💡Okay, so I have this inequality to prove: if ( x_{1} geq x_{2} geq x_{3} geq x_{4} geq x_{5} geq 0 ), thenleft(x_{1} + x_{2} + x_{3} + x_{4} + x_{5}right)^{2} geq frac{25}{2}left(x_{4}^{2} + x_{5}^{2}right).And I also need to figure out when equality holds. Hmm, let's break this down step by step.First, I notice that the left-hand side (LHS) is the square of the sum of all five variables, while the right-hand side (RHS) is a multiple of the sum of the squares of the last two variables. Since all variables are non-negative and ordered in decreasing order, maybe I can use some inequality like Cauchy-Schwarz or AM-QM or something similar.Let me recall the Cauchy-Schwarz inequality: for any real numbers ( a_i ) and ( b_i ), we haveleft( sum a_i b_i right)^2 leq left( sum a_i^2 right) left( sum b_i^2 right).But I'm not sure if that directly applies here. Maybe I can think about the sum of the variables on the LHS and relate it to the sum of squares on the RHS.Another idea is to use the fact that since ( x_1 geq x_2 geq x_3 geq x_4 geq x_5 geq 0 ), each of the first three variables is at least as large as ( x_4 ). So, maybe I can bound the sum ( x_1 + x_2 + x_3 ) from below by ( 3x_4 ). That might help in relating the LHS to the RHS.Let me try that. If ( x_1 geq x_2 geq x_3 geq x_4 ), then each of ( x_1, x_2, x_3 ) is at least ( x_4 ). So,x_1 + x_2 + x_3 geq 3x_4.Therefore, the total sum ( x_1 + x_2 + x_3 + x_4 + x_5 geq 3x_4 + x_4 + x_5 = 4x_4 + x_5 ).So, the LHS squared is at least ( (4x_4 + x_5)^2 ). Let me compute that:(4x_4 + x_5)^2 = 16x_4^2 + 8x_4x_5 + x_5^2.Now, I need to compare this to ( frac{25}{2}(x_4^2 + x_5^2) ). Let me compute ( frac{25}{2}(x_4^2 + x_5^2) ):frac{25}{2}x_4^2 + frac{25}{2}x_5^2.So, I need to show that:16x_4^2 + 8x_4x_5 + x_5^2 geq frac{25}{2}x_4^2 + frac{25}{2}x_5^2.Let me rearrange this inequality:16x_4^2 + 8x_4x_5 + x_5^2 - frac{25}{2}x_4^2 - frac{25}{2}x_5^2 geq 0.Simplify the coefficients:For ( x_4^2 ): ( 16 - frac{25}{2} = frac{32}{2} - frac{25}{2} = frac{7}{2} ).For ( x_5^2 ): ( 1 - frac{25}{2} = -frac{23}{2} ).So, the inequality becomes:frac{7}{2}x_4^2 + 8x_4x_5 - frac{23}{2}x_5^2 geq 0.Hmm, this seems a bit messy. Maybe I made a wrong turn somewhere. Let me think again.Wait, perhaps instead of trying to directly compare ( (4x_4 + x_5)^2 ) to ( frac{25}{2}(x_4^2 + x_5^2) ), I should consider another approach. Maybe using the Cauchy-Schwarz inequality in a different way.Let me consider the sum ( x_1 + x_2 + x_3 + x_4 + x_5 ). Since ( x_1 geq x_2 geq x_3 geq x_4 geq x_5 ), the sum is maximized when all variables are equal, but here we have a specific ordering. Maybe I can use the fact that the sum is at least ( 5x_5 ), but that might not be helpful.Alternatively, perhaps I can use the QM-AM inequality, which states that the quadratic mean is at least the arithmetic mean. But I'm not sure.Wait, another idea: since ( x_1 geq x_2 geq x_3 geq x_4 geq x_5 ), maybe I can bound ( x_1 + x_2 + x_3 ) in terms of ( x_4 ) and ( x_5 ). For example, ( x_1 + x_2 + x_3 geq 3x_4 ), as I thought earlier.So, the total sum ( S = x_1 + x_2 + x_3 + x_4 + x_5 geq 4x_4 + x_5 ).Now, I need to relate ( S^2 ) to ( x_4^2 + x_5^2 ). Maybe I can use the Cauchy-Schwarz inequality on the terms ( 4x_4 + x_5 ).Let me think: if I have ( 4x_4 + x_5 ), I can write this as ( 4x_4 + 1x_5 ). Then, using Cauchy-Schwarz,(4x_4 + x_5)^2 leq (4^2 + 1^2)(x_4^2 + x_5^2) = 17(x_4^2 + x_5^2).Wait, but that's an upper bound, and I need a lower bound for ( S^2 ). So, this might not help directly.Alternatively, maybe I can use the reverse Cauchy-Schwarz? Not sure.Wait, perhaps I can use the fact that ( (a + b)^2 geq 4ab ). Let me see:If I let ( a = 4x_4 ) and ( b = x_5 ), then(4x_4 + x_5)^2 geq 4 cdot 4x_4 cdot x_5 = 16x_4x_5.But I'm not sure if this helps either.Wait, maybe I should consider the ratio of the two sides. Let me define ( S = x_1 + x_2 + x_3 + x_4 + x_5 ), and I need to show that ( S^2 geq frac{25}{2}(x_4^2 + x_5^2) ).So, the ratio ( frac{S^2}{x_4^2 + x_5^2} geq frac{25}{2} ).Alternatively, ( frac{S}{sqrt{x_4^2 + x_5^2}} geq frac{5}{sqrt{2}} ).This looks like a ratio that might be maximized or minimized under certain conditions.Wait, another approach: perhaps use Lagrange multipliers to find the minimum of ( S^2 ) given the constraints ( x_1 geq x_2 geq x_3 geq x_4 geq x_5 geq 0 ) and fixed ( x_4 ) and ( x_5 ). But that might be complicated.Alternatively, maybe I can consider the case where all variables are equal, which would give me the equality condition. If ( x_1 = x_2 = x_3 = x_4 = x_5 = c ), then(5c)^2 = 25c^2,andfrac{25}{2}(c^2 + c^2) = frac{25}{2} cdot 2c^2 = 25c^2.So, equality holds when all variables are equal. That's a good sign.But I need to show that for any ordered variables, the inequality holds. Maybe I can consider perturbing the variables while keeping the order and see if the inequality still holds.Wait, perhaps I can use the fact that the sum ( S ) is minimized when ( x_1 = x_2 = x_3 = x_4 = x_5 ), but I'm not sure.Wait, another idea: use the Cauchy-Schwarz inequality in the form:(x_1 + x_2 + x_3 + x_4 + x_5)^2 leq (1^2 + 1^2 + 1^2 + 1^2 + 1^2)(x_1^2 + x_2^2 + x_3^2 + x_4^2 + x_5^2).But that gives an upper bound, not a lower bound.Alternatively, maybe use the fact that ( x_1 + x_2 + x_3 geq 3x_4 ), so ( S geq 4x_4 + x_5 ), and then try to bound ( S^2 ) in terms of ( x_4 ) and ( x_5 ).Wait, let's go back to the earlier step where I had:(4x_4 + x_5)^2 = 16x_4^2 + 8x_4x_5 + x_5^2.I need to show that this is at least ( frac{25}{2}(x_4^2 + x_5^2) ).So, let's write:16x_4^2 + 8x_4x_5 + x_5^2 geq frac{25}{2}x_4^2 + frac{25}{2}x_5^2.Subtracting the RHS from both sides:16x_4^2 + 8x_4x_5 + x_5^2 - frac{25}{2}x_4^2 - frac{25}{2}x_5^2 geq 0.Simplify:left(16 - frac{25}{2}right)x_4^2 + 8x_4x_5 + left(1 - frac{25}{2}right)x_5^2 geq 0.Calculate the coefficients:frac{7}{2}x_4^2 + 8x_4x_5 - frac{23}{2}x_5^2 geq 0.Hmm, this is a quadratic in terms of ( x_4 ) and ( x_5 ). Maybe I can write it as:frac{7}{2}x_4^2 + 8x_4x_5 - frac{23}{2}x_5^2 geq 0.Let me factor this expression or see if it can be written as a square. Alternatively, consider it as a quadratic in ( x_4 ):frac{7}{2}x_4^2 + 8x_5x_4 - frac{23}{2}x_5^2 geq 0.Multiply both sides by 2 to eliminate fractions:7x_4^2 + 16x_5x_4 - 23x_5^2 geq 0.This is a quadratic in ( x_4 ):7x_4^2 + 16x_5x_4 - 23x_5^2 geq 0.Let me compute the discriminant ( D ):D = (16x_5)^2 - 4 cdot 7 cdot (-23x_5^2) = 256x_5^2 + 644x_5^2 = 900x_5^2.So, the roots are:x_4 = frac{-16x_5 pm sqrt{900x_5^2}}{2 cdot 7} = frac{-16x_5 pm 30x_5}{14}.Calculating the two roots:1. ( x_4 = frac{-16x_5 + 30x_5}{14} = frac{14x_5}{14} = x_5 ).2. ( x_4 = frac{-16x_5 - 30x_5}{14} = frac{-46x_5}{14} = -frac{23}{7}x_5 ).Since ( x_4 geq x_5 geq 0 ), the second root is negative and thus irrelevant. So, the quadratic is positive when ( x_4 geq x_5 ) or ( x_4 leq -frac{23}{7}x_5 ). But since ( x_4 geq x_5 geq 0 ), the quadratic is positive for ( x_4 geq x_5 ).Therefore, the inequality ( 7x_4^2 + 16x_5x_4 - 23x_5^2 geq 0 ) holds when ( x_4 geq x_5 ), which is given.Thus, going back, we have:(4x_4 + x_5)^2 geq frac{25}{2}(x_4^2 + x_5^2).Since ( S = x_1 + x_2 + x_3 + x_4 + x_5 geq 4x_4 + x_5 ), it follows that:S^2 geq (4x_4 + x_5)^2 geq frac{25}{2}(x_4^2 + x_5^2).Therefore, the original inequality holds.Now, for equality conditions. Equality holds when all the inequalities used become equalities.First, in the step where ( x_1 + x_2 + x_3 geq 3x_4 ), equality holds when ( x_1 = x_2 = x_3 = x_4 ).Second, in the step where ( (4x_4 + x_5)^2 geq frac{25}{2}(x_4^2 + x_5^2) ), equality holds when the quadratic expression equals zero, which happens when ( x_4 = x_5 ).Therefore, combining these, equality holds when ( x_1 = x_2 = x_3 = x_4 = x_5 ).So, the conditions for equality are that all variables are equal.

💡First, I need to determine how many blocks Wayne initially collected. I know that after his father gave him 6 more blocks, Wayne has a total of 15 blocks.To find the initial number of blocks, I can subtract the 6 blocks his father gave him from the total number of blocks he has now.So, 15 blocks (total after receiving more) minus 6 blocks (given by his father) equals 9 blocks (initially collected by Wayne).Therefore, Wayne initially collected 9 blocks.

💡Okay, so I have two math problems here. The first one is about finding the domain of a function, and the second one is about finding the range of another function. Let me tackle them one by one.Starting with the first problem: Find the domain of the function ( y = sqrt{1 - cos frac{x}{2}} ). Hmm, the domain of a function refers to all the possible input values (x-values) for which the function is defined. Since this function involves a square root, I remember that the expression inside the square root must be greater than or equal to zero. So, I need to set up the inequality:( 1 - cos frac{x}{2} geq 0 )Simplifying this, I get:( cos frac{x}{2} leq 1 )Now, I know that the cosine function oscillates between -1 and 1 for all real numbers. So, ( cos theta leq 1 ) is always true for any real number ( theta ). In this case, ( theta = frac{x}{2} ), which is also a real number for any real x. Therefore, the inequality ( cos frac{x}{2} leq 1 ) holds for all real numbers x. Wait, does that mean the domain is all real numbers? Because if the expression inside the square root is always non-negative, then the function is defined for every real number x. So, the domain should be ( mathbb{R} ), which is all real numbers. But just to double-check, let me think about the cosine function. The maximum value of cosine is 1, and it occurs at multiples of ( 2pi ). So, ( cos frac{x}{2} ) will be 1 when ( frac{x}{2} = 2pi n ) for any integer n, which simplifies to ( x = 4pi n ). At these points, the expression inside the square root becomes zero, which is still acceptable because the square root of zero is zero. For other values of x, ( cos frac{x}{2} ) will be less than 1, making ( 1 - cos frac{x}{2} ) positive, so the square root is still defined. Therefore, I'm confident that the domain is all real numbers.Moving on to the second problem: Find the range of the function ( y = frac{3sin x + 1}{sin x - 2} ). The range of a function refers to all possible output values (y-values) it can take. To find the range, I need to determine all possible values of y given the constraints on ( sin x ).I remember that the sine function oscillates between -1 and 1. So, ( sin x ) can take any value in the interval [-1, 1]. Let me denote ( t = sin x ), so ( t in [-1, 1] ). Substituting this into the function, we get:( y = frac{3t + 1}{t - 2} )Now, I need to find the range of y as t varies from -1 to 1. To do this, I can treat this as a function of t and find its possible values. Let me rewrite the function to make it easier to analyze:( y = frac{3t + 1}{t - 2} )I can perform some algebraic manipulation here. Let me try to express this in terms of a constant plus a fraction. Let's see:First, divide 3t + 1 by t - 2. Let me perform polynomial long division or see if I can split the fraction.Alternatively, I can express y as:( y = frac{3t + 1}{t - 2} = frac{3(t - 2) + 7}{t - 2} = 3 + frac{7}{t - 2} )Yes, that works. So, ( y = 3 + frac{7}{t - 2} ). Now, since ( t in [-1, 1] ), let's find the range of ( t - 2 ). ( t - 2 ) will be in the interval:When t = -1: ( -1 - 2 = -3 )When t = 1: ( 1 - 2 = -1 )So, ( t - 2 in [-3, -1] )Therefore, ( frac{1}{t - 2} ) will be in the interval:Since ( t - 2 ) is negative, the reciprocal will also be negative. The smallest value of ( t - 2 ) is -3, so the reciprocal is ( -1/3 ). The largest value of ( t - 2 ) is -1, so the reciprocal is -1. Thus, ( frac{1}{t - 2} in [-1, -1/3] )Multiplying by 7, we get:( frac{7}{t - 2} in [-7, -7/3] )Now, adding 3 to each part of the inequality:Lower bound: ( 3 + (-7) = -4 )Upper bound: ( 3 + (-7/3) = 3 - 7/3 = 9/3 - 7/3 = 2/3 )So, ( y = 3 + frac{7}{t - 2} ) will be in the interval [-4, 2/3]Therefore, the range of the function is all real numbers from -4 to 2/3.But wait, let me verify this. Let me plug in the endpoints of t to see if y indeed takes those values.When t = -1:( y = frac{3(-1) + 1}{-1 - 2} = frac{-3 + 1}{-3} = frac{-2}{-3} = 2/3 )When t = 1:( y = frac{3(1) + 1}{1 - 2} = frac{4}{-1} = -4 )So, the maximum value of y is 2/3, and the minimum value is -4. Therefore, the range is indeed [-4, 2/3].I think that's correct. So, summarizing:1. The domain of ( y = sqrt{1 - cos frac{x}{2}} ) is all real numbers.2. The range of ( y = frac{3sin x + 1}{sin x - 2} ) is from -4 to 2/3.**Final Answer**(1) The domain is boxed{mathbb{R}}.(2) The range is boxed{[-4, frac{2}{3}]}.

💡Okay, so I have this trigonometry problem here. It says that sin θ equals 4/5, and also that sin θ minus cos θ is greater than 1. Then, I need to find tan 2θ and choose the correct answer from the options given: A is 24/7, B is -24/7, C is 4/3, and D is -4/3.Alright, let me start by recalling some basic trigonometric identities. I know that sin²θ + cos²θ = 1. Since sin θ is given as 4/5, I can use this identity to find cos θ. Let me write that down:sin θ = 4/5So, sin²θ = (4/5)² = 16/25Then, cos²θ = 1 - sin²θ = 1 - 16/25 = 9/25Therefore, cos θ could be either 3/5 or -3/5 because the square root of 9/25 is 3/5, but it could be positive or negative.Now, the problem also gives me another condition: sin θ - cos θ > 1. Let me plug in the values I have to see which one makes sense.If cos θ is positive, 3/5, then sin θ - cos θ would be 4/5 - 3/5 = 1/5, which is 0.2. But 0.2 is not greater than 1, so that can't be right.If cos θ is negative, -3/5, then sin θ - cos θ would be 4/5 - (-3/5) = 4/5 + 3/5 = 7/5, which is 1.4. That is indeed greater than 1, so cos θ must be -3/5.Alright, so now I know that cos θ is -3/5. Next, I need to find tan θ because I remember that tan 2θ can be found using the double-angle formula:tan 2θ = 2 tan θ / (1 - tan²θ)So, let's compute tan θ first. Tan θ is sin θ divided by cos θ, so that's (4/5) / (-3/5). The 5s cancel out, so it's 4 / (-3) = -4/3.So, tan θ is -4/3. Now, plug this into the double-angle formula:tan 2θ = 2*(-4/3) / (1 - (-4/3)²)Let me compute the numerator first: 2*(-4/3) = -8/3Now, the denominator: 1 - (-4/3)². The square of -4/3 is 16/9, so 1 - 16/9 = (9/9 - 16/9) = -7/9So, now tan 2θ = (-8/3) / (-7/9). Dividing two fractions is the same as multiplying by the reciprocal, so:(-8/3) * (-9/7) = (8*9)/(3*7) = 72/21Simplify this fraction by dividing numerator and denominator by 3: 72 ÷ 3 = 24, and 21 ÷ 3 = 7. So, 24/7.Therefore, tan 2θ is 24/7, which is option A.Wait, but let me double-check my steps to make sure I didn't make any mistakes. Starting from tan θ, which is -4/3, that seems correct because sin θ is positive and cos θ is negative, so θ is in the second quadrant where tangent is negative. Then, using the double-angle formula, I think I did that correctly.Numerator: 2*(-4/3) = -8/3Denominator: 1 - (16/9) = -7/9Dividing them: (-8/3) / (-7/9) = (8/3)*(9/7) = 72/21 = 24/7.Yes, that seems right. So, the answer should be A: 24/7.**Final Answer**boxed{A}

💡Okay, so I have this math problem here: What is the decimal equivalent of (1/4)^1? Alright, let's break this down step by step. First, I need to understand what each part of the expression means. The expression is (1/4) raised to the power of 1. Alright, so I know that when something is raised to the power of 1, it basically means that the number stays the same. So, if I have any number, say 5, and I raise it to the power of 1, it's still 5. Similarly, if I have a fraction like 1/4 and raise it to the power of 1, it should still be 1/4. So, (1/4)^1 is just 1/4.Now, the question is asking for the decimal equivalent of this. So, I need to convert the fraction 1/4 into a decimal. I remember that to convert a fraction to a decimal, I can divide the numerator by the denominator. In this case, that means dividing 1 by 4.Let me try to do that division. So, 1 divided by 4. I know that 4 goes into 1 zero times, but since 1 is less than 4, I need to add a decimal point and some zeros to continue the division. So, 1.000 divided by 4. 4 goes into 10 two times because 4 times 2 is 8. Subtracting 8 from 10 gives me 2. Then, bring down the next 0, making it 20. 4 goes into 20 five times because 4 times 5 is 20. Subtracting 20 from 20 gives me 0. So, I have no remainder, and the division is complete.Putting it all together, 1 divided by 4 equals 0.25. So, the decimal equivalent of (1/4)^1 is 0.25.Wait a second, let me double-check that to make sure I didn't make any mistakes. If I take 0.25 and multiply it by 4, I should get back to 1. Let's see: 0.25 times 4 is indeed 1. So, that checks out.Is there another way to think about this? Maybe by understanding what fractions represent. The fraction 1/4 means one part out of four equal parts. If I imagine dividing something into four equal parts, like a pizza, each slice would be 1/4 of the pizza. In decimal terms, each slice would be 0.25 of the whole pizza. That makes sense because 0.25 times 4 equals 1, which is the whole pizza.Also, I remember that some fractions have common decimal equivalents that are good to memorize. For example, 1/2 is 0.5, 1/4 is 0.25, and 1/8 is 0.125. So, 1/4 being 0.25 is a standard conversion that's useful to know.Another way to look at it is by using exponents. The expression (1/4)^1 is straightforward because any number to the power of 1 is itself. So, it's just 1/4, which we've already determined is 0.25 in decimal form.I could also think about this in terms of money to visualize it better. If I have a dollar, and I divide it into four equal parts, each part would be 25 cents. Since a dollar is 100 cents, 25 cents is a quarter of a dollar, which is 0.25 dollars. That's another way to see that 1/4 equals 0.25.Is there any chance I could have misread the original problem? Maybe if the exponent was different, like (1/4)^2 or something else, the answer would change. But since it's (1/4)^1, it's just 1/4, which is 0.25.Let me also consider if there's any alternative method to convert 1/4 to a decimal without division. I know that fractions can sometimes be converted by finding an equivalent fraction with a denominator that's a power of 10. For example, 1/4 can be converted to 25/100 by multiplying both the numerator and denominator by 25. Then, 25/100 is easily written as 0.25. That's another valid method to reach the same result.So, to summarize, whether I use division, money visualization, or converting to an equivalent fraction with a denominator of 100, I consistently arrive at the decimal 0.25 as the equivalent of (1/4)^1. This gives me confidence that the answer is correct.I think I've covered all the bases here. I've broken down the problem, checked my work through multiple methods, and even thought about real-world applications to ensure my understanding is solid. It's always good to verify your answers, especially when dealing with fractions and decimals, as small mistakes can happen easily. But in this case, everything lines up perfectly, so I'm pretty sure that 0.25 is the right answer.

💡Alright, so I've got this probability problem here, and I need to figure it out step by step. Let me read it again: "In a set of 10 balls, there are 6 red balls and 4 white balls (identical aside from their markings), two balls are drawn successively without replacement. Given that the first ball drawn is red, the probability that the second ball drawn is also red is _____"Okay, so there are 10 balls total, 6 red and 4 white. We're drawing two balls one after the other without putting the first one back. And we're given that the first ball drawn is red. We need to find the probability that the second ball is also red.Hmm, so this is a conditional probability problem. The probability of the second event happening given that the first event has already occurred. I remember that conditional probability is calculated as P(B|A) = P(A and B) / P(A). But in this case, since we're dealing with drawing without replacement, the probabilities might change after the first draw.Let me think. Initially, there are 6 red balls out of 10. So the probability of drawing a red ball first is 6/10, which simplifies to 3/5. But since we're given that the first ball is red, we don't need to worry about the white balls for the first draw anymore. Now, after drawing one red ball, there are now 5 red balls left and still 4 white balls, making a total of 9 balls remaining.So, the probability of drawing a red ball on the second draw, given that the first was red, should be 5/9. That makes sense because we've already removed one red ball from the set, so both the number of red balls and the total number of balls have decreased by one.Wait, let me double-check. If I use the formula for conditional probability, P(B|A) = P(A and B) / P(A). So, P(A) is the probability that the first ball is red, which is 6/10 or 3/5. P(A and B) is the probability that both the first and the second balls are red. To find that, I can multiply the probability of drawing a red first by the probability of drawing a red second given that the first was red. So, P(A and B) = P(A) * P(B|A) = (6/10) * (5/9) = 30/90 = 1/3.But wait, if I use the formula P(B|A) = P(A and B) / P(A), then P(B|A) = (1/3) / (3/5) = (1/3) * (5/3) = 5/9. So that confirms it. The probability is indeed 5/9.I think that's it. So, given that the first ball drawn is red, the probability that the second ball is also red is 5/9.