💡Okay, so I'm trying to help Tony figure out the best speeds to drive at for his trip to see his friends. Let me see if I understand the problem correctly. First, Tony drives from his town to a friend's town, which is 120 miles away, and it takes him 3 hours. That means his average speed is 40 mph because 120 miles divided by 3 hours is 40 mph. But on the next day, he's driving to another friend's town, which is 200 miles away from the first friend's town. However, on his way there, he stops twice at two different towns, adding extra distances of 60 miles and 90 miles respectively. So, the total distance he needs to cover the next day is 200 + 60 + 90 = 350 miles.He drives at different speeds for each leg of the journey: x mph for the first stop, y mph for the second stop, and z mph for the final leg. The constraints are:1. x varies between 30 and 60 mph.2. y varies between 40 and 70 mph.3. z varies between 50 and 90 mph.4. The total driving time should not exceed 10 hours.Our goal is to find the optimal speeds x, y, and z that minimize the total driving time while satisfying these constraints.Hmm, so to minimize the total driving time, Tony should drive as fast as possible within the given constraints because higher speed means less time to cover the same distance. That makes sense. So, ideally, he should drive at the maximum allowed speed for each leg of the journey.Let's break it down:1. **First Leg:** 60 miles at speed x (30 ≤ x ≤ 60 mph) - Time = Distance / Speed = 60 / x hours2. **Second Leg:** 90 miles at speed y (40 ≤ y ≤ 70 mph) - Time = 90 / y hours3. **Third Leg:** 200 miles at speed z (50 ≤ z ≤ 90 mph) - Time = 200 / z hoursTotal time = 60/x + 90/y + 200/zWe need to minimize this total time, given the constraints on x, y, and z.Since higher speeds reduce time, to minimize the total time, we should set x, y, and z to their maximum allowed values.So, set x = 60 mph, y = 70 mph, and z = 90 mph.Let's calculate the total time with these speeds:1. First Leg: 60 / 60 = 1 hour2. Second Leg: 90 / 70 ≈ 1.2857 hours3. Third Leg: 200 / 90 ≈ 2.2222 hoursAdding these up: 1 + 1.2857 + 2.2222 ≈ 4.5079 hoursThat's approximately 4.51 hours, which is well within the 10-hour constraint.But wait, is this the only way? What if driving at maximum speeds isn't always possible due to traffic or other factors? But the problem states that the speeds vary within these ranges, so assuming he can drive at the maximum speeds without any issues, this should be the optimal solution.Let me double-check the calculations:- 60 miles at 60 mph: 1 hour- 90 miles at 70 mph: 90/70 = 1.2857 hours- 200 miles at 90 mph: 200/90 = 2.2222 hoursTotal: 1 + 1.2857 + 2.2222 = 4.5079 hoursYes, that seems correct. So, the optimal speeds are x = 60 mph, y = 70 mph, and z = 90 mph, resulting in a total driving time of approximately 4.51 hours.

💡Alright, so I have this problem with complex numbers and vectors. Let me try to understand what it's asking for. We have three complex numbers: z₁ = -1 + 2i, z₂ = 1 - i, and z₃ = 3 - 2i. Each of these corresponds to points A, B, and C in the complex plane, right? So, point A is at (-1, 2), point B is at (1, -1), and point C is at (3, -2). The problem says that the vector OC is equal to x times vector OA plus y times vector OB. So, in equation form, that would be:OC = x * OA + y * OBI need to find the value of x + y. Okay, so let's break this down. Vectors OA, OB, and OC correspond to the position vectors of points A, B, and C from the origin. So, OA is the vector from the origin to point A, which is (-1, 2), OB is (1, -1), and OC is (3, -2). So, if I write this out in terms of components, the equation becomes:(3, -2) = x*(-1, 2) + y*(1, -1)This gives me two equations, one for the x-component and one for the y-component:For the x-component:3 = x*(-1) + y*(1)For the y-component:-2 = x*(2) + y*(-1)So, simplifying these equations:1. -x + y = 32. 2x - y = -2Now, I have a system of two equations with two variables, x and y. I need to solve for x and y and then add them together to get x + y.Let me write down the equations again:1. -x + y = 32. 2x - y = -2Hmm, maybe I can solve this system using substitution or elimination. Let's try elimination because the coefficients look like they might cancel out nicely.If I add equation 1 and equation 2 together, the y terms will cancel out:(-x + y) + (2x - y) = 3 + (-2)Simplifying the left side:(-x + 2x) + (y - y) = x + 0 = xAnd the right side:3 - 2 = 1So, x = 1Now that I have x, I can plug it back into one of the original equations to find y. Let's use equation 1:-1 + y = 3Adding 1 to both sides:y = 4So, x = 1 and y = 4. Therefore, x + y = 1 + 4 = 5.Wait, let me double-check my work to make sure I didn't make any mistakes. Starting with the equations:1. -x + y = 32. 2x - y = -2Adding them:(-x + 2x) + (y - y) = 3 - 2Which simplifies to:x = 1Plugging x = 1 into equation 1:-1 + y = 3So, y = 4Yes, that seems correct. So, x + y is indeed 5.I think I've got it. The value of x + y is 5.**Final Answer**The value of ( x + y ) is boxed{5}.

💡Alright, so I have this problem here: "How old is Florence? Try to determine it yourself; we will only indicate that the difference between the fifth power of the sought age and the age itself is divisible by 10." Hmm, okay, let's break this down.First, I need to find Florence's age, which we'll call 'a'. The key piece of information is that the difference between the fifth power of her age and her age itself is divisible by 10. So, mathematically, that means:a^5 - a is divisible by 10.Okay, so I need to find an integer 'a' such that when I subtract 'a' from its fifth power, the result is a multiple of 10. In other words:a^5 - a ≡ 0 mod 10.Alright, modular arithmetic might be useful here. Let's recall that if a number is divisible by 10, it must be divisible by both 2 and 5. So, maybe I can check the divisibility by 2 and 5 separately.First, let's consider modulo 2. For any integer 'a', it's either even or odd.- If 'a' is even, then a ≡ 0 mod 2. So, a^5 ≡ 0^5 ≡ 0 mod 2, and a ≡ 0 mod 2. Therefore, a^5 - a ≡ 0 - 0 ≡ 0 mod 2.- If 'a' is odd, then a ≡ 1 mod 2. So, a^5 ≡ 1^5 ≡ 1 mod 2, and a ≡ 1 mod 2. Therefore, a^5 - a ≡ 1 - 1 ≡ 0 mod 2.So, regardless of whether 'a' is even or odd, a^5 - a is divisible by 2. That's good to know.Now, let's check modulo 5. For this, I'll use Fermat's Little Theorem, which states that if 'p' is a prime number, then for any integer 'a', a^p ≡ a mod p. Since 5 is a prime number, we can apply this theorem.According to Fermat's Little Theorem:a^5 ≡ a mod 5.Therefore, a^5 - a ≡ 0 mod 5.So, a^5 - a is divisible by 5 as well.Since a^5 - a is divisible by both 2 and 5, it must be divisible by 10. That means, for any integer 'a', a^5 - a is divisible by 10.Wait a minute, does that mean that any age 'a' satisfies this condition? That can't be right because the problem is asking me to determine Florence's age based on this condition. If every integer satisfies this, then the condition doesn't help narrow down the age.Let me double-check my reasoning. I used Fermat's Little Theorem for modulo 5, which seems correct. And for modulo 2, I considered both even and odd cases, which also seem to hold. So, it appears that for any integer 'a', a^5 - a is divisible by 10.If that's the case, then the problem doesn't provide enough information to determine Florence's specific age. It could be any age, and the condition would still hold.But maybe I'm missing something. Perhaps the problem expects me to consider only certain ranges of ages or additional constraints. For example, if Florence is a person, her age is likely to be between 1 and, say, 100. But even within that range, every age would satisfy the condition a^5 - a being divisible by 10.Alternatively, maybe the problem is testing my understanding of modular arithmetic and wants me to recognize that the condition is always true, hence no unique solution exists.In that case, the answer would be that Florence's age cannot be determined from the given information because the condition holds for all integers.Yeah, that makes sense. The problem is designed to highlight that the condition a^5 - a being divisible by 10 is universally true for any integer 'a', so it doesn't help in finding a specific age.I think I've got it. The key takeaway is recognizing that the condition provided doesn't narrow down the age to a specific value because it's always satisfied.

💡First, the program takes an input value ( x ).Next, it checks if ( x ) is less than 0. Since ( x = 3 ) is not less than 0, the program proceeds to the next condition.Then, it checks if ( x ) is equal to 0. Since ( x = 3 ) is not equal to 0, the program moves to the final condition.Finally, since ( x ) is neither less than 0 nor equal to 0, it must be greater than 0. Therefore, ( y ) is assigned the value 1.The output of the program is 1.

💡First, I need to calculate the population increase in Region A. For the 0-14 age group in Region A:- There are 2000 people, and there is one birth every 20 seconds.- In 25 minutes, there are 25 * 60 = 1500 seconds.- The number of births in this age group is 1500 seconds / 20 seconds/birth = 75 births.For the 15-64 age group in Region A:- There are 6000 people, and there is one birth every 30 seconds.- In 25 minutes, there are 1500 seconds.- The number of births in this age group is 1500 seconds / 30 seconds/birth = 50 births.Next, I'll calculate the population increase in Region B.For the 0-14 age group in Region B:- There are 1500 people, and there is one birth every 25 seconds.- In 25 minutes, there are 1500 seconds.- The number of births in this age group is 1500 seconds / 25 seconds/birth = 60 births.For the 15-64 age group in Region B:- There are 5000 people, and there is one birth every 35 seconds.- In 25 minutes, there are 1500 seconds.- The number of births in this age group is 1500 seconds / 35 seconds/birth ≈ 42.86 births, but since we can't have a fraction of a birth, we'll round down to 42 births.Now, I'll add up the total population increase for both regions:Region A total births = 75 (0-14 age group) + 50 (15-64 age group) = 125 birthsRegion B total births = 60 (0-14 age group) + 42 (15-64 age group) = 102 birthsTotal population increase in both regions = Region A births + Region B birthsTotal population increase = 125 births + 102 births = 227 birthsTherefore, the total population increase in both regions after 25 minutes is boxed{227}.

💡Okay, so I have these two equations to solve: (1) x minus 4 equals negative 5, and (2) one half of x plus 2 equals 6. Alright, let's take them one at a time.Starting with the first equation: x - 4 = -5. I need to find the value of x that makes this equation true. Hmm, so if I have x minus 4 equals negative 5, I guess I need to get x by itself on one side of the equation. To do that, I should probably get rid of that -4 that's being subtracted from x.How do I get rid of a negative number? Well, I remember that whatever I do to one side of the equation, I have to do to the other to keep it balanced. So if I add 4 to both sides, that should cancel out the -4 on the left side. Let me try that:x - 4 + 4 = -5 + 4Simplifying both sides, the -4 and +4 on the left cancel each other out, leaving just x. On the right side, -5 plus 4 is -1. So, x equals -1. That seems straightforward. Let me double-check by plugging it back into the original equation:(-1) - 4 = -5Yep, that works because -1 minus 4 is indeed -5. Okay, so the first equation is solved, and x is -1.Now, moving on to the second equation: (1/2)x + 2 = 6. Again, I need to solve for x. This one looks a bit trickier because there's a fraction involved, but I think I can handle it.First, I want to isolate the term with x, which is (1/2)x. To do that, I should get rid of the +2 that's being added to it. How do I do that? I'll subtract 2 from both sides of the equation to keep it balanced:(1/2)x + 2 - 2 = 6 - 2Simplifying both sides, the +2 and -2 on the left cancel out, leaving just (1/2)x. On the right side, 6 minus 2 is 4. So now I have:(1/2)x = 4Now, I need to solve for x, which means I need to get rid of that fraction. Since (1/2)x means x multiplied by 1/2, I can undo that by multiplying both sides of the equation by the reciprocal of 1/2, which is 2. Let's do that:2 * (1/2)x = 4 * 2Simplifying both sides, 2 times (1/2)x is just x, because 2 and 1/2 cancel each other out. On the right side, 4 times 2 is 8. So, x equals 8.Let me check that by plugging it back into the original equation:(1/2)*8 + 2 = 6Calculating that, (1/2)*8 is 4, and 4 plus 2 is 6. Yep, that works. So, x is indeed 8.Wait a minute, did I do everything correctly? Let me go through the steps again for both equations to make sure I didn't make any mistakes.For the first equation:1. Start with x - 4 = -52. Add 4 to both sides: x - 4 + 4 = -5 + 43. Simplify: x = -14. Check: (-1) - 4 = -5, which is correct.For the second equation:1. Start with (1/2)x + 2 = 62. Subtract 2 from both sides: (1/2)x + 2 - 2 = 6 - 23. Simplify: (1/2)x = 44. Multiply both sides by 2: 2*(1/2)x = 4*25. Simplify: x = 86. Check: (1/2)*8 + 2 = 6, which is correct.Okay, I think I did everything right. Both solutions satisfy their respective equations when I plug them back in. I guess I've solved both equations correctly!