answer：First, calculate the initial number of red and green balls in the urn. Since 40% of the 150 balls are red, there are 60 red balls and 90 green balls.To determine how many green balls need to be removed so that red balls make up 80% of the urn, set up the equation where the number of red balls divided by the new total number of balls equals 80%. Let ( x ) be the number of green balls to remove. The equation becomes:[frac{60}{150 - x} = 0.80]Solving for ( x ):[60 = 0.80 times (150 - x)][60 = 120 - 0.80x][0.80x = 60][x = frac{60}{0.80} = 75]Therefore, 75 green balls must be removed.

answer：First, Adam starts with 5.He spends an unknown amount, which we'll call ( C ), on a new game.After spending, he receives an additional 5 from his allowance.Now, he has a total of 8.We can set up the equation:[5 - C + 5 = 8]Combining the amounts:[10 - C = 8]Solving for ( C ):[10 - 8 = C][C = 2]Therefore, the new game costs 2.

answer：Okay, so I have this problem here. It says that the function f(x) = 5 cos(wx + φ) satisfies f(π/3 + x) = f(π/3 - x) for any real number x. Then, there's another function g(x) = 4 sin(wx + φ) + 1, and I need to find g(π/3). The options are A:1, B:5, C:-3, D:0.Hmm, let's break this down. The first part is about the function f(x). It says that f(π/3 + x) equals f(π/3 - x) for any x. That sounds like a symmetry condition. In other words, the function f(x) is symmetric about the line x = π/3. So, the graph of f(x) has a vertical line of symmetry at x = π/3.Since f(x) is a cosine function, which is normally symmetric about its midline, but here it's shifted or something. So, this symmetry condition must tell us something about the phase shift or the frequency or something else about the cosine function.Let me recall that for a function to be symmetric about a vertical line x = a, it must satisfy f(a + x) = f(a - x) for all x. So, in this case, a is π/3. So, f(π/3 + x) = f(π/3 - x). That means that x = π/3 is a line of symmetry for the cosine function.Now, for a cosine function, the standard form is f(x) = A cos(Bx + C) + D. In this case, it's f(x) = 5 cos(wx + φ). So, the amplitude is 5, the frequency is w, and the phase shift is -φ/w. The vertical shift is zero because there's no constant term added or subtracted.Since the function is symmetric about x = π/3, that should correspond to a maximum or a minimum point of the cosine function. Because cosine functions have their maxima and minima at points where the argument is a multiple of π. So, if x = π/3 is a line of symmetry, then it must be either a maximum or a minimum.Therefore, f(π/3) must be either the maximum value or the minimum value of the function. The maximum value of f(x) is 5, and the minimum is -5. So, f(π/3) is either 5 or -5.Let me write that down: f(π/3) = 5 cos(w*(π/3) + φ) = ±5. So, cos(w*(π/3) + φ) = ±1. That implies that the argument inside the cosine function must be an integer multiple of π. So, w*(π/3) + φ = kπ, where k is an integer.So, w*(π/3) + φ = kπ. Let me write that as equation (1): w*(π/3) + φ = kπ.Okay, now moving on to the function g(x) = 4 sin(wx + φ) + 1. We need to find g(π/3). Let's plug in x = π/3 into g(x):g(π/3) = 4 sin(w*(π/3) + φ) + 1.But from equation (1), we know that w*(π/3) + φ = kπ. So, sin(kπ) is equal to zero because sine of any integer multiple of π is zero. Therefore, sin(w*(π/3) + φ) = sin(kπ) = 0.So, plugging that back into g(π/3):g(π/3) = 4*0 + 1 = 1.So, the answer should be 1, which is option A.Wait, let me double-check. If f(x) is symmetric about x = π/3, that means that x = π/3 is either a maximum or a minimum. So, f(π/3) is either 5 or -5. Therefore, the argument inside the cosine is an integer multiple of π, which makes the sine of that argument zero. Therefore, g(π/3) is 1.Yes, that seems consistent. So, I think the answer is A:1.**Final Answer**boxed{A}

answer：Okay, so I have this problem where I need to write a three-digit number after 1220 to form a seven-digit number, and this seven-digit number has to be a multiple of 2014. Hmm, let me try to figure this out step by step.First, I need to understand what exactly is being asked. I have the number 1220, and I need to add a three-digit number to it, making it a seven-digit number. So, if I take 1220 and add a three-digit number, say ABC, it would look like 1220ABC. This number, 1220ABC, needs to be divisible by 2014.Alright, so my goal is to find the three-digit number ABC such that 1220ABC is a multiple of 2014. To do this, I think I need to use some division and modular arithmetic. Let me recall how that works.First, I can represent the seven-digit number as 1220000 + ABC, where ABC is the three-digit number I need to find. So, 1220000 + ABC should be divisible by 2014. That means when I divide 1220000 + ABC by 2014, there should be no remainder.To find ABC, I can start by finding out what 1220000 is when divided by 2014. If I can find the remainder of that division, then I can figure out what ABC needs to be to make the total number divisible by 2014.Let me calculate 1220000 divided by 2014. I can do this by long division or maybe use a calculator for accuracy. Let me try to do it step by step.First, 2014 times 600 is 1,208,400. That's because 2014 * 600 = 1,208,400. Now, subtracting that from 1,220,000 gives me 1,220,000 - 1,208,400 = 11,600. So, after subtracting 2014 * 600, I have 11,600 left.Now, how many times does 2014 go into 11,600? Let me divide 11,600 by 2014. 2014 * 5 is 10,070. Subtracting that from 11,600 gives me 1,530. So, 2014 goes into 11,600 five times with a remainder of 1,530.So, putting it all together, 1220000 divided by 2014 is 605 with a remainder of 1,530. That means 1220000 = 2014 * 605 + 1,530.Now, I need the total number, which is 1220000 + ABC, to be divisible by 2014. That means the remainder when 1220000 is divided by 2014 plus ABC should be zero. So, 1,530 + ABC should be a multiple of 2014.But ABC is a three-digit number, so it's between 0 and 999. Wait, actually, ABC is a three-digit number, so it's between 100 and 999. Hmm, but 1,530 is already more than 1,000, so maybe I need to think differently.Wait, if 1220000 has a remainder of 1,530 when divided by 2014, then to make the total number divisible by 2014, ABC must be equal to 2014 - 1,530. Let me calculate that.2014 - 1,530 = 484. So, ABC should be 484. Let me check if that makes sense.If I add 484 to 1220000, I get 1,220,484. Now, let's divide 1,220,484 by 2014 to see if it's a whole number.2014 times 600 is 1,208,400. Subtracting that from 1,220,484 gives me 12,084. Now, 2014 times 6 is 12,084. So, 2014 * 606 = 1,220,484. That's exactly the number we have. So, yes, 1,220,484 is divisible by 2014, and the quotient is 606.Therefore, the three-digit number ABC that needs to be appended to 1220 is 484.Wait, let me just double-check my calculations to make sure I didn't make any mistakes. So, 2014 * 605 = 1,218,470. Adding 484 gives me 1,218,470 + 484 = 1,218,954. Wait, that doesn't seem right. Hmm, maybe I messed up somewhere.Wait, no, actually, 1220000 + 484 is 1,220,484. And 2014 * 606 is 1,220,484. So, that's correct. Earlier, when I thought 2014 * 605 was 1,218,470, that's correct, but then adding 484 to 1,220,000 gives 1,220,484, which is 2014 * 606. So, the calculations are correct.I think I confused myself for a moment there, but it all checks out. So, the three-digit number is indeed 484.

answer：Okay, so I have this problem here: Let ( a, b in mathbb{R} ). Then, "( a + b > 4 )" is a ( ) condition for "( a > 2 ) and ( b > 2 )". The options are A, B, C, D, which correspond to different types of conditions: sufficient but not necessary, necessary but not sufficient, necessary and sufficient, or neither sufficient nor necessary.Hmm, I need to figure out whether "( a + b > 4 )" is a sufficient condition, a necessary condition, both, or neither for "( a > 2 ) and ( b > 2 )". Let me recall what these terms mean.A sufficient condition means that if the condition is met, then the statement must be true. In other words, if "( a + b > 4 )", then "( a > 2 ) and ( b > 2 )" must hold. But wait, is that always the case? Let me think of an example.Suppose ( a = 3 ) and ( b = 2 ). Then, ( a + b = 5 ), which is greater than 4. However, ( b = 2 ) is not greater than 2; it's equal to 2. So, in this case, "( a + b > 4 )" is true, but "( a > 2 ) and ( b > 2 )" is false. Therefore, "( a + b > 4 )" is not a sufficient condition because it doesn't guarantee that both ( a ) and ( b ) are greater than 2.Now, what about a necessary condition? A necessary condition means that if the statement "( a > 2 ) and ( b > 2 )" is true, then the condition "( a + b > 4 )" must also be true. Let me test this.If ( a > 2 ) and ( b > 2 ), then adding these inequalities gives ( a + b > 4 ). So, yes, if both ( a ) and ( b ) are greater than 2, their sum must be greater than 4. Therefore, "( a + b > 4 )" is a necessary condition for "( a > 2 ) and ( b > 2 )".Putting it all together, "( a + b > 4 )" is necessary but not sufficient for "( a > 2 ) and ( b > 2 )". So, the correct answer should be option B.Wait, let me double-check. If "( a + b > 4 )" is necessary, that means it's required for "( a > 2 ) and ( b > 2 )", but it's not enough on its own. That makes sense because, as I saw earlier, there are cases where ( a + b > 4 ) but one of ( a ) or ( b ) is not greater than 2. So, yes, it's necessary but not sufficient.I think that's solid. I don't see any flaws in this reasoning. So, I'm confident that the answer is B.

answer：To determine the value of ( C ), we need to ensure that both given 7-digit numbers are divisible by 3. A number is divisible by 3 if the sum of its digits is divisible by 3.**First Number: ( 74A52B1 )**- Sum of known digits: ( 7 + 4 + 5 + 2 + 1 = 19 )- Let ( S_1 = A + B )- Total sum: ( 19 + S_1 )- For divisibility by 3: ( (19 + S_1) mod 3 = 0 )- Since ( 19 mod 3 = 1 ), ( S_1 mod 3 = 2 )- Possible values of ( S_1 ): 2, 5, 8, ...**Second Number: ( 326AB4C )**- Sum of known digits: ( 3 + 2 + 6 + 4 = 15 )- Let ( S_2 = A + B + C )- Total sum: ( 15 + S_2 )- For divisibility by 3: ( (15 + S_2) mod 3 = 0 )- Since ( 15 mod 3 = 0 ), ( S_2 mod 3 = 0 )- Possible values of ( S_2 ): 0, 3, 6, ...**Combining Conditions:**- ( S_1 = A + B ) must be 2, 5, 8, ...- ( S_2 = A + B + C ) must be 0, 3, 6, ...- If ( S_1 = 2 ), then ( S_2 = 2 + C ) must be a multiple of 3. The smallest non-negative ( C ) is 1 (since ( 2 + 1 = 3 )).- Checking ( C = 1 ) with other ( S_1 ) values: - ( S_1 = 5 ): ( 5 + 1 = 6 ) (multiple of 3) - ( S_1 = 8 ): ( 8 + 1 = 9 ) (multiple of 3)Thus, ( C = 1 ) works for all valid ( S_1 ) values, ensuring both numbers are divisible by 3.**Final Answer:**(boxed{textbf{(A) }1})