three colors, and this pattern would repeat every three bears.Therefore, the incorrect guess is the 20th bear being brown. It should be white to maintain the repeating pattern and satisfy the condition.I think this makes sense. The key was recognizing the repeating pattern based on modulo 3 and checking for consistency among the guesses. Since positions 2, 20, and 800 share the same modulo value, their colors should align, and the discrepancy pointed to the 20th bear as the incorrect guess.