💡Alright, so I have this problem about graph theory, and I need to prove two things: a) That the sum of the degrees of all vertices in any graph is equal to twice the number of edges, and therefore, this sum is even.b) That the number of vertices with odd degrees is even.Okay, let's start with part (a). I remember that in a graph, each edge connects two vertices. So, if I think about the degree of a vertex, it's just the number of edges connected to it. So, if I add up all the degrees of every vertex, I'm essentially counting each edge twice because each edge contributes to the degree of two vertices. Wait, so if I have a graph with, say, three vertices connected in a triangle, each vertex has degree 2. So, the sum of degrees would be 2 + 2 + 2 = 6. And the number of edges is 3, so twice the number of edges is 6. That matches. But what if the graph is more complicated? Like, suppose I have a graph with some vertices connected by multiple edges or maybe even loops? Hmm, in simple graphs, loops aren't allowed, right? So, each edge just connects two distinct vertices. So, in that case, each edge will contribute exactly 2 to the sum of degrees—once for each endpoint.So, in general, for any graph, the sum of degrees is equal to twice the number of edges. That makes sense. So, mathematically, if I denote the sum of degrees as Σdeg(v) for all vertices v, and the number of edges as |E|, then:Σdeg(v) = 2|E|And since 2|E| is obviously even, the sum of degrees must be even.Okay, so that seems straightforward for part (a). Now, moving on to part (b): the number of vertices with odd degrees is even.Hmm, so if I think about the sum of degrees being even, as established in part (a), and if I consider that the sum of degrees is even, then the number of vertices with odd degrees must be even. Because if I have an odd number of odd degrees, their sum would be odd, and adding even degrees (which don't affect the parity) would still leave the total sum odd, which contradicts part (a). Let me test this with an example. Suppose I have a graph with four vertices, and two of them have degree 1 (which is odd), and the other two have degree 2 (which is even). So, the sum of degrees is 1 + 1 + 2 + 2 = 6, which is even. That works.What if I have three vertices with odd degrees? Let's say degrees 1, 1, 1, and the fourth vertex has degree 3. So, the sum is 1 + 1 + 1 + 3 = 6, which is even. Wait, but I have three vertices with odd degrees, which is odd, but the sum is still even. That contradicts my earlier thought.Wait, no, in this case, the sum is still even because 1 + 1 + 1 + 3 = 6, which is even. But I have three vertices with odd degrees, which is odd. So, that seems to contradict the statement that the number of vertices with odd degrees must be even.Hmm, maybe I made a mistake in my reasoning. Let me think again. If I have an odd number of vertices with odd degrees, then the sum of their degrees would be odd, right? Because the sum of an odd number of odd numbers is odd. Then, adding the degrees of the even-degree vertices, which are even, wouldn't change the parity. So, the total sum would be odd, which contradicts part (a) where the sum is even.Wait, but in my example, I had three vertices with odd degrees: 1, 1, 1, and one vertex with degree 3. So, the sum is 1 + 1 + 1 + 3 = 6, which is even. But 1 + 1 + 1 is 3, which is odd, and adding 3 (which is odd) gives 6, which is even. So, in this case, the number of odd-degree vertices is 4, which is even. Wait, no, in my example, I have four vertices, three with degree 1 and one with degree 3. So, actually, the number of odd-degree vertices is four, which is even. Wait, but I thought I had three. No, in my example, I have four vertices: three with degree 1 and one with degree 3. So, that's four odd-degree vertices, which is even. So, the sum is even.Wait, I think I confused myself earlier. Let me clarify. If I have an odd number of odd-degree vertices, then the sum of their degrees would be odd, and adding any even numbers (from the even-degree vertices) would still leave the total sum odd, which contradicts part (a). Therefore, the number of odd-degree vertices must be even.So, in my previous example, I actually had four odd-degree vertices, which is even, and the sum was even. If I tried to have three odd-degree vertices, the sum would be odd, which would contradict part (a). Therefore, the number of vertices with odd degrees must be even.Okay, that makes sense now. So, to summarize:a) The sum of the degrees of all vertices is equal to twice the number of edges because each edge contributes to the degree of two vertices. Therefore, the sum is even.b) Since the sum of degrees is even, the number of vertices with odd degrees must be even because the sum of an odd number of odd numbers is odd, which would contradict the even sum.Yeah, that seems to cover both parts. I think I understand it now.

💡Alright, so I have this algebra problem here, and I need to figure out the value of this expression: 2m² + 13mn + 6n² - 444. The problem gives me two equations: m² + 2mn = 384 and 3mn + 2n² = 560. Hmm, okay, let's see how I can use these to find the value of the expression.First, I need to understand what I'm being asked. I have two equations with two variables, m and n, and I need to find the value of a third expression. It might help if I can express the third expression in terms of the given equations. Let me write down the given equations:1. m² + 2mn = 3842. 3mn + 2n² = 560And the expression I need to find is:2m² + 13mn + 6n² - 444Hmm, maybe I can manipulate the given equations to get something that looks like the expression I need. Let me see.Looking at the first equation, m² + 2mn = 384. If I multiply both sides by 2, I get:2m² + 4mn = 768Okay, that's equation (1) multiplied by 2. Let's call this equation (3):3. 2m² + 4mn = 768Now, looking at the second equation, 3mn + 2n² = 560. If I multiply both sides by 3, I get:9mn + 6n² = 1680Let's call this equation (4):4. 9mn + 6n² = 1680Hmm, now if I add equation (3) and equation (4), what do I get?2m² + 4mn + 9mn + 6n² = 768 + 1680Simplify the left side:2m² + (4mn + 9mn) + 6n² = 2m² + 13mn + 6n²And the right side:768 + 1680 = 2448So, adding equations (3) and (4) gives me:2m² + 13mn + 6n² = 2448Wait a minute, that's exactly the expression I need, except for the -444 part. So, if I subtract 444 from both sides, I get:2m² + 13mn + 6n² - 444 = 2448 - 444Let me calculate 2448 - 444:2448 - 400 = 20482048 - 44 = 2004So, 2448 - 444 = 2004Therefore, the value of the expression 2m² + 13mn + 6n² - 444 is 2004.Let me double-check my steps to make sure I didn't make any mistakes. I multiplied the first equation by 2 to get 2m² + 4mn = 768, which seems correct. Then I multiplied the second equation by 3 to get 9mn + 6n² = 1680, which also looks right. Adding these two equations together gives me 2m² + 13mn + 6n² = 2448. Subtracting 444 from both sides gives me 2004. That seems consistent.I think that's the correct answer. So, among the options given, D is 2004, which must be the correct choice.

💡Okay, so I need to find the range of the function G(x) = |x + 2| - 2|x - 2|. Hmm, I remember that the range of a function is all the possible output values it can take. Since this function involves absolute values, it's probably piecewise linear, meaning it has different expressions depending on the value of x. First, I should figure out where the expressions inside the absolute values change their signs because that's where the function's behavior will change. The expressions inside the absolute values are (x + 2) and (x - 2). So, (x + 2) changes sign at x = -2, and (x - 2) changes sign at x = 2. That means I should consider three intervals: x < -2, -2 ≤ x < 2, and x ≥ 2. Let me handle each interval one by one.**1. When x < -2:**In this case, both (x + 2) and (x - 2) are negative because x is less than -2. So, the absolute values will flip their signs.G(x) = |x + 2| - 2|x - 2| becomes:G(x) = -(x + 2) - 2*(-(x - 2)) = -x - 2 + 2x - 4 = ( -x + 2x ) + ( -2 - 4 ) = x - 6.Wait, that doesn't seem right. Let me check my algebra again.G(x) = -(x + 2) - 2*(-(x - 2)) = -x - 2 + 2x - 4 = ( -x + 2x ) + ( -2 - 4 ) = x - 6.Hmm, actually, that seems correct. So for x < -2, G(x) simplifies to x - 6.**2. When -2 ≤ x < 2:**Here, (x + 2) is non-negative because x is greater than or equal to -2, but (x - 2) is still negative because x is less than 2.So, G(x) = |x + 2| - 2|x - 2| becomes:G(x) = (x + 2) - 2*(-(x - 2)) = x + 2 + 2x - 4 = (x + 2x) + (2 - 4) = 3x - 2.Okay, that looks correct. So for -2 ≤ x < 2, G(x) is 3x - 2.**3. When x ≥ 2:**Both (x + 2) and (x - 2) are non-negative since x is greater than or equal to 2.So, G(x) = |x + 2| - 2|x - 2| becomes:G(x) = (x + 2) - 2*(x - 2) = x + 2 - 2x + 4 = (x - 2x) + (2 + 4) = -x + 6.Alright, so for x ≥ 2, G(x) is -x + 6.Now, I have three linear pieces for G(x):1. G(x) = x - 6 for x < -22. G(x) = 3x - 2 for -2 ≤ x < 23. G(x) = -x + 6 for x ≥ 2To find the range, I need to see what values G(x) can take in each interval and then combine them.**Analyzing Each Interval:**1. **For x < -2: G(x) = x - 6** - As x approaches negative infinity, G(x) approaches negative infinity. - At x = -2, G(-2) = (-2) - 6 = -8. So, in this interval, G(x) goes from negative infinity up to -8.2. **For -2 ≤ x < 2: G(x) = 3x - 2** - At x = -2, G(-2) = 3*(-2) - 2 = -6 - 2 = -8. - At x = 2, G(2) = 3*(2) - 2 = 6 - 2 = 4. Since this is a linear function increasing from -8 to 4 as x goes from -2 to 2, the range here is [-8, 4].3. **For x ≥ 2: G(x) = -x + 6** - At x = 2, G(2) = -2 + 6 = 4. - As x approaches positive infinity, G(x) approaches negative infinity. So, in this interval, G(x) decreases from 4 to negative infinity.**Combining the Ranges:**Now, let's see the overall range by combining the three intervals:- From x < -2: G(x) goes from -∞ to -8.- From -2 ≤ x < 2: G(x) goes from -8 to 4.- From x ≥ 2: G(x) goes from 4 to -∞.Wait, that seems a bit confusing. Let me think again.Actually, when combining these, the function can take any value from -∞ up to 4 and from -∞ up to -8, but the overlapping part is from -8 to 4. However, since each interval contributes to the overall range, the function can actually take any value from -∞ to 4 because:- For values less than -8, it's covered by the first interval (x < -2).- For values between -8 and 4, it's covered by the second interval.- For values greater than 4, wait, no, in the third interval, G(x) starts at 4 and goes down to -∞, so it doesn't cover anything above 4.Wait, that's not correct. Let me correct that.Actually, in the first interval, as x approaches -∞, G(x) approaches -∞, and at x = -2, G(x) is -8.In the second interval, G(x) goes from -8 to 4.In the third interval, G(x) starts at 4 and goes down to -∞ as x increases.So, putting it all together, the function G(x) can take any value from -∞ up to 4. But wait, in the second interval, it goes from -8 to 4, and in the first interval, it goes from -∞ to -8, so overall, the range is from -∞ to 4.But hold on, in the first interval, G(x) is x - 6. So as x approaches -∞, G(x) approaches -∞, and at x = -2, it's -8.In the second interval, G(x) increases from -8 to 4 as x goes from -2 to 2.In the third interval, G(x) decreases from 4 to -∞ as x increases beyond 2.So, the maximum value G(x) can take is 4, and it can go down to -∞. Therefore, the range is (-∞, 4].Wait, but in the initial problem, the user had a different answer. Let me check my calculations again.Wait, in the initial problem, the user wrote that the range is [-8, 8], but according to my calculations, it's (-∞, 4]. That seems conflicting. Maybe I made a mistake.Let me re-examine each interval.1. **x < -2: G(x) = x - 6** - At x = -2, G(-2) = -8. - As x decreases, G(x) decreases without bound.2. **-2 ≤ x < 2: G(x) = 3x - 2** - At x = -2, G(-2) = -8. - At x = 2, G(2) = 4.3. **x ≥ 2: G(x) = -x + 6** - At x = 2, G(2) = 4. - As x increases, G(x) decreases without bound.So, the function reaches a maximum value of 4 at x = 2 and x = -2, but wait, at x = -2, it's -8, which is a minimum. So, the maximum is 4, and it can go down to -∞.Wait, but in the initial problem, the user said the range is [-8, 8], which suggests that the function can reach up to 8. But according to my calculations, the maximum is 4. Did I make a mistake in simplifying the expressions?Let me double-check the simplification for each interval.1. **x < -2:** G(x) = |x + 2| - 2|x - 2| = -(x + 2) - 2*(-(x - 2)) = -x - 2 + 2x - 4 = x - 6. That seems correct.2. **-2 ≤ x < 2:** G(x) = (x + 2) - 2*(-(x - 2)) = x + 2 + 2x - 4 = 3x - 2. Correct.3. **x ≥ 2:** G(x) = (x + 2) - 2*(x - 2) = x + 2 - 2x + 4 = -x + 6. Correct.So, the expressions are correct. Therefore, the maximum value is indeed 4, not 8. So, the range should be (-∞, 4].But the initial problem's answer was [-8, 8]. That suggests that maybe I missed something. Let me think again.Wait, perhaps I made a mistake in evaluating the expressions at the endpoints.Wait, at x = -2, G(-2) = |-2 + 2| - 2| -2 - 2| = |0| - 2| -4| = 0 - 2*4 = -8. Correct.At x = 2, G(2) = |2 + 2| - 2|2 - 2| = |4| - 2*0 = 4 - 0 = 4. Correct.So, the function reaches -8 at x = -2 and 4 at x = 2. Beyond that, it goes to -∞ in both directions.Wait, but in the first interval, as x approaches -∞, G(x) approaches -∞, and in the third interval, as x approaches ∞, G(x) approaches -∞.So, the function's maximum value is 4, and it can go down to -∞. Therefore, the range is (-∞, 4].But the initial problem's answer was [-8, 8], which is different. Maybe the user made a mistake, or perhaps I misinterpreted the function.Wait, let me check the function again: G(x) = |x + 2| - 2|x - 2|. Yes, that's correct.Wait, perhaps I should graph the function to visualize it better.Let me sketch the graph mentally.- For x < -2: G(x) = x - 6. This is a straight line with a slope of 1, starting from (-∞, -∞) and reaching (-2, -8).- For -2 ≤ x < 2: G(x) = 3x - 2. This is a straight line with a slope of 3, starting at (-2, -8) and going up to (2, 4).- For x ≥ 2: G(x) = -x + 6. This is a straight line with a slope of -1, starting at (2, 4) and going down to (∞, -∞).So, the graph has a peak at (2, 4) and a trough at (-2, -8). But wait, at x = -2, it's -8, and as x decreases, it goes to -∞. So, the function's maximum value is 4, and it can go down to -∞. Therefore, the range is (-∞, 4].But the initial answer was [-8, 8], which suggests that the function can reach up to 8. Maybe I made a mistake in the simplification.Wait, let me re-examine the simplification for x < -2.G(x) = |x + 2| - 2|x - 2|.For x < -2, both x + 2 and x - 2 are negative.So, |x + 2| = -(x + 2) = -x - 2.|x - 2| = -(x - 2) = -x + 2.Therefore, G(x) = (-x - 2) - 2*(-x + 2) = -x - 2 + 2x - 4 = x - 6. Correct.Wait, but if I plug in x = -3, which is less than -2, G(-3) = |-3 + 2| - 2|-3 - 2| = | -1 | - 2| -5 | = 1 - 2*5 = 1 - 10 = -9.Using the expression x - 6, G(-3) = -3 - 6 = -9. Correct.Similarly, at x = -4, G(-4) = |-4 + 2| - 2|-4 - 2| = | -2 | - 2| -6 | = 2 - 12 = -10.Using x - 6, G(-4) = -4 - 6 = -10. Correct.So, the expression for x < -2 is correct.Wait, but in the initial problem, the user had the range as [-8, 8], which suggests that the function can reach up to 8. But according to my calculations, the maximum is 4. Maybe the user made a mistake.Alternatively, perhaps I misread the function. Let me check again.The function is G(x) = |x + 2| - 2|x - 2|. Yes, that's correct.Wait, maybe I should consider the possibility that the function can reach higher values. Let me test some points.At x = 0, G(0) = |0 + 2| - 2|0 - 2| = 2 - 2*2 = 2 - 4 = -2.At x = 1, G(1) = |1 + 2| - 2|1 - 2| = 3 - 2*1 = 3 - 2 = 1.At x = 3, G(3) = |3 + 2| - 2|3 - 2| = 5 - 2*1 = 5 - 2 = 3.Wait, but according to my earlier analysis, for x ≥ 2, G(x) = -x + 6. So, at x = 3, G(3) = -3 + 6 = 3. Correct.Wait, but if I go to x = 4, G(4) = |4 + 2| - 2|4 - 2| = 6 - 2*2 = 6 - 4 = 2.Similarly, G(5) = |5 + 2| - 2|5 - 2| = 7 - 2*3 = 7 - 6 = 1.G(6) = |6 + 2| - 2|6 - 2| = 8 - 2*4 = 8 - 8 = 0.G(7) = |7 + 2| - 2|7 - 2| = 9 - 2*5 = 9 - 10 = -1.So, as x increases beyond 2, G(x) decreases from 4 to -∞.Wait, but in the initial problem, the user had the range as [-8, 8], which suggests that the function can reach up to 8. But according to my calculations, the maximum is 4. Maybe the user made a mistake in their initial answer.Alternatively, perhaps I made a mistake in the simplification for x < -2.Wait, let me try x = -5.G(-5) = |-5 + 2| - 2|-5 - 2| = | -3 | - 2| -7 | = 3 - 14 = -11.Using the expression x - 6, G(-5) = -5 - 6 = -11. Correct.Wait, but if I go to x = -10, G(-10) = |-10 + 2| - 2|-10 - 2| = | -8 | - 2| -12 | = 8 - 24 = -16.Using x - 6, G(-10) = -10 - 6 = -16. Correct.So, the function does go to -∞ as x approaches -∞, and the maximum value is 4 at x = 2.Therefore, the range should be (-∞, 4].But the initial problem's answer was [-8, 8]. Maybe the user confused the domain with the range or made a mistake in their calculations.Alternatively, perhaps I misread the function. Let me check again.The function is G(x) = |x + 2| - 2|x - 2|. Yes, that's correct.Wait, maybe I should consider the possibility that the function can reach higher values by considering different intervals or perhaps considering the function's behavior at different points.Wait, let me consider x = 2. G(2) = |2 + 2| - 2|2 - 2| = 4 - 0 = 4.x = 1: G(1) = 3 - 2 = 1.x = 0: G(0) = 2 - 4 = -2.x = -1: G(-1) = | -1 + 2| - 2| -1 - 2| = 1 - 2*3 = 1 - 6 = -5.x = -2: G(-2) = 0 - 2*4 = -8.x = -3: G(-3) = 1 - 2*5 = 1 - 10 = -9.x = -4: G(-4) = 2 - 2*6 = 2 - 12 = -10.So, as x decreases, G(x) decreases without bound.Therefore, the function's maximum value is 4, and it can go down to -∞. So, the range is (-∞, 4].But the initial problem's answer was [-8, 8], which is incorrect based on my calculations.Wait, perhaps the user made a mistake in their initial answer. Alternatively, maybe I misread the function.Wait, let me check the function again: G(x) = |x + 2| - 2|x - 2|. Yes, that's correct.Alternatively, perhaps the function is G(x) = |x + 2| - 2|x - 2|, which is what I have.Wait, maybe the user intended to write G(x) = |x + 2| - 2|x - 2|, but perhaps there was a typo in the initial problem.Alternatively, perhaps I should consider the function G(x) = |x + 2| - 2|x - 2| and see if it can reach 8.Wait, let me solve for G(x) = 8.So, |x + 2| - 2|x - 2| = 8.Let me consider the intervals again.1. For x < -2:G(x) = x - 6 = 8 => x = 14. But x = 14 is not in x < -2. So, no solution here.2. For -2 ≤ x < 2:G(x) = 3x - 2 = 8 => 3x = 10 => x = 10/3 ≈ 3.333. But x = 10/3 is not in -2 ≤ x < 2. So, no solution here.3. For x ≥ 2:G(x) = -x + 6 = 8 => -x = 2 => x = -2. But x = -2 is not in x ≥ 2. So, no solution here.Therefore, there is no x such that G(x) = 8. So, the function cannot reach 8.Therefore, the initial answer of [-8, 8] is incorrect. The correct range is (-∞, 4].Wait, but the initial problem's answer was [-8, 8], which suggests that the function can reach up to 8. Maybe the user made a mistake in their initial answer.Alternatively, perhaps I made a mistake in my calculations. Let me double-check.Wait, let me consider x = 2: G(2) = 4.x = 3: G(3) = 3.x = 4: G(4) = 2.x = 5: G(5) = 1.x = 6: G(6) = 0.x = 7: G(7) = -1.So, as x increases beyond 2, G(x) decreases from 4 to -∞.Similarly, as x decreases below -2, G(x) decreases from -8 to -∞.Therefore, the function's maximum value is 4, and it can go down to -∞. So, the range is (-∞, 4].Therefore, the initial answer of [-8, 8] is incorrect. The correct range is (-∞, 4].Wait, but the initial problem's answer was [-8, 8]. Maybe the user intended to write a different function, such as G(x) = |x + 2| + 2|x - 2|, which would have a different range.Alternatively, perhaps the user made a mistake in their initial answer.In conclusion, based on my calculations, the range of G(x) = |x + 2| - 2|x - 2| is (-∞, 4].

💡Okay, so I have this problem here about an arithmetic sequence and a geometric sequence. Let me try to understand what it's asking. First, it says that {a_n} is an arithmetic sequence with a common difference not equal to zero. That means each term increases by a fixed amount, right? So, the nth term can be written as a_n = a_1 + (n-1)d, where a_1 is the first term and d is the common difference. And since d is not zero, the sequence is either increasing or decreasing, but not constant.Then, it says that a_1, a_2, a_4 form a geometric sequence. Hmm, okay. So, in a geometric sequence, each term is multiplied by a common ratio to get the next term. That means if I have three terms, say x, y, z, then y/x should be equal to z/y, and that ratio is the common ratio, r.So, applying that to a_1, a_2, a_4, I should have a_2 / a_1 = a_4 / a_2. That ratio should be the same. Let me write that down:a_2 / a_1 = a_4 / a_2Which implies that (a_2)^2 = a_1 * a_4Okay, so I can use the expressions for a_2 and a_4 from the arithmetic sequence.Since it's an arithmetic sequence, a_2 = a_1 + d, and a_4 = a_1 + 3d.So substituting these into the equation:(a_1 + d)^2 = a_1 * (a_1 + 3d)Let me expand the left side:(a_1)^2 + 2a_1d + d^2 = a_1^2 + 3a_1dNow, subtract a_1^2 from both sides:2a_1d + d^2 = 3a_1dHmm, let's subtract 2a_1d from both sides:d^2 = a_1dSo, d^2 = a_1dSince d is not zero, I can divide both sides by d:d = a_1So, the first term of the arithmetic sequence is equal to the common difference.That's interesting. So, a_1 = d.Now, let me think about the geometric sequence part. The common ratio r is given by a_2 / a_1.Since a_2 = a_1 + d, and a_1 = d, then a_2 = d + d = 2d.So, r = a_2 / a_1 = 2d / d = 2.So, the common ratio is 2.Wait, let me double-check that. If a_1 = d, then a_2 = a_1 + d = d + d = 2d, and a_4 = a_1 + 3d = d + 3d = 4d.So, the geometric sequence is d, 2d, 4d, which is clearly a geometric sequence with a common ratio of 2.Yes, that makes sense. So, the common ratio is 2.I think that's the answer. Let me see if there's another way to approach this or if I missed something.Alternatively, I could have set up the terms as a, a + d, a + 3d and then set up the ratio:(a + d)/a = (a + 3d)/(a + d)Cross-multiplying gives (a + d)^2 = a(a + 3d)Which is the same equation as before. So, it leads to the same result.Therefore, I'm confident that the common ratio is 2.

💡Okay, so I have this problem about a function f(x) with some specific properties, and I need to compare the magnitudes of f(-x₁) and f(-x₂) given certain conditions on x₁ and x₂. Let me try to unpack this step by step.First, let me list out the given conditions to make sure I understand them:1. **Symmetry about x = 1:** The function satisfies f(1 + x) = f(1 - x). This means that the function is symmetric around the line x = 1. So, if I pick any point x, the value of the function at 1 + x is the same as at 1 - x. This is like reflecting the function over the line x = 1.2. **Monotonicity on [1, ∞):** The function is increasing on the interval [1, ∞). That means as x increases beyond 1, the value of f(x) also increases. So, for any two points a and b in [1, ∞) where a < b, we have f(a) ≤ f(b).3. **Conditions on x₁ and x₂:** We have x₁ < 0, x₂ > 0, and x₁ + x₂ < -2. So, x₁ is negative, x₂ is positive, and their sum is less than -2.Our goal is to compare f(-x₁) and f(-x₂). That is, we need to determine whether f(-x₁) is greater than, less than, or equal to f(-x₂).Let me start by trying to visualize or understand the implications of the given conditions.**Understanding the Symmetry:**The first condition tells us that the function is symmetric about x = 1. So, if I take any point x, the function's value at 1 + x is the same as at 1 - x. This suggests that the graph of f(x) is a mirror image around the vertical line x = 1.For example, f(1 + 1) = f(1 - 1) implies f(2) = f(0). Similarly, f(1 + 2) = f(1 - 2) implies f(3) = f(-1), and so on.This symmetry might help us relate f(-x₁) and f(-x₂) by expressing them in terms of points on the other side of the symmetry axis, which is x = 1.**Understanding the Monotonicity:**The second condition tells us that f(x) is increasing for all x ≥ 1. So, as we move to the right on the x-axis starting from x = 1, the function's value never decreases; it either stays the same or increases.This is important because it allows us to compare the function's values at different points in the interval [1, ∞). If we can express -x₁ and -x₂ in terms of points in this interval, we can use the monotonicity to compare f(-x₁) and f(-x₂).**Analyzing the Conditions on x₁ and x₂:**We have x₁ < 0 and x₂ > 0, so -x₁ > 0 and -x₂ < 0. The sum x₁ + x₂ < -2. Let's see what this tells us about -x₁ and -x₂.Starting with x₁ + x₂ < -2:Since x₁ is negative and x₂ is positive, their sum being less than -2 means that the magnitude of x₁ is greater than the magnitude of x₂ plus 2. In other words, |x₁| > |x₂| + 2.But since x₁ is negative, -x₁ = |x₁|, and x₂ is positive, so |x₂| = x₂. Therefore, we can rewrite the inequality as:-x₁ > x₂ + 2This is because:x₁ + x₂ < -2 => x₁ < -2 - x₂ Since x₁ is negative, multiplying both sides by -1 (and reversing the inequality):-x₁ > 2 + x₂So, we have:-x₁ > 2 + x₂This is a crucial inequality because it relates -x₁ and 2 + x₂, which might be useful when applying the symmetry condition.**Expressing f(-x₁) and f(-x₂) Using Symmetry:**Given that f(1 + x) = f(1 - x), let's try to express f(-x₁) and f(-x₂) in terms of f evaluated at points related to 1.Let's start with f(-x₁):We can write -x₁ as 1 + ( -x₁ - 1 ). So, let me set:Let x = -x₁ - 1Then, 1 + x = 1 + (-x₁ - 1) = -x₁And 1 - x = 1 - (-x₁ - 1) = 1 + x₁ + 1 = x₁ + 2Therefore, by the symmetry condition:f(-x₁) = f(1 + x) = f(1 - x) = f(x₁ + 2)Similarly, let's do the same for f(-x₂):Let x = -x₂ - 1Then, 1 + x = 1 + (-x₂ - 1) = -x₂And 1 - x = 1 - (-x₂ - 1) = 1 + x₂ + 1 = x₂ + 2Therefore, by the symmetry condition:f(-x₂) = f(1 + x) = f(1 - x) = f(x₂ + 2)So, now we have:f(-x₁) = f(x₁ + 2)f(-x₂) = f(x₂ + 2)**Comparing f(x₁ + 2) and f(x₂ + 2):**Now, we need to compare f(x₁ + 2) and f(x₂ + 2). To do this, let's analyze the arguments x₁ + 2 and x₂ + 2.Given that x₁ < 0 and x₂ > 0, let's see where x₁ + 2 and x₂ + 2 lie.First, x₁ + 2:Since x₁ < 0, x₁ + 2 < 2. But we need to know if x₁ + 2 is greater than or equal to 1 because the function is increasing on [1, ∞).Given that x₁ + x₂ < -2, and x₂ > 0, we can find a relationship between x₁ and x₂.From x₁ + x₂ < -2, since x₂ > 0, we have x₁ < -2 - x₂.But x₂ > 0, so -2 - x₂ < -2.Therefore, x₁ < -2 - x₂ < -2.So, x₁ < -2.Therefore, x₁ + 2 < 0.Wait, that can't be right because x₁ + 2 is less than 0? But earlier, we had -x₁ > 2 + x₂, which is positive.Wait, perhaps I made a miscalculation.Let me double-check.Given x₁ + x₂ < -2.Since x₂ > 0, then x₁ < -2 - x₂.But x₂ is positive, so -2 - x₂ is less than -2.Therefore, x₁ < -2 - x₂ < -2.So, x₁ < -2.Therefore, x₁ + 2 < 0.Wait, but earlier, we had -x₁ > 2 + x₂.Since x₁ < -2, -x₁ > 2.And x₂ > 0, so 2 + x₂ > 2.Therefore, -x₁ > 2 + x₂.But x₁ + 2 < 0, which is less than 1, so f(x₁ + 2) is not in the interval where f is increasing.Hmm, this seems conflicting.Wait, perhaps I need to reconsider.We have f(-x₁) = f(x₁ + 2), and f(-x₂) = f(x₂ + 2).But x₁ + 2 is less than 0, as x₁ < -2.Similarly, x₂ + 2 is greater than 2, since x₂ > 0.So, x₁ + 2 < 0, and x₂ + 2 > 2.But f is increasing on [1, ∞), so f(x₂ + 2) is greater than f(1) because x₂ + 2 > 1.But what about f(x₁ + 2)? Since x₁ + 2 < 0, which is less than 1, and we don't know the behavior of f on (-∞, 1).However, we have the symmetry condition f(1 + x) = f(1 - x). So, perhaps we can relate f(x₁ + 2) to f at some point greater than 1.Wait, let's try that.Let me set y = x₁ + 2.Since x₁ < -2, y = x₁ + 2 < 0.We can write y = 1 + (y - 1). So, let me set x = y - 1.Then, 1 + x = y, and 1 - x = 2 - y.Therefore, by the symmetry condition:f(y) = f(1 + x) = f(1 - x) = f(2 - y)So, f(x₁ + 2) = f(2 - (x₁ + 2)) = f(-x₁)Wait, that's circular because f(x₁ + 2) = f(-x₁), which is what we started with.Hmm, perhaps another approach.Wait, maybe I should consider that since f is symmetric about x = 1, and it's increasing on [1, ∞), then it must be decreasing on (-∞, 1].Is that necessarily true?Wait, not necessarily. The function could have any behavior on (-∞, 1], but given the symmetry, perhaps it's also symmetric in terms of increasing/decreasing.Wait, let me think.If f is symmetric about x = 1, then the behavior on the left side of 1 is a mirror image of the behavior on the right side.So, if f is increasing on [1, ∞), then on (-∞, 1], it should be decreasing.Because, for example, f(1 + x) = f(1 - x). So, as x increases beyond 1, f increases, and as x decreases below 1, f decreases.Therefore, f is decreasing on (-∞, 1].So, now, knowing that f is decreasing on (-∞, 1], and increasing on [1, ∞).Given that, let's see where x₁ + 2 and x₂ + 2 lie.We have x₁ + 2 < 0, as x₁ < -2.And x₂ + 2 > 2, as x₂ > 0.So, x₁ + 2 is in (-∞, 1], and x₂ + 2 is in [1, ∞).Therefore, f(x₁ + 2) is in the decreasing part of the function, and f(x₂ + 2) is in the increasing part.But how does this help us compare f(x₁ + 2) and f(x₂ + 2)?Well, since f is decreasing on (-∞, 1], and x₁ + 2 < 0 < 1, we can say that as x increases towards 1, f(x) decreases.Similarly, on [1, ∞), as x increases, f(x) increases.But we need to relate f(x₁ + 2) and f(x₂ + 2).Wait, perhaps we can find a relationship between x₁ + 2 and x₂ + 2.From earlier, we have:-x₁ > 2 + x₂Which can be rewritten as:x₁ < -2 - x₂But x₁ + 2 < 0, and x₂ + 2 > 2.Wait, perhaps we can consider the distance from 1.Let me think.Since f is symmetric about 1, the value of f at a point a distance d to the right of 1 is equal to the value at a distance d to the left of 1.So, f(1 + d) = f(1 - d).Given that, and knowing that f is increasing on [1, ∞), then for points to the right of 1, the further right you go, the higher the function value.Similarly, for points to the left of 1, the further left you go, the lower the function value because it's decreasing.So, if we have two points, one to the right of 1 and one to the left of 1, how do their function values compare?Well, if the distance from 1 is the same, their function values are equal.But if one is further away, then depending on the direction, the function value will be higher or lower.Wait, but in our case, x₁ + 2 is to the left of 1, and x₂ + 2 is to the right of 1.So, let's compute the distances from 1.For x₁ + 2:Distance from 1 is |(x₁ + 2) - 1| = |x₁ + 1|Similarly, for x₂ + 2:Distance from 1 is |(x₂ + 2) - 1| = |x₂ + 1|But since x₁ + 2 < 0, |x₁ + 1| = -(x₁ + 1) because x₁ + 1 < 0.Similarly, x₂ + 2 > 2, so |x₂ + 1| = x₂ + 1.Now, from the earlier inequality:-x₁ > 2 + x₂Which can be rewritten as:x₁ < -2 - x₂Let me add 1 to both sides:x₁ + 1 < -2 - x₂ + 1 x₁ + 1 < -1 - x₂Taking absolute values:|x₁ + 1| > | -1 - x₂ | = |x₂ + 1|Because x₁ + 1 < -1 - x₂, and since x₁ + 1 is negative, |x₁ + 1| = -(x₁ + 1), and |x₂ + 1| = x₂ + 1.So, we have:|x₁ + 1| > |x₂ + 1|This means that the distance of x₁ + 2 from 1 is greater than the distance of x₂ + 2 from 1.But since f is symmetric about 1, the function values depend on the distance from 1.However, on the right side of 1, as distance increases, f increases, and on the left side of 1, as distance increases, f decreases.Therefore, since |x₁ + 1| > |x₂ + 1|, and x₁ + 2 is on the left side, f(x₁ + 2) < f(1 - |x₁ + 1|) = f(1 + |x₁ + 1|).Wait, no, that might not be the right way to think about it.Wait, let's consider that f(1 + d) = f(1 - d). So, f(x₁ + 2) = f(1 - (1 - (x₁ + 2))) = f(1 - ( -x₁ -1 )) = f(1 + x₁ + 1) = f(x₁ + 2). Hmm, that's not helpful.Wait, perhaps another approach.Since f is decreasing on (-∞, 1], and x₁ + 2 < 0 < 1, then as x increases towards 1, f(x) decreases.Similarly, on [1, ∞), as x increases, f(x) increases.But we need to compare f(x₁ + 2) and f(x₂ + 2).Given that x₁ + 2 is to the left of 1, and x₂ + 2 is to the right of 1.But how do their distances from 1 compare?We have |x₁ + 1| > |x₂ + 1|, as established earlier.So, x₁ + 2 is further away from 1 on the left side, and x₂ + 2 is closer to 1 on the right side.But since f is decreasing on the left and increasing on the right, the further left you go, the smaller f(x) becomes, and the further right you go, the larger f(x) becomes.But in our case, x₁ + 2 is further left, so f(x₁ + 2) is smaller than f(1), and x₂ + 2 is further right, so f(x₂ + 2) is larger than f(1).But we need to compare f(x₁ + 2) and f(x₂ + 2).Wait, perhaps we can relate them through the symmetry.Since f(x₁ + 2) = f(-x₁) and f(x₂ + 2) = f(-x₂), and we have -x₁ > 2 + x₂, which is greater than 1, and -x₂ < 0, which is less than 1.But f is increasing on [1, ∞), so f(-x₁) > f(2 + x₂).Wait, but f(2 + x₂) = f(-x₂) by symmetry.Therefore, f(-x₁) > f(-x₂).Wait, that seems to be the conclusion.Let me recap:We have:f(-x₁) = f(x₁ + 2)f(-x₂) = f(x₂ + 2)From the given condition x₁ + x₂ < -2, we derived that -x₁ > 2 + x₂.Since -x₁ > 2 + x₂ and both -x₁ and 2 + x₂ are greater than 1 (because -x₁ > 2 + x₂ > 2), and f is increasing on [1, ∞), it follows that f(-x₁) > f(2 + x₂).But by symmetry, f(2 + x₂) = f(-x₂).Therefore, f(-x₁) > f(-x₂).So, the conclusion is that f(-x₁) is greater than f(-x₂).I think that makes sense.To summarize:1. Use the symmetry condition to express f(-x₁) and f(-x₂) in terms of f evaluated at points x₁ + 2 and x₂ + 2.2. Use the given condition x₁ + x₂ < -2 to establish that -x₁ > 2 + x₂.3. Recognize that both -x₁ and 2 + x₂ are in the interval [1, ∞), where f is increasing.4. Therefore, since -x₁ > 2 + x₂, and f is increasing, f(-x₁) > f(2 + x₂).5. By symmetry, f(2 + x₂) = f(-x₂).6. Hence, f(-x₁) > f(-x₂).I think that covers all the steps logically. I didn't see any gaps or errors in this reasoning.

💡Alright, so I've got this problem here about sets and functions. Let me try to wrap my head around it step by step. We have the set M which is {1, 2, ..., 20}. For any 9-element subset S of M, there's a function f(S) that gives an integer between 1 and 20, inclusive. The task is to prove that there exists a 10-element subset T of M such that for every element k in T, f(T {k}) is not equal to k. Hmm, okay. So, first, let me make sure I understand what's being asked. We need to find a 10-element subset T where if you remove any single element k from T, the function f applied to the resulting 9-element subset doesn't give back k. That is, f(T {k}) ≠ k for all k in T.I think this is a problem that might involve some combinatorial arguments or maybe the pigeonhole principle. Let me think about how to approach it.Maybe I should consider the number of possible 10-element subsets versus the number of possible 9-element subsets. Since f maps each 9-element subset to an integer between 1 and 20, perhaps there's a way to show that not all 10-element subsets can be "bad" in the sense that for each, there's some k where f(T {k}) = k.Wait, so if I can show that the number of 10-element subsets is greater than the number of 9-element subsets, then by the pigeonhole principle, there must be at least one 10-element subset that isn't "bad." That is, it doesn't have any k such that f(T {k}) = k.Let me check the numbers. The number of 10-element subsets of M is C(20,10), which is 184,756. The number of 9-element subsets is C(20,9), which is 167,960. So, actually, C(20,10) is larger than C(20,9). That might be useful.But wait, each 10-element subset T can be associated with 10 different 9-element subsets, each missing one element. So, for each T, there are 10 different S = T {k} where k is in T. If we think about the function f, for each S, f(S) is some element in M. So, for each S, f(S) could potentially be one of the 20 elements. But in our case, for a 10-element subset T to be "bad," there must exist at least one k in T such that f(T {k}) = k.So, if we consider all possible 10-element subsets, each can be mapped to 10 different 9-element subsets. If each of these 9-element subsets is mapped via f to some element, then the question is whether all these mappings can cover all possible 10-element subsets as "bad."But since the number of 10-element subsets is larger than the number of 9-element subsets, maybe we can argue that not all 10-element subsets can be "bad." Because each 9-element subset can only be associated with one "bad" 10-element subset? Wait, no, actually, each 9-element subset can be part of multiple 10-element subsets.Hmm, maybe I need to think differently. Let's consider that for each 9-element subset S, f(S) is some element in M. So, for each S, f(S) could be an element not in S or in S. But in our problem, we're concerned with 10-element subsets T where for each k in T, f(T {k}) ≠ k.So, if I fix a 10-element subset T, then for each k in T, f(T {k}) is some element. We want none of these f(T {k}) to equal k.Alternatively, if we think about it, for each 9-element subset S, f(S) is some element. So, if we have a 10-element subset T, and for each k in T, S = T {k}, then f(S) is some element. We want that for all k in T, f(S) ≠ k.So, if we can show that there's some T where none of the f(S) for S = T {k} equals k, then we're done.Maybe another approach is to use the probabilistic method or something like that, but I'm not sure. Alternatively, maybe we can model this as a graph problem.Let me think about it as a graph. Suppose we have a bipartite graph where one partition is the set of 10-element subsets and the other partition is the set of 9-element subsets. We can draw edges between a 10-element subset T and a 9-element subset S if S is a subset of T, i.e., S = T {k} for some k in T.Then, the function f assigns to each 9-element subset S an element f(S) in M. So, for each edge from T to S, we can think of f(S) as a label on that edge. We want to find a 10-element subset T such that for all edges from T to S, the label f(S) is not equal to the element k that was removed to get S from T.Wait, that might be a bit abstract. Let me see if I can formalize it.Each 10-element subset T has 10 edges connecting it to its 10 different 9-element subsets S. Each of these edges has a label f(S). We want that for none of these edges, the label f(S) equals the element k that was removed to get S from T.So, in other words, for T to be "good," none of the labels on its edges should correspond to the element that was removed.So, the problem reduces to showing that in this bipartite graph, there exists a node T in the 10-element subset partition such that none of its adjacent edges have labels equal to the corresponding removed elements.Hmm, okay. So, perhaps we can use some combinatorial argument here. Maybe we can count the number of possible "bad" assignments and show that it's less than the total number of possible assignments, hence there must exist a "good" T.Wait, but f is arbitrary, right? So, f is any function from 9-element subsets to elements of M. So, we have to show that regardless of how f is defined, there exists such a T.So, maybe it's better to think in terms of the pigeonhole principle. Since the number of 10-element subsets is larger than the number of 9-element subsets, perhaps each 9-element subset can only be responsible for making a limited number of 10-element subsets "bad."Let me try to formalize this. Suppose that for each 9-element subset S, f(S) is some element in M. If f(S) is in S, then S cannot be used to make a 10-element subset T "bad" because T would have to include f(S), but S is missing f(S) if f(S) is in S. Wait, no, S is a 9-element subset, so if f(S) is in S, then S union {f(S)} would be S itself, which is not possible because S is already 9 elements. Wait, maybe I'm getting confused.Alternatively, if f(S) is not in S, then f(S) could be an element that, when added to S, forms a 10-element subset T. So, for each S, if f(S) is not in S, then T = S union {f(S)} is a 10-element subset, and for this T, f(S) = f(T {f(S)}) = f(S). So, in this case, T would be "bad" because f(T {f(S)}) = f(S) = k, where k = f(S). But wait, k is f(S), which is not necessarily in T.Wait, no, T is S union {f(S)}, so f(S) is in T if and only if f(S) is not in S. So, if f(S) is not in S, then f(S) is in T, and f(T {f(S)}) = f(S) = k, which is in T. So, T is "bad" because there exists k in T such that f(T {k}) = k.On the other hand, if f(S) is in S, then T = S union {f(S)} is just S, which is a 9-element subset, so that doesn't make sense. So, in that case, f(S) being in S doesn't contribute to a "bad" 10-element subset.Therefore, for each 9-element subset S, if f(S) is not in S, then it defines a unique "bad" 10-element subset T = S union {f(S)}. If f(S) is in S, then it doesn't define any "bad" 10-element subset.So, the number of "bad" 10-element subsets is at most the number of 9-element subsets S where f(S) is not in S. Since for each such S, we get a unique T.But how many such S are there? Each S has 9 elements, so the number of elements not in S is 11. So, for each S, f(S) can be any of the 20 elements, but only 11 of them would make T = S union {f(S)} a "bad" 10-element subset.Therefore, the number of "bad" 10-element subsets is at most 11 times the number of 9-element subsets, but actually, since each S can only contribute at most one "bad" T, it's at most the number of 9-element subsets where f(S) is not in S.But wait, actually, each "bad" T can be formed in multiple ways. For example, a single T can be formed by multiple S's. Specifically, each T has 10 different S's, each missing one element. So, for each T, there are 10 different S's that could potentially map to it if f(S) is set appropriately.But in our case, each S can map to at most one T, namely T = S union {f(S)} if f(S) is not in S. So, the total number of "bad" T's is at most the number of 9-element subsets S where f(S) is not in S.But how many such S's are there? Since f is arbitrary, f(S) can be anything, so potentially, all S's could have f(S) not in S, making the number of "bad" T's equal to the number of 9-element subsets, which is C(20,9).But wait, the number of 10-element subsets is C(20,10), which is larger than C(20,9). So, even if every 9-element subset S with f(S) not in S contributes a unique "bad" T, the total number of "bad" T's is at most C(20,9), which is less than C(20,10). Therefore, there must be some T's that are not "bad," meaning they are "good."Therefore, such a T exists.Wait, let me double-check that. The number of "bad" T's is at most C(20,9), and the total number of T's is C(20,10). Since C(20,10) > C(20,9), there must be some T's that are not "bad," i.e., for all k in T, f(T {k}) ≠ k.Yes, that makes sense. So, by the pigeonhole principle, since the number of 10-element subsets exceeds the number of 9-element subsets, and each "bad" T is associated with at least one S, but the number of S's is less, there must be some T's left that are not "bad."Therefore, such a T exists.I think that's the gist of it. So, the key idea is to count the number of possible "bad" 10-element subsets and show that it's less than the total number of 10-element subsets, hence at least one "good" T must exist.