💡Okay, so I have this function here: ( g(x) = sqrt{2 - sqrt{4 - sqrt{5 - x}}} ). I need to find its domain. Hmm, the domain of a function is basically all the real numbers x for which the function is defined. Since this function has multiple square roots, I think I need to make sure that everything inside each square root is non-negative because you can't take the square root of a negative number in the set of real numbers.Let me start from the innermost square root and work my way out. The innermost part is ( sqrt{5 - x} ). For this to be defined, the expression inside the square root, which is ( 5 - x ), must be greater than or equal to zero. So, I can write that as:( 5 - x geq 0 )Solving for x, I subtract 5 from both sides:( -x geq -5 )Then, I multiply both sides by -1, remembering to flip the inequality sign because multiplying by a negative number reverses the inequality:( x leq 5 )Okay, so that gives me one condition: x has to be less than or equal to 5.Now, moving one layer out, the next square root is ( sqrt{4 - sqrt{5 - x}} ). For this to be defined, the expression inside it, which is ( 4 - sqrt{5 - x} ), must also be greater than or equal to zero. So:( 4 - sqrt{5 - x} geq 0 )Let me solve this inequality. I can rearrange it to:( sqrt{5 - x} leq 4 )Now, since the square root function always gives a non-negative result, ( sqrt{5 - x} ) is already greater than or equal to zero. So, this inequality tells me that ( sqrt{5 - x} ) must be less than or equal to 4. To find the values of x that satisfy this, I can square both sides of the inequality because both sides are non-negative, and squaring preserves the inequality.Squaring both sides:( (sqrt{5 - x})^2 leq 4^2 )Simplifying:( 5 - x leq 16 )Now, solving for x:Subtract 5 from both sides:( -x leq 11 )Multiply both sides by -1 and reverse the inequality:( x geq -11 )Alright, so now I have another condition: x has to be greater than or equal to -11.So far, combining these two conditions, I have:( -11 leq x leq 5 )But wait, I need to check the outermost square root as well. The outermost function is ( sqrt{2 - sqrt{4 - sqrt{5 - x}}} ). For this to be defined, the expression inside it, which is ( 2 - sqrt{4 - sqrt{5 - x}} ), must be greater than or equal to zero. So:( 2 - sqrt{4 - sqrt{5 - x}} geq 0 )Let me solve this inequality. I can rearrange it to:( sqrt{4 - sqrt{5 - x}} leq 2 )Again, since the square root is non-negative, I can square both sides without worrying about reversing the inequality:( (sqrt{4 - sqrt{5 - x}})^2 leq 2^2 )Simplifying:( 4 - sqrt{5 - x} leq 4 )Subtract 4 from both sides:( -sqrt{5 - x} leq 0 )Multiply both sides by -1, which reverses the inequality:( sqrt{5 - x} geq 0 )But wait, ( sqrt{5 - x} ) is always greater than or equal to zero because square roots can't be negative. So, this inequality doesn't give me any new information. It's always true as long as the expression inside the square root is defined, which we've already handled in the first step.Therefore, the only conditions I have are from the first two square roots: ( -11 leq x leq 5 ).Let me just double-check to make sure I haven't missed anything. Starting from the innermost:1. ( 5 - x geq 0 ) gives ( x leq 5 ).2. ( 4 - sqrt{5 - x} geq 0 ) gives ( x geq -11 ).3. ( 2 - sqrt{4 - sqrt{5 - x}} geq 0 ) doesn't add any new constraints.So, combining these, the domain of g(x) is all real numbers x such that ( -11 leq x leq 5 ).I think that's it. I don't see any other constraints. Let me just visualize the function to make sure. Starting from the inside, as x increases from -11 to 5, ( 5 - x ) decreases from 16 to 0, so ( sqrt{5 - x} ) decreases from 4 to 0. Then, ( 4 - sqrt{5 - x} ) increases from 0 to 4, so ( sqrt{4 - sqrt{5 - x}} ) increases from 0 to 2. Finally, ( 2 - sqrt{4 - sqrt{5 - x}} ) decreases from 2 to 0, which is always non-negative in this interval. So, yes, everything seems to check out.I don't think I made any mistakes here. The key was to work from the innermost square root outward, ensuring each expression inside the square roots is non-negative, and then combining the resulting inequalities. The outermost square root didn't impose any additional restrictions beyond what was already found.So, I feel confident that the domain of g(x) is from -11 to 5, inclusive.**Final Answer**The domain of the function is boxed{[-11, 5]}.

💡First, I need to identify which part of the ratio corresponds to Summer. Since Summer has the highest number of degrees and the ratio is 5:4:6:3, the highest number in the ratio is 6. Therefore, Summer corresponds to 6 parts.Next, I know that Summer has a total of 150 degrees. To find the value of one part, I divide Summer's degrees by the number of parts she represents: 150 degrees ÷ 6 parts = 25 degrees per part.Now, I can determine the number of degrees each person has by multiplying their respective ratio parts by 25 degrees per part:- Summer: 6 parts × 25 degrees/part = 150 degrees- Jolly: 4 parts × 25 degrees/part = 100 degrees- Winter: 5 parts × 25 degrees/part = 125 degrees- Autumn: 3 parts × 25 degrees/part = 75 degreesFinally, to find the combined number of degrees, I add up all their degrees:150 degrees + 100 degrees + 125 degrees + 75 degrees = 450 degrees

💡Okay, so I have this problem about light reflecting off two mirrors that form an angle α. The goal is to prove that if α is equal to 90 degrees divided by an integer n, then any light ray will eventually travel in the opposite direction after several reflections. Hmm, interesting. I remember that when light reflects off a mirror, the angle of incidence equals the angle of reflection, kind of like how a billiard ball bounces off the sides of a table. Maybe I can use some similar reasoning here.First, let me visualize the setup. There are two straight mirrors forming an angle α. So, imagine two lines intersecting at a point, creating an angle α between them. A light ray comes in and reflects off one mirror, then off the other, and so on. I need to figure out how the direction of the light changes with each reflection.I think it might help to model the reflections using some coordinate system or maybe using angles. Let me denote the initial direction of the light ray as some angle θ with respect to one of the mirrors. When it reflects off the first mirror, the angle of incidence is θ, so the angle of reflection is also θ. That means the light ray's direction changes by 2θ relative to its original path, right?Wait, no, maybe I should think about it differently. If the light ray hits the first mirror at an angle θ, then after reflection, it will be traveling at an angle θ on the other side of the normal. So, the total change in direction is 2θ. But since the mirrors form an angle α, the light ray will then approach the second mirror at some angle, and reflect again.This seems a bit complicated. Maybe I can use the concept of multiple reflections and see how the angle changes each time. If I can find a pattern or a periodicity, that might help me figure out when the light ray will reverse its direction.Let me consider the case where α is 90 degrees divided by an integer n. So, α = 90° / n. For example, if n = 1, α is 90°, which is like two mirrors perpendicular to each other. If n = 2, α is 45°, and so on. I wonder how the number of reflections affects the direction of the light ray in these cases.Maybe I can model the reflections as transformations of the angle. Each reflection changes the angle by some amount, and after a certain number of reflections, the total change should be 180 degrees to reverse the direction. Let me try to formalize this.Suppose the light ray starts at some angle θ with respect to the first mirror. After reflecting off the first mirror, it changes direction by 2θ. Then, it approaches the second mirror, which is at an angle α from the first. The angle of incidence on the second mirror would be θ - α, right? Because the light ray has already changed direction by 2θ, and the second mirror is at an angle α.Wait, no, maybe I need to think about the angle between the light ray and the second mirror. If the first mirror is at angle 0°, and the second mirror is at angle α, then the light ray, after reflecting off the first mirror, is traveling at an angle 2θ from the first mirror. So, the angle between the light ray and the second mirror would be α - 2θ.Hmm, this is getting a bit tangled. Maybe I should use a different approach. I remember that in billiards, when a ball reflects off a cushion, the angle of incidence equals the angle of reflection, and sometimes you can model the table as being reflected instead of the ball. Maybe I can use a similar idea here.If I reflect the mirrors instead of the light ray, it might simplify the problem. So, instead of thinking about the light ray bouncing off the mirrors, I can think of the mirrors being reflected, and the light ray traveling straight through these reflections. This way, the problem becomes about the light ray traveling in a straight line through a series of mirrored rooms.If I do that, then the angle between the mirrors becomes important in determining how the reflections tile the plane. If the angle α is 90° / n, then the number of reflections needed to make a full 360° rotation would be related to n. Specifically, the number of reflections needed to reverse the direction would be 2n, because each reflection contributes a certain angle change.Let me try to calculate the total angle change after several reflections. If each reflection off a mirror changes the direction by 2α, then after k reflections, the total change would be 2kα. We want this total change to be 180°, so that the light ray is traveling in the opposite direction. Therefore, we need 2kα = 180°, which simplifies to kα = 90°. Since α = 90° / n, substituting gives k * (90° / n) = 90°, so k = n. Therefore, after n reflections, the light ray would have changed direction by 180°, reversing its path.Wait, but I thought earlier that it would take 2n reflections. Maybe I made a mistake in counting. Let me double-check. If each reflection off a mirror contributes 2α to the total angle change, then after n reflections, the total change is 2nα. But since α = 90° / n, then 2nα = 2n * (90° / n) = 180°, which is correct. So, after n reflections, the light ray has reversed direction.But hold on, in reality, the light ray reflects off both mirrors alternately. So, each pair of reflections (one off each mirror) contributes a total angle change of 2α + 2α = 4α? Or is it different?No, actually, each reflection off a mirror contributes 2α to the total angle change relative to the original direction. So, if the light ray reflects off the first mirror, changing direction by 2α, then reflects off the second mirror, changing direction by another 2α, the total change after two reflections is 4α. Continuing this way, after k reflections, the total change is 2kα.But wait, that doesn't seem right because the angle between the mirrors is α, so the reflections are not independent. Maybe I need to consider the angle between the mirrors more carefully.Let me think of the mirrors as two lines intersecting at angle α. The light ray reflects off one mirror, then the other, and so on. Each reflection can be thought of as a rotation of the light ray's direction by 2α. So, after each reflection, the direction of the light ray is rotated by 2α. Therefore, after k reflections, the total rotation is 2kα.We want this total rotation to be 180°, so that the light ray is traveling in the opposite direction. Therefore, 2kα = 180°, which simplifies to kα = 90°. Since α = 90° / n, substituting gives k * (90° / n) = 90°, so k = n. Therefore, after n reflections, the light ray has been rotated by 180°, reversing its direction.But wait, in reality, the light ray alternates between reflecting off the two mirrors. So, each reflection alternates between the two mirrors, each contributing a rotation of 2α. Therefore, after n reflections, regardless of which mirror it reflects off, the total rotation is 2nα. Since α = 90° / n, 2nα = 180°, which is the desired reversal.Therefore, the light ray will reverse direction after n reflections. But the problem states "after several reflections from both mirrors," so it's not specifying whether it's n reflections or 2n reflections. But from the calculation, it seems that n reflections are sufficient.However, I need to ensure that this works for any initial direction. The key here is that the angle α is a divisor of 90°, meaning that the reflections will cycle through a finite number of directions before repeating. Since 90° is a quarter of a full circle, and α divides it into n equal parts, the reflections will cycle through n different directions before returning to the original direction. But we want the opposite direction, which is halfway through the cycle.Therefore, after n reflections, the light ray will have cycled halfway through the n directions, resulting in the opposite direction. Hence, it will eventually travel in the direction opposite to its initial direction.Conversely, if α is not equal to 90° / n for some integer n, then the reflections will not cycle through a finite number of directions, and the light ray may never exactly reverse its direction. It might get arbitrarily close, but never exactly opposite.So, putting it all together, if α is equal to 90° divided by an integer n, then after n reflections, the light ray will have reversed its direction. This is because the total rotation after n reflections is 180°, which is half of a full circle, effectively reversing the direction. If α is not of this form, the light ray won't reverse its direction exactly after any finite number of reflections.I think this makes sense. The key idea is that the angle between the mirrors must divide evenly into 90°, allowing the reflections to cycle through a finite number of directions, including the opposite direction. If the angle doesn't divide evenly, the reflections won't align to reverse the direction exactly.Let me try to summarize the steps:1. The light ray reflects off each mirror, changing its direction by 2α each time.2. After k reflections, the total change in direction is 2kα.3. To reverse the direction, we need 2kα = 180°, so kα = 90°.4. Given α = 90° / n, substituting gives k = n.5. Therefore, after n reflections, the light ray will have reversed direction.This seems to cover the proof. I should also consider whether this works for any initial direction, which it does because the angle change is consistent regardless of the initial angle θ. The reflections will always contribute a fixed rotation, leading to the reversal after n steps.I think I've got it. The crucial point is that the angle α must be a divisor of 90°, allowing the reflections to cycle through a finite number of directions, including the opposite one. If α isn't of that form, the light ray won't reverse direction exactly.

💡Alright, so I have this problem where I need to figure out the relationship between three values: a, b, and c. The problem gives me expressions for each of them, and I need to determine which one is bigger than the others. The options are A, B, C, or D, each with a different order. Let me try to break this down step by step.First, let's look at what each of a, b, and c is equal to.Starting with a: it's given as cos17° multiplied by cos23° minus sin17° multiplied by sin23°. Hmm, that looks familiar. I think that's the formula for the cosine of a sum of two angles. Let me recall: cos(A + B) equals cosA cosB minus sinA sinB. Yes, that's right! So, a is actually cos(17° + 23°). Let me calculate that sum: 17 plus 23 is 40. So, a is equal to cos40°. Okay, that simplifies things a bit.Next, let's look at b: it's given as 2 times cos²25° minus 1. Hmm, that also seems familiar. I think that's one of the double-angle formulas. Let me recall: cos2θ equals 2cos²θ minus 1. So, if I let θ be 25°, then 2cos²25° minus 1 is equal to cos50°. Therefore, b is equal to cos50°. That makes sense.Lastly, c is given as sqrt(3)/2. I remember that sqrt(3)/2 is the cosine of 30°, because cos30° equals sqrt(3)/2. So, c is equal to cos30°. Got it.So now, I have a, b, and c expressed in terms of cosines of different angles:- a = cos40°- b = cos50°- c = cos30°Now, I need to figure out the relationship between a, b, and c. Since all of them are cosine values, and cosine is a function that decreases as the angle increases from 0° to 90°, I can use this property to compare them.Let me recall: in the first quadrant (angles from 0° to 90°), the cosine function decreases as the angle increases. That means that as the angle gets larger, the cosine of that angle gets smaller. So, cos0° is 1, cos30° is about 0.866, cos45° is about 0.707, cos60° is 0.5, and cos90° is 0. So, the larger the angle, the smaller the cosine value.Given that, let's list the angles we have:- c is cos30°, which is 30°- a is cos40°, which is 40°- b is cos50°, which is 50°Since 30° is less than 40°, which is less than 50°, and cosine decreases as the angle increases, the cosine values will be in the reverse order. So, cos30° is greater than cos40°, which is greater than cos50°. Therefore, c > a > b.Let me double-check to make sure I didn't make a mistake. So, a is cos40°, which is approximately 0.766. b is cos50°, which is approximately 0.6428. c is cos30°, which is approximately 0.866. So, 0.866 > 0.766 > 0.6428, which confirms that c > a > b.Therefore, the correct order is c > a > b, which corresponds to option C.I think that's solid. I used the cosine addition formula and the double-angle formula to simplify a and b, recognized the cosine values for c, and then used the property of the cosine function decreasing in the first quadrant to order them. Everything seems to check out.**Final Answer**boxed{C}

💡Okay, so I have this problem about a bacterial colony growing in a petri dish. Every day, the colony increases by 50% of its size from the previous day. It fills the entire dish in 40 days. I need to find out on which day the colony was only 10% of the dish filled.Hmm, let's break this down. The colony grows by 50% each day. That means every day, the size is 1.5 times the size of the previous day. So, if I denote the size of the colony on day n as S_n, then S_n = 1.5 * S_{n-1}.Given that on day 40, the colony fills the entire dish, so S_40 = 100% (let's say the dish is 100% filled on day 40). I need to find the day when the colony was only 10% of the dish.Since the growth is exponential, I can work backwards from day 40 to find when the colony was 10%. If on day 40 it's 100%, then on day 39, it must have been 100% / 1.5, because it grows by 50% each day. Similarly, on day 38, it would be (100% / 1.5) / 1.5, and so on.Let me write this as a formula. If S_n = 1.5 * S_{n-1}, then S_{n-1} = S_n / 1.5. So, starting from day 40, I can keep dividing by 1.5 to go back each day.Let me calculate this step by step:- Day 40: 100%- Day 39: 100% / 1.5 ≈ 66.67%- Day 38: 66.67% / 1.5 ≈ 44.44%- Day 37: 44.44% / 1.5 ≈ 29.63%- Day 36: 29.63% / 1.5 ≈ 19.75%- Day 35: 19.75% / 1.5 ≈ 13.17%- Day 34: 13.17% / 1.5 ≈ 8.78%Okay, so on day 34, the colony is approximately 8.78%, which is less than 10%. On day 35, it's about 13.17%, which is more than 10%. So, the colony was approximately 10% filled on day 35.Wait, but the question asks for the day when it was only 10%. Since on day 34 it's below 10% and on day 35 it's above 10%, the exact day when it was 10% would be somewhere between day 34 and day 35. But since we're dealing with whole days, we can say that on day 35, it was just over 10%, so that's the day when it reached 10%.But let me double-check my calculations to make sure I didn't make a mistake.Starting from day 40:- Day 40: 100%- Day 39: 100 / 1.5 ≈ 66.6667%- Day 38: 66.6667 / 1.5 ≈ 44.4444%- Day 37: 44.4444 / 1.5 ≈ 29.6296%- Day 36: 29.6296 / 1.5 ≈ 19.7531%- Day 35: 19.7531 / 1.5 ≈ 13.1687%- Day 34: 13.1687 / 1.5 ≈ 8.7791%Yes, that seems consistent. So, on day 35, it's approximately 13.17%, which is just over 10%. Therefore, the day when the colony was approximately 10% filled is day 35.I think that makes sense because exponential growth means that the colony grows rapidly as it approaches the full capacity. So, the 10% mark would be relatively close to the day when it fills the dish.Just to be thorough, maybe I can set up an equation to solve for the exact day when the colony is 10%. Let's denote the initial size on day 0 as S_0. Then, on day n, the size is S_n = S_0 * (1.5)^n.We know that on day 40, S_40 = S_0 * (1.5)^40 = 100%. We want to find n such that S_n = 10%.So, 10% = S_0 * (1.5)^nBut we also know that S_0 * (1.5)^40 = 100%, so S_0 = 100% / (1.5)^40Substituting back into the equation for S_n:10% = (100% / (1.5)^40) * (1.5)^nSimplify:10% = 100% * (1.5)^{n - 40}Divide both sides by 100%:0.1 = (1.5)^{n - 40}Now, take the natural logarithm of both sides:ln(0.1) = (n - 40) * ln(1.5)Solve for n:n - 40 = ln(0.1) / ln(1.5)n = 40 + (ln(0.1) / ln(1.5))Calculate the value:ln(0.1) ≈ -2.302585093ln(1.5) ≈ 0.4054651081So,n ≈ 40 + (-2.302585093 / 0.4054651081)n ≈ 40 - 5.68n ≈ 34.32So, approximately 34.32 days. Since we can't have a fraction of a day, and on day 34 it's below 10% and on day 35 it's above 10%, the day when it was approximately 10% is day 35.Therefore, the answer is day 35.

💡Okay, so I have this function f(x) = tan(2x + π/3), and I need to figure out which of the four propositions about it are correct. Let me take it step by step.First, I remember that the tangent function, tan(θ), has a period of π. That means it repeats every π units. But in this case, the function is tan(2x + π/3). The 2x inside the tangent affects the period. I think the period of tan(Bx + C) is π/|B|, so here, B is 2. So, the period should be π/2. That means the function repeats every π/2 units. So, the period is π/2. I'll note that down.Now, looking at proposition (1): If x₁ and x₂ are two distinct roots of f(x) = 3, then |x₁ - x₂| ≥ π/2. Hmm, okay. So, roots of f(x) = 3 would be the solutions to tan(2x + π/3) = 3. Let me solve that equation.tan(θ) = 3 implies θ = arctan(3) + kπ, where k is any integer. So, substituting back, 2x + π/3 = arctan(3) + kπ. Solving for x, we get x = (arctan(3) + kπ - π/3)/2. So, the solutions are x = (arctan(3) - π/3)/2 + kπ/2.So, the roots are spaced π/2 apart. That means the difference between any two distinct roots is at least π/2. So, |x₁ - x₂| is exactly π/2 times some integer. Since they are distinct, the smallest difference is π/2. So, |x₁ - x₂| ≥ π/2. That makes proposition (1) correct.Moving on to proposition (2): (-π/6, 0) is the symmetry center of the function f(x). Hmm, symmetry center. For tangent functions, I remember that they are symmetric about their points of inflection, which are the points where the function crosses the x-axis. So, the centers of symmetry are the points where the function has vertical asymptotes shifted by half the period.Wait, let me think. The general form is tan(Bx + C). The symmetry centers occur at the midpoints between consecutive vertical asymptotes. The vertical asymptotes of tan(2x + π/3) occur where 2x + π/3 = π/2 + kπ, so solving for x, x = (π/2 - π/3 + kπ)/2 = (π/6 + kπ)/2 = π/12 + kπ/2.So, the vertical asymptotes are at x = π/12 + kπ/2. The midpoints between these asymptotes would be halfway between π/12 and π/12 + π/2, which is π/12 + π/4 = (π/12 + 3π/12) = 4π/12 = π/3. Wait, that doesn't seem right. Let me recast it.Wait, the vertical asymptotes are at x = π/12 + kπ/2. So, the first asymptote is at x = π/12, then the next is at x = π/12 + π/2 = 7π/12, then 13π/12, etc. The midpoint between π/12 and 7π/12 is (π/12 + 7π/12)/2 = (8π/12)/2 = (2π/3)/2 = π/3. So, the symmetry center is at (π/3, 0). But proposition (2) says (-π/6, 0). Hmm, that doesn't match.Wait, maybe I made a mistake. Let's see. Let me solve for the points where the function crosses the x-axis. That would be where tan(2x + π/3) = 0, which is when 2x + π/3 = kπ, so x = (kπ - π/3)/2. For k=0, x = -π/6. For k=1, x = (π - π/3)/2 = (2π/3)/2 = π/3. For k=2, x = (2π - π/3)/2 = (5π/3)/2 = 5π/6. So, the function crosses the x-axis at x = -π/6, π/3, 5π/6, etc.So, the midpoints between these x-intercepts would be the centers of symmetry. The midpoint between -π/6 and π/3 is (-π/6 + π/3)/2 = (π/6)/2 = π/12. The midpoint between π/3 and 5π/6 is (π/3 + 5π/6)/2 = (7π/6)/2 = 7π/12. Wait, but earlier, I found the midpoints between asymptotes as π/3, 7π/12, etc.I think I'm confusing two different things. The centers of symmetry for the tangent function are indeed the midpoints between consecutive vertical asymptotes. So, if the asymptotes are at π/12 + kπ/2, then the midpoints are at π/12 + kπ/4. Wait, no. Let me clarify.The vertical asymptotes are at x = π/12 + kπ/2. So, between x = π/12 and x = 7π/12, the midpoint is (π/12 + 7π/12)/2 = (8π/12)/2 = (2π/3)/2 = π/3. Similarly, between x = 7π/12 and x = 13π/12, the midpoint is (7π/12 + 13π/12)/2 = (20π/12)/2 = (5π/3)/2 = 5π/6. Wait, that can't be right because 5π/6 is greater than π.Wait, maybe I need to think differently. The period is π/2, so the function repeats every π/2. The vertical asymptotes are spaced π/2 apart. So, the midpoints between asymptotes would be π/4 apart. So, starting from π/12, the next midpoint would be π/12 + π/4 = (π/12 + 3π/12) = 4π/12 = π/3. Then, π/3 + π/4 = 7π/12, and so on.So, the centers of symmetry are at x = π/12 + kπ/4, where k is an integer. So, for k=0, x=π/12; k=1, x=π/3; k=2, x=7π/12; etc. So, the symmetry centers are at (π/12 + kπ/4, 0). So, (-π/6, 0) would correspond to k=-1: π/12 - π/4 = π/12 - 3π/12 = -2π/12 = -π/6. So, yes, (-π/6, 0) is a symmetry center. So, proposition (2) is correct.Now, proposition (3): (1/4 kπ - π/6, 0) for k ∈ Z is also the symmetry center of the function. From what I just figured out, the symmetry centers are at x = π/12 + kπ/4, which can be rewritten as x = (kπ)/4 + π/12. Let me factor that: x = (kπ)/4 + π/12 = (3kπ + π)/12 = π(3k + 1)/12. Hmm, that doesn't look like 1/4 kπ - π/6.Wait, let me see. If I take k as any integer, then (1/4 kπ - π/6) can be rewritten as (kπ)/4 - π/6. Let me express both terms with a common denominator: (3kπ - 2π)/12 = π(3k - 2)/12. Comparing this with the symmetry centers I found: π(3k + 1)/12. These are different. So, unless there's a shift in k, they might represent the same set.Wait, let me check for specific k values. For k=0, (1/4 *0 - π/6) = -π/6, which we saw is a symmetry center. For k=1, (1/4 π - π/6) = (3π/12 - 2π/12) = π/12, which is also a symmetry center. For k=2, (1/4 *2π - π/6) = (π/2 - π/6) = (3π/6 - π/6) = 2π/6 = π/3, which is another symmetry center. Similarly, for k=3, (3π/4 - π/6) = (9π/12 - 2π/12) = 7π/12, which is also a symmetry center. So, it seems that (1/4 kπ - π/6) for k ∈ Z does indeed cover all the symmetry centers. So, proposition (3) is correct.Finally, proposition (4): x = (1/2 kπ - π/12) for k ∈ Z is the symmetry axis of the function f(x). Wait, the function f(x) is a tangent function, which is symmetric about its centers, not about vertical lines. So, it doesn't have a symmetry axis like a parabola or something. The symmetry axis would imply reflection symmetry across a vertical line, but the tangent function is symmetric about its centers, meaning it's symmetric about points, not lines.To confirm, let's think about the graph of tan(2x + π/3). It has vertical asymptotes and is symmetric about its centers, which we found earlier. It doesn't have a vertical line of symmetry because it's periodic and repeats every π/2. So, proposition (4) is incorrect.So, summarizing:(1) Correct.(2) Correct.(3) Correct.(4) Incorrect.Therefore, the correct propositions are (1), (2), and (3).