💡Okay, so I have these two problems to solve. Let me take them one at a time. Starting with the first one: Given real numbers x and y satisfying |x - y| < 1 and |2x + y| < 1, I need to prove that |y| < 1. Hmm, okay. I remember that absolute value inequalities can sometimes be manipulated by adding or subtracting them. Maybe I can express y in terms of x or vice versa.Let me write down the given inequalities:1. |x - y| < 12. |2x + y| < 1I need to find a way to combine these to get |y| < 1. Maybe I can solve for y in one inequality and substitute into the other. Let's see.From the first inequality, |x - y| < 1, which implies that -1 < x - y < 1. If I rearrange this, I get y - 1 < x < y + 1. So x is within 1 unit of y.From the second inequality, |2x + y| < 1, which implies that -1 < 2x + y < 1.Hmm, maybe I can express x from the first inequality and plug it into the second. Let me solve for x in terms of y.From |x - y| < 1, we have x = y + t, where |t| < 1. So t is some number between -1 and 1.Substituting x = y + t into the second inequality:|2(y + t) + y| < 1 |2y + 2t + y| < 1 |3y + 2t| < 1So, |3y + 2t| < 1. Since |t| < 1, 2t is between -2 and 2. So 3y + 2t is between 3y - 2 and 3y + 2.But I need to find |y|. Maybe I can bound y by considering the maximum and minimum values.Let me think differently. Maybe I can use the triangle inequality. The triangle inequality says that |a + b| ≤ |a| + |b|. So, |3y| = |(2x + y) - 2(x - y)|.Wait, that might be a bit convoluted, but let's try it.Let me express 3y as (2x + y) - 2(x - y). So,3y = (2x + y) - 2(x - y)Taking absolute values on both sides:|3y| = |(2x + y) - 2(x - y)| ≤ |2x + y| + 2|x - y|We know that |2x + y| < 1 and |x - y| < 1, so:|3y| ≤ |2x + y| + 2|x - y| < 1 + 2*1 = 3Therefore, |3y| < 3, which implies |y| < 1. That seems to work!Okay, so that was the first problem. Now, moving on to the second one.Given that a > b > c > d, I need to prove that:1/(a - b) + 1/(b - c) + 1/(c - d) ≥ 9/(a - d)Hmm, this looks like an inequality involving reciprocals and sums. I remember that inequalities like AM ≥ GM or Cauchy-Schwarz might be useful here.Let me denote the differences as follows:Let x = a - b, y = b - c, z = c - d. Since a > b > c > d, all of x, y, z are positive.So, the left-hand side becomes 1/x + 1/y + 1/z, and the right-hand side is 9/(a - d). But a - d is equal to x + y + z. So, the inequality becomes:1/x + 1/y + 1/z ≥ 9/(x + y + z)Hmm, this looks familiar. I think this is related to the Cauchy-Schwarz inequality or perhaps the AM-HM inequality.Recall that for positive real numbers, the harmonic mean is less than or equal to the arithmetic mean. But here, we have the sum of reciprocals on the left and a multiple of the reciprocal of the sum on the right.Wait, maybe Cauchy-Schwarz can be applied here. The Cauchy-Schwarz inequality in the form:(1/x + 1/y + 1/z)(x + y + z) ≥ (1 + 1 + 1)^2 = 9Yes, that seems right. Let me write it out.By Cauchy-Schwarz,(1/x + 1/y + 1/z)(x + y + z) ≥ (1 + 1 + 1)^2 = 9Therefore,1/x + 1/y + 1/z ≥ 9/(x + y + z)But x + y + z = a - d, so substituting back,1/(a - b) + 1/(b - c) + 1/(c - d) ≥ 9/(a - d)Which is exactly what we needed to prove. So that works!Wait, let me double-check. If I apply Cauchy-Schwarz to the vectors (1/sqrt(x), 1/sqrt(y), 1/sqrt(z)) and (sqrt(x), sqrt(y), sqrt(z)), then their dot product is 3, and the product of their norms is sqrt(1/x + 1/y + 1/z) * sqrt(x + y + z). So, by Cauchy-Schwarz, 3 ≤ sqrt(1/x + 1/y + 1/z) * sqrt(x + y + z). Squaring both sides gives 9 ≤ (1/x + 1/y + 1/z)(x + y + z), which is the same as before. So yes, that's correct.Alternatively, using the AM-HM inequality: For positive real numbers, (x + y + z)/3 ≥ 3/(1/x + 1/y + 1/z). Rearranging, 1/x + 1/y + 1/z ≥ 9/(x + y + z). So that's another way to see it.Either way, the inequality holds. So both problems are solved!**Final Answer**(1) boxed{|y| < 1}(2) boxed{frac{1}{a-b} + frac{1}{b-c} + frac{1}{c-d} geq frac{9}{a-d}}

💡Okay, so I have this problem where I need to use four boards, each with four squares, to form a shape shown in Figure 2. Then, I have to fill all 16 squares with the numbers 1, 3, 5, and 7. The catch is that each row, each column, and each board must have different numbers. Finally, I need to find the average of the numbers in squares A, B, C, and D.First, I need to visualize what's going on. Figure 1 has four boards, each with four squares. Figure 2 is a shape made up of these four boards. Since each board has four squares, and there are four boards, that makes 16 squares in total, which matches the 16 squares in Figure 2.Now, the numbers I can use are 1, 3, 5, and 7. Each square must contain one of these numbers, and no number can repeat in any row, column, or board. So, it's similar to a Sudoku puzzle but with different numbers and a different grid arrangement.Since each board is a 4-square board, and each must contain the numbers 1, 3, 5, and 7 without repetition, the sum of the numbers in each board is 1 + 3 + 5 + 7, which is 16. Similarly, each row and each column in Figure 2 must also contain these four numbers, so each row and column will also sum to 16.Now, the question is about the average of the numbers in squares A, B, C, and D. I need to figure out where these squares are located. Since the problem doesn't specify their exact positions, I have to assume they are four specific squares in Figure 2. However, without seeing the figure, I can't know their exact positions. But maybe I don't need to know their exact positions because of the constraints.Since each board, row, and column must contain all four numbers, the sum of any four squares that form a board, a row, or a column will always be 16. If A, B, C, and D are part of such a group (either a board, a row, or a column), their sum will be 16, and the average will be 16 divided by 4, which is 4.But wait, what if A, B, C, and D are not all in the same board, row, or column? Then their sum might not necessarily be 16. However, given the constraints, it's likely that A, B, C, and D are positioned in such a way that they form a complete set of numbers from 1, 3, 5, and 7. Therefore, their sum would still be 16, and the average would be 4.Alternatively, if A, B, C, and D are spread out across different boards, rows, and columns, their individual numbers could vary, but since each number must appear exactly once in each row, column, and board, the average should still be consistent.To double-check, let's consider that each number 1, 3, 5, and 7 appears exactly four times in the entire grid. Since there are 16 squares, each number appears four times. Therefore, the total sum of all numbers in the grid is 4*(1 + 3 + 5 + 7) = 4*16 = 64. The average of all numbers is 64/16 = 4. So, regardless of which four squares we pick, if they are a representative sample, their average should also be 4.Therefore, without knowing the exact positions of A, B, C, and D, based on the constraints and the uniform distribution of numbers, the average of these four numbers is likely 4.

💡Okay, so I have this problem here where I need to find the necessary and sufficient condition for the product of two complex numbers, (a + bi) and (c + di), to be a real number. The options are given as A, B, C, D, each with a different equation. Hmm, let me try to figure this out step by step.First, I remember that when you multiply two complex numbers, you use the distributive property, right? So, let me write that out. If I have (a + bi)(c + di), I can expand this by multiplying each term in the first complex number by each term in the second one.So, that would be a*c + a*di + bi*c + bi*di. Let me write that down:(a + bi)(c + di) = a*c + a*di + bi*c + bi*di.Now, simplifying each term:- a*c is just ac, which is a real number.- a*di is ad*i, which is an imaginary term.- bi*c is bc*i, another imaginary term.- bi*di is b*d*i^2. Wait, i squared is -1, so this becomes -bd.So, putting it all together, the product is:ac + adi + bci - bd.Now, let me group the real parts and the imaginary parts:Real parts: ac - bd.Imaginary parts: adi + bci. I can factor out the i, so that becomes (ad + bc)i.So, the entire expression is (ac - bd) + (ad + bc)i.Now, the problem says that this product needs to be a real number. For a complex number to be real, its imaginary part must be zero. That means the coefficient of i has to be zero.So, the imaginary part is (ad + bc)i, which means the coefficient is (ad + bc). Therefore, for the product to be real, we must have:ad + bc = 0.Looking back at the options given:A: ad - bc = 0B: ac - bd = 0C: ac + bd = 0D: ad + bc = 0So, the condition we found is ad + bc = 0, which is option D.Wait, let me double-check my steps to make sure I didn't make a mistake.1. Expanded the product correctly: (a + bi)(c + di) = ac + adi + bci + bdi^2.2. Simplified each term: ac is real, adi and bci are imaginary, and bdi^2 is -bd, which is real.3. Grouped real and imaginary parts: (ac - bd) + (ad + bc)i.4. Set the imaginary part to zero: ad + bc = 0.Yes, that seems correct. So, the necessary and sufficient condition is ad + bc = 0, which is option D.I think that's it. I don't see any mistakes in my reasoning. It all checks out.**Final Answer**boxed{D}

💡First, I need to expand ((x - 2)^3) using the binomial theorem. The binomial theorem states that ((a + b)^n = sum_{k=0}^{n} binom{n}{k} a^{n-k} b^k). Applying this to ((x - 2)^3), I get:[(x - 2)^3 = x^3 - 6x^2 + 12x - 8]Next, I multiply this expanded form by ((2x + 1)):[(2x + 1)(x^3 - 6x^2 + 12x - 8)]To find the coefficient of (x^2) in the final expansion, I focus on the terms that will contribute to (x^2) when multiplied:- Multiplying (2x) by (-6x^2) gives (-12x^3), which is not needed for the (x^2) term.- Multiplying (2x) by (12x) gives (24x^2).- Multiplying (1) by (-6x^2) gives (-6x^2).Adding these contributions together:[24x^2 - 6x^2 = 18x^2]Therefore, the coefficient of (x^2) in the expansion is 18.

💡Okay, so I have this function f(x) that satisfies the condition f(x + 2) = 1 / f(x) for any real number x. They also told me that f(1) = -5, and I need to find f(f(5)). Hmm, let's break this down step by step.First, I need to understand what the condition f(x + 2) = 1 / f(x) means. It seems like every time I increase x by 2, the function value becomes the reciprocal of the original function value. So, if I know f(x), then f(x + 2) is just 1 divided by f(x). Interesting.Let me try to compute some values step by step. I know f(1) = -5. So, if I plug x = 1 into the condition, I get f(1 + 2) = f(3) = 1 / f(1) = 1 / (-5) = -1/5. Okay, so f(3) is -1/5.Now, what about f(5)? If I plug x = 3 into the condition, f(3 + 2) = f(5) = 1 / f(3) = 1 / (-1/5) = -5. Wait, so f(5) is -5? That's the same as f(1). Hmm, that's interesting. So, f(5) = f(1). Maybe this function has some periodicity?Let me check. If I go two steps ahead each time, starting from 1:f(1) = -5f(3) = 1 / f(1) = -1/5f(5) = 1 / f(3) = -5f(7) = 1 / f(5) = -1/5So, it seems like every time I add 4 to x, the function value repeats. So, f(x + 4) = f(x). That means the function is periodic with period 4. That's a key insight!So, knowing that f is periodic with period 4, I can find f(5) by noting that 5 is 1 + 4, so f(5) = f(1) = -5. That's consistent with what I found earlier.Now, the problem asks for f(f(5)). Since f(5) is -5, I need to find f(-5). Hmm, f(-5). How can I find that?Well, since the function is periodic with period 4, f(-5) is the same as f(-5 + 4 * k) where k is an integer. Let me choose k = 2, so -5 + 8 = 3. Therefore, f(-5) = f(3). From earlier, I know f(3) is -1/5. So, f(-5) = -1/5.Therefore, f(f(5)) = f(-5) = -1/5.Wait, let me double-check that. So, f(5) is -5, and f(-5) is f(3) because of periodicity, which is -1/5. So, yes, f(f(5)) is -1/5. That seems correct.But just to be thorough, let me see if I can find f(-5) another way without relying on periodicity. Maybe using the original condition.Starting from f(x + 2) = 1 / f(x), if I set x = -1, then f(-1 + 2) = f(1) = 1 / f(-1). We know f(1) is -5, so:f(1) = 1 / f(-1) => -5 = 1 / f(-1) => f(-1) = -1/5.But how does that help me find f(-5)? Well, since the function is periodic with period 4, f(-5) = f(-5 + 4) = f(-1). So, f(-5) = f(-1) = -1/5. Yep, same result.So, whether I use periodicity directly or break it down step by step, I get the same answer. That gives me more confidence that it's correct.Just to recap:1. Given f(x + 2) = 1 / f(x), which implies periodicity with period 4.2. f(1) = -5.3. Therefore, f(3) = 1 / f(1) = -1/5.4. Then, f(5) = f(1) = -5.5. To find f(f(5)) = f(-5), use periodicity: f(-5) = f(-1).6. From f(-1 + 2) = f(1) = 1 / f(-1), we find f(-1) = -1/5.7. Thus, f(f(5)) = -1/5.So, the answer should be -1/5, which is option B.**Final Answer**boxed{B}

💡Okay, so I have this problem here about the positional relationship between a line and a circle. The line is given by the equation y = kx + 1, and the circle is x² + y² = 4. The options are A: Intersect, B: Tangent, C: Separate, D: Depends on the value of k. Hmm, I need to figure out whether the line intersects the circle, is tangent to it, doesn't intersect at all, or if it depends on the value of k.First, let me recall what each of these terms means. If two geometric figures intersect, that means they cross each other at one or more points. If they are tangent, they touch at exactly one point. If they are separate, they don't touch or intersect at all. So, the question is asking about the relationship between this line and the circle.I remember that to determine the relationship between a line and a circle, we can substitute the equation of the line into the equation of the circle and solve for the points of intersection. Then, by looking at the discriminant of the resulting quadratic equation, we can determine whether they intersect, are tangent, or are separate.So, let's try that approach. The circle equation is x² + y² = 4. The line is y = kx + 1. I can substitute y from the line into the circle equation.Substituting y = kx + 1 into x² + y² = 4 gives:x² + (kx + 1)² = 4Let me expand that:x² + (k²x² + 2kx + 1) = 4Combine like terms:x² + k²x² + 2kx + 1 = 4Factor the x² terms:(1 + k²)x² + 2kx + 1 = 4Subtract 4 from both sides to set the equation to zero:(1 + k²)x² + 2kx + 1 - 4 = 0Simplify:(1 + k²)x² + 2kx - 3 = 0So now I have a quadratic equation in terms of x: (1 + k²)x² + 2kx - 3 = 0To find out how many real solutions this equation has, I can compute the discriminant. The discriminant of a quadratic equation ax² + bx + c = 0 is b² - 4ac. If the discriminant is positive, there are two real solutions (the line intersects the circle at two points). If it's zero, there's exactly one solution (the line is tangent to the circle). If it's negative, there are no real solutions (the line doesn't intersect the circle).So, let's compute the discriminant for our quadratic equation.Here, a = (1 + k²), b = 2k, and c = -3.Discriminant D = b² - 4ac = (2k)² - 4*(1 + k²)*(-3)Compute each part:(2k)² = 4k²4*(1 + k²)*(-3) = -12*(1 + k²) = -12 -12k²So, D = 4k² - (-12 -12k²) = 4k² +12 +12k² = (4k² +12k²) +12 = 16k² +12So, D = 16k² +12Now, let's analyze this discriminant. Since k is a real number, k² is always non-negative. Therefore, 16k² is non-negative, and 16k² +12 is always positive because 12 is positive. So, D is always greater than zero, regardless of the value of k.Since the discriminant is always positive, the quadratic equation always has two distinct real roots. That means the line y = kx +1 always intersects the circle x² + y² =4 at two distinct points, no matter what the value of k is.Wait, but let me think again. The line is y = kx +1. The circle is centered at the origin with radius 2. The point (0,1) is on the line because when x=0, y=1. Is (0,1) inside the circle? Let's check.The distance from the origin to (0,1) is 1, which is less than the radius 2. So, the point (0,1) is inside the circle. Therefore, the line passes through a point inside the circle. So, does that mean it must intersect the circle at two points?Yes, that makes sense. If a line passes through a point inside a circle, it must intersect the circle at two points. So, regardless of the slope k, as long as the line passes through (0,1), which is inside the circle, it will intersect the circle at two points.Therefore, the positional relationship is always intersecting, regardless of the value of k. So, the answer should be A: Intersect.But wait, let me make sure I didn't make a mistake in my discriminant calculation. Let me go through it again.Starting with the substitution:x² + (kx +1)² =4Expanding:x² + k²x² + 2kx +1 =4Combine like terms:(1 +k²)x² +2kx +1 -4=0Which is:(1 +k²)x² +2kx -3=0Discriminant D = (2k)^2 -4*(1 +k²)*(-3)Compute:4k² -4*(1 +k²)*(-3) =4k² +12 +12k²=16k² +12Yes, that's correct. So D=16k² +12, which is always positive because 16k² is non-negative and 12 is positive. So, D>0 for all real k.Therefore, the line always intersects the circle at two points, so the answer is A: Intersect.Alternatively, thinking geometrically, since the line passes through (0,1), which is inside the circle of radius 2 centered at (0,0), the line must intersect the circle at two points. So, regardless of the slope, it will always cut through the circle.Therefore, the correct answer is A: Intersect.**Final Answer**boxed{A}