💡Okay, so I have this problem about an arithmetic sequence. Let me try to understand it step by step. First, the arithmetic sequence is given as {a_n}, with the first term a₁ = -5/2. The common difference is d, which I don't know yet. The sum of the first n terms is S_n, and it's given that S₄ = 2S₂ + 4. Then, there's another sequence {b_n} defined by b_n = (1 + a_n)/a_n. I need to find the value of n when b_n reaches its maximum value.Alright, let's start with the arithmetic sequence. I remember that the sum of the first n terms of an arithmetic sequence is given by S_n = n/2 [2a₁ + (n - 1)d]. So, maybe I can use this formula to find the common difference d.Given that S₄ = 2S₂ + 4, let's write expressions for S₄ and S₂ using the sum formula.For S₄:S₄ = 4/2 [2*(-5/2) + (4 - 1)d] Simplify that:S₄ = 2 [ -5 + 3d ] So, S₄ = 2*(-5 + 3d) = -10 + 6d.For S₂:S₂ = 2/2 [2*(-5/2) + (2 - 1)d] Simplify:S₂ = 1 [ -5 + d ] So, S₂ = -5 + d.Now, according to the problem, S₄ = 2S₂ + 4. Let's substitute the expressions we found:-10 + 6d = 2*(-5 + d) + 4 Let me compute the right side:2*(-5 + d) = -10 + 2d So, right side becomes -10 + 2d + 4 = (-10 + 4) + 2d = -6 + 2d.Now, set the left side equal to the right side:-10 + 6d = -6 + 2d Let me solve for d:Subtract 2d from both sides: -10 + 4d = -6 Add 10 to both sides: 4d = 4 Divide both sides by 4: d = 1.Okay, so the common difference d is 1. That means each term increases by 1 from the previous term.Now, let's write the general term a_n of the arithmetic sequence. The formula is a_n = a₁ + (n - 1)d.Substituting the known values:a_n = -5/2 + (n - 1)*1 Simplify:a_n = -5/2 + n - 1 Combine constants:-5/2 - 1 = -5/2 - 2/2 = -7/2 So, a_n = n - 7/2.Alright, so a_n is n - 7/2. Now, let's move on to the sequence {b_n} which is defined as b_n = (1 + a_n)/a_n.Let me substitute a_n into this formula:b_n = (1 + (n - 7/2)) / (n - 7/2) Simplify the numerator:1 + n - 7/2 = n + (1 - 7/2) = n - 5/2 So, b_n = (n - 5/2)/(n - 7/2).Hmm, this looks like a rational function. Maybe I can simplify it further or analyze its behavior.Let me write it as:b_n = (2n - 5)/(2n - 7) I multiplied numerator and denominator by 2 to eliminate the fractions.Now, let's see if I can rewrite this to make it easier to analyze. Maybe split the fraction:b_n = (2n - 7 + 2)/(2n - 7) Which is equal to:b_n = 1 + 2/(2n - 7).Ah, that's a nice form. So, b_n is 1 plus 2 divided by (2n - 7). Now, to find the maximum value of b_n, I need to analyze how this expression behaves as n increases.Since n is a positive integer, starting from 1, let's see what happens with each term.First, let's note that the denominator 2n - 7 will determine the behavior of the fraction 2/(2n - 7). Let's consider when 2n - 7 is positive or negative.2n - 7 > 0 when n > 7/2, which is n > 3.5. Since n is an integer, this means for n ≥ 4, the denominator is positive.For n < 4, the denominator is negative. So, let's compute b_n for n = 1, 2, 3, 4, etc., to see how it behaves.Compute b_n for n = 1:a₁ = 1 - 7/2 = -5/2 b₁ = (1 + (-5/2))/(-5/2) = (-3/2)/(-5/2) = (3/2)/(5/2) = 3/5 = 0.6Wait, but according to our earlier expression, b_n = 1 + 2/(2n - 7). Let's check that:For n = 1: 1 + 2/(2*1 - 7) = 1 + 2/(-5) = 1 - 2/5 = 3/5, which matches.Similarly, for n = 2:b₂ = 1 + 2/(4 - 7) = 1 + 2/(-3) = 1 - 2/3 = 1/3 ≈ 0.333n = 3:b₃ = 1 + 2/(6 - 7) = 1 + 2/(-1) = 1 - 2 = -1n = 4:b₄ = 1 + 2/(8 - 7) = 1 + 2/1 = 3n = 5:b₅ = 1 + 2/(10 - 7) = 1 + 2/3 ≈ 1.666n = 6:b₆ = 1 + 2/(12 - 7) = 1 + 2/5 = 1.4n = 7:b₇ = 1 + 2/(14 - 7) = 1 + 2/7 ≈ 1.2857n = 8:b₈ = 1 + 2/(16 - 7) = 1 + 2/9 ≈ 1.222n = 9:b₉ = 1 + 2/(18 - 7) = 1 + 2/11 ≈ 1.1818And so on. It seems that as n increases beyond 4, the value of b_n decreases from 3 towards 1.Looking at the computed values:n | b_n---|---1 | 0.62 | 0.3333 | -14 | 35 | ≈1.6666 | 1.47 | ≈1.28578 | ≈1.2229 | ≈1.1818So, the maximum value occurs at n = 4, where b₄ = 3. After that, the value decreases and approaches 1 as n increases.But wait, let me think again. For n = 3, b₃ is -1, which is lower than n=1 and n=2. So, the sequence {b_n} first decreases from n=1 to n=3, then jumps up to 3 at n=4, and then starts decreasing again towards 1.Therefore, the maximum value of b_n is at n=4.But just to be thorough, let me analyze the function b_n = 1 + 2/(2n -7). Let's consider it as a function of a real variable x, so f(x) = 1 + 2/(2x -7). The behavior of this function can tell us about the maximum.The function f(x) = 1 + 2/(2x -7) is a hyperbola. The denominator 2x -7 is zero at x = 7/2 = 3.5, so there's a vertical asymptote there. For x < 3.5, the denominator is negative, so f(x) is 1 minus a positive number, making f(x) less than 1. For x > 3.5, the denominator is positive, so f(x) is 1 plus a positive number, making f(x) greater than 1.As x increases beyond 3.5, the term 2/(2x -7) decreases towards zero, so f(x) approaches 1 from above. Therefore, the function f(x) is decreasing for x > 3.5.Since n must be an integer greater than or equal to 1, the closest integer greater than 3.5 is 4. Therefore, the maximum value of b_n occurs at n=4.Wait, but let me check the behavior around n=3.5. Since n must be integer, n=3 is less than 3.5, and n=4 is greater. So, n=4 is the first integer where the function starts decreasing from a higher value.But looking at the computed values, at n=4, b_n is 3, which is higher than any other n. So, yes, n=4 is the maximum.Is there any possibility that for some higher n, b_n could be higher? Let's see, as n increases, 2/(2n -7) approaches zero, so b_n approaches 1. So, it's always decreasing after n=4, approaching 1. So, 3 is indeed the maximum.Therefore, the value of n when b_n reaches its maximum is 4.**Final Answer**The value of ( n ) when ( b_n ) reaches its maximum value is (boxed{4}).

💡Okay, so I have this problem about two congruent triangles inscribed in a circle. The triangles are ABC and A₁B₁C₁, and their vertices are ordered oppositely. I need to show that the points where the sides AB and A₁B₁ intersect, BC and B₁C₁ intersect, and CA and C₁A₁ intersect all lie on a straight line. Hmm, interesting.First, let me visualize the problem. There's a circle, and two congruent triangles inscribed in it. Since they're congruent, all their corresponding sides and angles are equal. But their vertices are ordered oppositely, which probably means that if ABC is going clockwise, A₁B₁C₁ is going counterclockwise or something like that.I remember something about triangles inscribed in a circle and their symmetries. Maybe reflection symmetry? If the triangles are congruent and their vertices are oppositely ordered, perhaps one is a reflection of the other across some line. That line might be the perpendicular bisector of a segment connecting corresponding vertices.Let me try to formalize this. Let's denote the circle as k with center O. Suppose I take a line t that passes through O and is the perpendicular bisector of segment AA₁. If A and A₁ coincide, then t would just be the line AO. Reflecting points B and C over line t should give me points B₂ and C₂, which should lie on the circle k because the circle is symmetric with respect to t.Since the triangles are congruent and oppositely ordered, the angles at the center corresponding to the sides should be equal but in the opposite direction. So, the angle ∠A₁OB₁ should be equal to ∠AOB but in the opposite direction. This makes me think that B₁ is the reflection of B over t, so B₁ = B₂. Similarly, C₁ should be the reflection of C over t, so C₁ = C₂.Therefore, triangle A₁B₁C₁ is the reflection of triangle ABC across line t. That seems right. So, if I reflect ABC over t, I get A₁B₁C₁.Now, considering the intersections of the sides. If I take AB and A₁B₁, their intersection point should lie on the line t because of the reflection symmetry. Similarly, the intersection of BC and B₁C₁, and the intersection of CA and C₁A₁ should also lie on t. Therefore, all three intersection points lie on the same straight line t.Wait, but I need to make sure that these intersections actually exist. The problem statement says "if they exist," so maybe sometimes they might be parallel? But since the triangles are inscribed in the same circle and are congruent, their sides shouldn't be parallel unless the triangles are overlapping in a certain way. But since they are oppositely ordered, their sides should intersect at some point.Let me think about this more carefully. If ABC and A₁B₁C₁ are reflections over t, then each side of ABC is reflected to the corresponding side of A₁B₁C₁. So, the sides AB and A₁B₁ are reflections over t, meaning they intersect at a point on t. Similarly for the other sides. So, all three intersection points must lie on t.Is there a theorem that directly states this? Maybe something related to the radical axis or something in projective geometry? Wait, no, radical axis is about circles. Maybe it's a case of the Desargues theorem?Desargues' theorem states that if two triangles are perspective from a point, they are perspective from a line. But here, the triangles are perspective from line t, so maybe the intersections lie on a line? Hmm, not sure if that's directly applicable.Alternatively, since the triangles are reflections over t, the intersections of their corresponding sides must lie on t. That seems straightforward because reflection preserves lines and their intersections. So, if two lines are reflections over t, their intersection must lie on t.Therefore, the points where AB intersects A₁B₁, BC intersects B₁C₁, and CA intersects C₁A₁ all lie on the line t. Hence, they are collinear.I think that makes sense. So, the key idea is that the triangles are reflections over a line t, and the intersections of corresponding sides must lie on that line. Therefore, the three points are collinear on t.**Final Answer**boxed{}

💡Okay, so I have this problem here: Given that ( a ) and ( b ) are real numbers, then "( a > b > 0 )" is ( ). The options are A, B, C, D, which correspond to whether it's a sufficient but not necessary condition, a necessary but not sufficient condition, a necessary and sufficient condition, or neither necessary nor sufficient.Alright, first, I need to understand what the question is asking. It seems like it's about determining the nature of the condition "( a > b > 0 )" in relation to some statement. But wait, the original problem didn't specify what the condition is for. Hmm, maybe it's implied? In the previous example, it was about ( |a| > |b| ). Maybe this is a similar question? Or perhaps it's about something else. Wait, the user just repeated the same problem, so maybe it's the same context.Assuming it's about ( |a| > |b| ), let me think. So, if ( a > b > 0 ), does that imply ( |a| > |b| )? Well, if both ( a ) and ( b ) are positive, then ( |a| = a ) and ( |b| = b ), so yes, ( |a| > |b| ) would follow directly. So, in that case, "( a > b > 0 )" is a sufficient condition for ( |a| > |b| ).But is it a necessary condition? That is, does ( |a| > |b| ) necessarily mean that ( a > b > 0 )? Hmm, not necessarily. For example, if ( a = -3 ) and ( b = 2 ), then ( |a| = 3 ) and ( |b| = 2 ), so ( |a| > |b| ) is true, but ( a > b > 0 ) is not true because ( a ) is negative. So, "( a > b > 0 )" isn't a necessary condition for ( |a| > |b| ).Therefore, "( a > b > 0 )" is a sufficient but not necessary condition for ( |a| > |b| ). So, the answer should be A.Wait, but let me double-check. Maybe I'm misinterpreting the question. If the question is about something else, like ( a^2 > b^2 ), then similar logic applies. If ( a > b > 0 ), then ( a^2 > b^2 ) because squaring preserves the inequality for positive numbers. But again, ( a^2 > b^2 ) doesn't necessarily mean ( a > b > 0 ) because ( a ) could be negative with a larger magnitude than ( b ).Alternatively, if the condition was for ( a + b > 0 ), then "( a > b > 0 )" would imply ( a + b > 0 ), but ( a + b > 0 ) doesn't necessarily require both ( a ) and ( b ) to be positive. For example, ( a = 3 ) and ( b = -1 ) would still give ( a + b = 2 > 0 ).But since the original context was about ( |a| > |b| ), I think that's the intended comparison here. So, in that case, my initial conclusion holds: "( a > b > 0 )" is a sufficient but not necessary condition for ( |a| > |b| ).Just to make sure, let's think of another angle. Suppose we're talking about some other condition, like ( a ) being greater than ( b ) in some operation or property. But without more context, it's hard to say. However, given the structure of the question and the options, it's likely referring to a comparison involving absolute values or similar.Also, considering the options, if it was a necessary and sufficient condition, then both directions would have to hold, which they don't in the case of ( |a| > |b| ). If it was neither, then neither direction would hold, but we know that the condition does imply ( |a| > |b| ), so it can't be neither.So, yeah, I'm pretty confident that the answer is A: a sufficient but not necessary condition.

💡Alright, so I have this problem about an infinite set of natural numbers, S. It says that S contains four distinct integers a, b, c, d such that the gcd of a and b is not equal to the gcd of c and d. I need to show that S contains three distinct integers x, y, z where the gcd of x and y equals the gcd of y and z, but this is not equal to the gcd of z and x.Hmm, okay. Let me try to unpack this step by step. First, S is an infinite set of nonzero natural numbers. So, it's got infinitely many numbers, none of which are zero. And within this set, there are four distinct numbers a, b, c, d. The key thing here is that the gcd of a and b is different from the gcd of c and d. So, gcd(a, b) ≠ gcd(c, d).I need to find three distinct numbers x, y, z in S such that gcd(x, y) = gcd(y, z), but this is not equal to gcd(z, x). So, kind of a chain where the first two gcds are equal, but the last one is different.Let me think about how to approach this. Maybe I can use the fact that S is infinite. Since it's infinite, it must have some structure or repetition in terms of gcds. If I can find a way to leverage the four numbers a, b, c, d with different gcds, maybe I can extend that to find the required three numbers.Wait, the problem mentions that S contains four distinct integers a, b, c, d with gcd(a, b) ≠ gcd(c, d). So, perhaps I can use these four numbers to construct the three numbers x, y, z. Or maybe not directly, but their properties can help.Let me consider the gcds. If I have four numbers, two pairs with different gcds, maybe I can find a common element or something that connects them. Or perhaps, since S is infinite, I can find more numbers that share certain gcd properties.Another thought: maybe I can fix one number and find two others that have the same gcd with it but different gcd with each other. For example, if I fix y, then find x and z such that gcd(x, y) = gcd(z, y), but gcd(x, z) is different.But how do I ensure that such x and z exist? Since S is infinite, maybe I can find infinitely many numbers with the same gcd with y, and among those, find two that have a different gcd with each other.Wait, but I need to relate this back to the four numbers a, b, c, d. Maybe I can use their gcds to structure this.Let me try to outline a possible approach:1. Since S is infinite, and it contains four numbers with two different gcds, perhaps I can find infinitely many numbers in S that share a common gcd with one of these four numbers.2. Then, within those infinitely many numbers, I can find two numbers that have the same gcd with a fixed number but a different gcd with each other.But I'm not sure if that's the right way. Maybe I need to think about the possible gcds in S. Since S is infinite, the gcds can't all be the same, otherwise, S would have a common divisor greater than 1, which might not necessarily be the case.Wait, actually, if S had a common divisor greater than 1, say d, then every element in S would be divisible by d. But since S is a set of natural numbers, and it's infinite, it's possible, but not necessarily required.But in the problem statement, S is just an infinite set of nonzero natural numbers. So, it might have a common divisor, or it might not. But the fact that it contains four numbers with different gcds suggests that it's not the case that all elements share a common divisor.Wait, actually, if all elements shared a common divisor d > 1, then the gcd of any two elements would be at least d. So, in that case, all gcds would be multiples of d, but they could still be different. For example, if d=2, then gcds could be 2, 4, 6, etc.But in our case, we have four numbers with two different gcds. So, maybe S doesn't have a common divisor greater than 1. Or maybe it does, but the four numbers a, b, c, d have gcds that are different multiples of that common divisor.Hmm, this is getting a bit tangled. Maybe I should consider the case where S has a common divisor first, and then see if I can adjust the problem accordingly.Suppose that S has a greatest common divisor k > 1. Then, every element in S is divisible by k. So, if I divide every element by k, I get a new set S' where the gcd of any two elements is the gcd of the original elements divided by k. So, gcd(a/k, b/k) = gcd(a, b)/k, and similarly for c and d.But since gcd(a, b) ≠ gcd(c, d), then gcd(a/k, b/k) ≠ gcd(c/k, d/k). So, S' still has four distinct numbers with different gcds. And S' is also infinite because S is infinite.Therefore, without loss of generality, I can assume that the greatest common divisor of all elements in S is 1. Because if it's not, I can just divide everything by k and work with S' instead.Okay, so now I can assume that gcd of all elements in S is 1. That might simplify things because it means that there are elements in S that are coprime to each other.But wait, not necessarily all elements are coprime, but the overall gcd is 1. So, there might be some elements that share common divisors, but not all.Given that, I can think about the prime factors in S. Since the overall gcd is 1, there must be elements in S that are not divisible by some primes. So, for example, if S had a common prime divisor p, then all elements would be divisible by p, which would contradict the gcd being 1. Therefore, S must have elements not divisible by any particular prime.Wait, no, that's not necessarily true. It just means that there is no prime that divides every element of S. So, for every prime p, there exists at least one element in S not divisible by p.But individual elements can still share prime factors. So, for example, some elements might be even, some might be multiples of 3, etc., but not all elements share the same prime factor.Given that, maybe I can use the four numbers a, b, c, d to find the required three numbers.Let me think about the four numbers a, b, c, d. We know that gcd(a, b) ≠ gcd(c, d). Let's denote d1 = gcd(a, b) and d2 = gcd(c, d), with d1 ≠ d2.Since S is infinite, and it contains a, b, c, d, it must contain infinitely many other numbers. Maybe I can find a number z such that gcd(a, z) = d1 and gcd(b, z) = d1, but gcd(c, z) ≠ d2.Wait, but I need to relate this to the three numbers x, y, z with the specific gcd conditions.Alternatively, perhaps I can fix y as one of the four numbers, say b, and then find x and z such that gcd(x, b) = gcd(z, b), but gcd(x, z) is different.But how do I ensure that such x and z exist?Since S is infinite, and if I fix b, then the set S {b} is still infinite. So, maybe I can find infinitely many numbers in S that have the same gcd with b. Then, among those, I can find two numbers x and z such that gcd(x, z) is different.But I need to make sure that gcd(x, y) = gcd(y, z), which would be the same as gcd(x, b) = gcd(z, b). So, if I can find x and z such that both have the same gcd with b, but their gcd with each other is different, then I'm done.But how do I ensure that such x and z exist?Wait, since S is infinite, and the function mapping each element to its gcd with b can only take on finitely many values (since gcds are bounded by b), by the pigeonhole principle, there must be infinitely many elements in S with the same gcd with b.So, let's say that there are infinitely many elements in S with gcd k with b, where k is some divisor of b.Then, within these infinitely many elements, I can look at their pairwise gcds. Since there are infinitely many, and the number of possible gcds is finite (since they are all divisors of b), again by the pigeonhole principle, there must be two elements x and z among them such that gcd(x, z) is the same as some other pair.Wait, but I need gcd(x, z) to be different from k. Hmm, not sure.Alternatively, maybe I can use the four numbers a, b, c, d to construct such a trio.Let me think: since gcd(a, b) = d1 and gcd(c, d) = d2, and d1 ≠ d2, maybe I can use a, b, c, d to form the trio.For example, take x = a, y = b, z = c. Then, gcd(a, b) = d1, gcd(b, c) could be something else, and gcd(c, a) could be something else.But I need gcd(x, y) = gcd(y, z), which would mean gcd(a, b) = gcd(b, c). But we don't know if that's the case.Alternatively, maybe I can take x = a, y = c, z = d. Then, gcd(a, c) could be something, gcd(c, d) = d2, and gcd(d, a) could be something else.But again, I don't know if gcd(a, c) equals gcd(c, d).Wait, maybe I need to find a number that shares the same gcd with two different numbers.Alternatively, since S is infinite, maybe I can find a number that is a multiple of d1 and also a multiple of d2, but that might not necessarily help.Wait, another idea: since S is infinite, and it contains a, b, c, d, maybe I can find a number z such that z is a multiple of d1 but not a multiple of d2, or vice versa.But I'm not sure how that would help.Wait, let's think about the prime factors. Since d1 and d2 are different, they must have different prime factors or different exponents in their prime factorization.So, suppose d1 has a prime factor p that d2 doesn't have, or vice versa.Then, maybe I can find a number z in S that is divisible by p, but not by some other prime that's in d2.But I'm not sure.Alternatively, maybe I can use the fact that S has infinitely many numbers, so I can find numbers that share certain gcd properties.Wait, let's go back to the earlier idea. If I fix y as one of the numbers, say b, and look at the gcds of other numbers with b.Since S is infinite, and the number of possible gcds with b is finite (as they are divisors of b), there must be some gcd value that occurs infinitely often. Let's say that gcd(k, b) = d for infinitely many k in S.Then, among these infinitely many k's, I can look at their pairwise gcds. Since there are infinitely many, and the number of possible gcds is finite, there must be two numbers x and z among them such that gcd(x, z) is the same as some other pair.But I need gcd(x, z) to be different from d.Wait, but if all these k's have gcd d with b, then their gcd with each other could be d or some multiple of d, but not necessarily.Wait, no, because if x and z both have gcd d with b, then gcd(x, z) could be d or a multiple of d, depending on their other factors.But since S has gcd 1 overall, there must be some numbers in S that are coprime to b. So, maybe among the infinitely many numbers with gcd d with b, some of them are coprime to each other.Wait, but if they are coprime to each other, then gcd(x, z) = 1, which is different from d (assuming d > 1). So, if d > 1, then we have gcd(x, y) = d, gcd(y, z) = d, and gcd(z, x) = 1, which is different.But wait, is that always possible? If d > 1, then yes, because among infinitely many numbers with gcd d with b, some of them must be coprime to each other.But what if d = 1? Then, gcd(x, y) = 1, gcd(y, z) = 1, and we need gcd(z, x) ≠ 1. But if d = 1, then y = b has gcd 1 with x and z, but x and z might still have gcd 1 or greater.Wait, but if d = 1, then y is coprime to both x and z, but x and z could still share a common factor. So, in that case, we might have gcd(x, z) > 1, which would satisfy the condition.So, in either case, whether d > 1 or d = 1, we can find x and z such that gcd(x, y) = gcd(y, z) = d, and gcd(x, z) ≠ d.Therefore, by fixing y as b, and using the infinite nature of S, we can find such x and z.Wait, but I need to make sure that x, y, z are distinct. Since S is infinite, and we're choosing from infinitely many numbers, we can ensure that x, y, z are distinct.So, putting it all together:1. Since S is infinite and contains four numbers a, b, c, d with gcd(a, b) ≠ gcd(c, d), we can assume without loss of generality that the overall gcd of S is 1.2. Fix y as one of the numbers, say b.3. Since S is infinite, there are infinitely many numbers in S with the same gcd with b, say d.4. Among these infinitely many numbers, there must be two numbers x and z such that gcd(x, z) ≠ d.5. Therefore, we have gcd(x, y) = gcd(y, z) = d, and gcd(x, z) ≠ d, which satisfies the required condition.So, that seems like a valid approach. I think this works because of the infinite nature of S, which allows us to apply the pigeonhole principle multiple times to find the necessary numbers.I should probably check if there are any edge cases or if I missed something. For example, what if d = 1? Then, gcd(x, y) = gcd(y, z) = 1, and we need gcd(x, z) ≠ 1. Since S has infinitely many numbers, and the overall gcd is 1, there must be numbers in S that share common factors, so such x and z must exist.Another edge case: what if all the gcds are 1? But the problem states that gcd(a, b) ≠ gcd(c, d), so at least two different gcds exist. So, d can't be 1 for both pairs, or if d is 1 for one pair, it's different for the other.Wait, actually, if d1 = 1 and d2 > 1, then we can still apply the same reasoning. Fix y as b, which has gcd d1 = 1 with a, and look for x and z such that gcd(x, y) = gcd(y, z) = 1, but gcd(x, z) ≠ 1.Since S is infinite and has overall gcd 1, there are infinitely many numbers coprime to b, and among them, some must share a common factor, so gcd(x, z) > 1.Therefore, regardless of the specific values of d1 and d2, as long as they are different, we can find such x, y, z.I think this covers all cases. So, the key idea is leveraging the infinite nature of S to find multiple numbers sharing the same gcd with a fixed element, and then using the pigeonhole principle to find two numbers among them with a different gcd.Yeah, I think that's the way to go.

💡To determine the true temperature when the malfunctioning thermometer shows (+17^{circ}), we can follow these steps:1. **Identify the reference points:** - Freezing point of water: (0^{circ}text{C}) corresponds to (+1^{circ}) on the faulty thermometer. - Boiling point of water: (100^{circ}text{C}) corresponds to (+105^{circ}) on the faulty thermometer.2. **Determine the range of the faulty thermometer:** - The faulty thermometer ranges from (+1^{circ}) to (+105^{circ}), which is a span of (105 - 1 = 104) degrees.3. **Calculate the conversion factor:** - The actual temperature range is (100^{circ}text{C}) (from (0^{circ}text{C}) to (100^{circ}text{C})). - The conversion factor is (frac{100}{104} = frac{25}{26}).4. **Set up the equation for the actual temperature:** - Let (T_{text{actual}}) be the true temperature corresponding to a faulty reading of (+17^{circ}). - The equation is: [ T_{text{actual}} = (17 - 1) times frac{25}{26} ]5. **Perform the calculation:** - Subtract the offset: [ 17 - 1 = 16 ] - Multiply by the conversion factor: [ T_{text{actual}} = 16 times frac{25}{26} = frac{400}{26} = frac{200}{13} approx 15.38 ]### Conclusion:The true temperature corresponding to the faulty thermometer reading of (+17^{circ}) is approximately:[boxed{15.38^{circ} text{C}}]

💡Alright, so I have this problem where there are two functions, f(x) and g(x), defined on all real numbers. They satisfy the condition that for any real numbers x and y, f(x - y) equals f(x)g(y) minus f(y)g(x). Also, it's given that f(1) is not zero, and f(1) equals f(2). I need to find the value of g(-1) plus g(1). Hmm, okay, let's break this down step by step.First, let me write down the given condition to make it clearer:For all x, y ∈ ℝ, f(x - y) = f(x)g(y) - f(y)g(x).And we know that f(1) ≠ 0 and f(1) = f(2). So, f(1) is equal to f(2), which is interesting. I need to find g(-1) + g(1). Let me see. Maybe I can plug in specific values for x and y to get more information about f and g. Since the equation holds for any x and y, I can choose x and y strategically to simplify things.Let me try setting x = y. Then, the equation becomes:f(x - x) = f(x)g(x) - f(x)g(x).Simplifying the left side, f(0) = 0, because x - x is 0. On the right side, f(x)g(x) - f(x)g(x) is 0. So, that tells me that f(0) = 0. Okay, that's a useful piece of information.Next, maybe I can set y = 0. Let's try that. If y = 0, then the equation becomes:f(x - 0) = f(x)g(0) - f(0)g(x).Simplifying, f(x) = f(x)g(0) - 0*g(x), since f(0) is 0. So, f(x) = f(x)g(0). Hmm, so f(x) = f(x)g(0). If f(x) is not zero for some x, which it isn't because f(1) ≠ 0, then we can divide both sides by f(x), right? So, 1 = g(0). Therefore, g(0) must be 1. Okay, that's another useful piece of information.Now, let's see if I can find more about the function f. Maybe I can set x = 0. Let's try that. If x = 0, then the equation becomes:f(0 - y) = f(0)g(y) - f(y)g(0).Simplifying, f(-y) = 0*g(y) - f(y)*1, since f(0) is 0 and g(0) is 1. So, f(-y) = -f(y). Oh, interesting! That means f is an odd function. So, f(-y) = -f(y) for all y. That's a key property. So, f is odd. That might be helpful later on.Now, the problem states that f(1) = f(2). Let's write that down: f(1) = f(2). Since f is odd, f(-1) = -f(1). Maybe I can use this in some way.Let me think about how to relate f(2) to f(1). Maybe I can express f(2) in terms of f(1) and g(-1) or something like that. Let's try plugging in specific values for x and y to get an expression involving f(2).Let me set x = 1 and y = -1. Then, the equation becomes:f(1 - (-1)) = f(1)g(-1) - f(-1)g(1).Simplifying the left side, f(1 + 1) = f(2). On the right side, f(1)g(-1) - f(-1)g(1). But since f is odd, f(-1) = -f(1). So, substituting that in, we get:f(2) = f(1)g(-1) - (-f(1))g(1) = f(1)g(-1) + f(1)g(1).Factor out f(1):f(2) = f(1)(g(-1) + g(1)).But we know from the problem statement that f(2) = f(1). So, substituting that in:f(1) = f(1)(g(-1) + g(1)).Since f(1) ≠ 0, we can divide both sides by f(1):1 = g(-1) + g(1).So, that's the value we're looking for: g(-1) + g(1) equals 1. Therefore, the answer is 1.Wait, let me double-check to make sure I didn't make any mistakes. So, I set x = 1 and y = -1, which gave me f(2) in terms of f(1), g(-1), and g(1). Then, since f is odd, I substituted f(-1) as -f(1). That led me to f(2) = f(1)(g(-1) + g(1)). Given that f(2) = f(1), dividing both sides by f(1) (which is non-zero) gives 1 = g(-1) + g(1). Yep, that seems correct.I think that's solid. I don't see any errors in the reasoning. So, the value of g(-1) + g(1) is indeed 1.**Final Answer**The value of ( g(-1) + g(1) ) is boxed{1}.