💡Okay, so I have this problem where I'm given a set M which contains the first 2017 positive integers, right? So M is {1, 2, 3, ..., 2017}. The task is to remove one element from this set such that the sum of the remaining elements is a perfect square. I need to figure out which element to remove.First, I think I should calculate the total sum of all elements in M before any removal. Since it's the sum of the first n positive integers, there's a formula for that: S = n(n + 1)/2. In this case, n is 2017, so plugging that into the formula, I get S = 2017 * 2018 / 2. Let me compute that.2017 multiplied by 2018... Hmm, that's a big number. Let me see. 2017 * 2000 is 4,034,000, and 2017 * 18 is 36,306. Adding those together gives 4,034,000 + 36,306 = 4,070,306. Then, dividing by 2, I get 2,035,153. So the total sum S is 2,035,153.Now, if I remove an element x from M, the new sum S' will be S - x. We want S' to be a perfect square. So, S' = k^2 for some integer k. That means 2,035,153 - x = k^2, which implies x = 2,035,153 - k^2.My goal is to find such a k that x is an integer between 1 and 2017, inclusive. So I need to find a perfect square k^2 that is just less than 2,035,153, such that when I subtract it from 2,035,153, the result is within the range of 1 to 2017.To find k, I can take the square root of 2,035,153 to get an approximate value. Let me calculate that. The square root of 2,035,153... Well, 1426 squared is 2,033,476 because 1400 squared is 1,960,000, and adding 26 squared gives 676, so 1426 squared is 1,960,000 + 2*1400*26 + 26^2 = 1,960,000 + 72,800 + 676 = 2,033,476.Wait, that's less than 2,035,153. Let me check 1427 squared. 1427 squared would be 1426 squared plus 2*1426 + 1, which is 2,033,476 + 2,852 + 1 = 2,036,329. Hmm, that's more than 2,035,153. So k must be 1426 because 1426 squared is less than S, and 1427 squared is more than S.So, k = 1426. Then, k squared is 2,033,476. Therefore, x = 2,035,153 - 2,033,476 = 1,677.Now, I need to check if 1,677 is indeed within the set M. Since M contains numbers from 1 to 2017, 1,677 is definitely within that range. So, removing 1,677 from the set M will result in the sum of the remaining elements being a perfect square, specifically 1426 squared.Let me just double-check my calculations to be sure. The total sum S is 2,035,153. Subtracting 1,677 gives 2,035,153 - 1,677 = 2,033,476, which is indeed 1426 squared. Yep, that seems correct.So, the element that needs to be removed is 1,677.**Final Answer**The removed element is boxed{1677}.

💡Let's break down the problem step by step to find the price of the book.1. **Define Variables:** - Let ( D ) be the amount given by Peter's father. - Let ( B ) be the amount given by Peter's elder brother. - The younger brother gave 10 euros.2. **Set Up Equations Based on the Problem Statement:** - **Father's Contribution:** The father gave half of what the brothers gave together. [ D = frac{1}{2}(B + 10) ] - **Elder Brother's Contribution:** The elder brother gave one third of what the others (father and younger brother) gave. [ B = frac{1}{3}(D + 10) ]3. **Substitute the Expression for ( D ) into the Equation for ( B ):** [ B = frac{1}{3}left(frac{1}{2}(B + 10) + 10right) ] Simplify the equation: [ B = frac{1}{3}left(frac{1}{2}B + 5 + 10right) ] [ B = frac{1}{3}left(frac{1}{2}B + 15right) ] [ B = frac{1}{6}B + 5 ]4. **Solve for ( B ):** [ B - frac{1}{6}B = 5 ] [ frac{5}{6}B = 5 ] [ B = 5 times frac{6}{5} = 6 ]5. **Find ( D ) Using the Value of ( B ):** [ D = frac{1}{2}(6 + 10) = frac{1}{2} times 16 = 8 ]6. **Calculate the Total Amount Given to Peter:** [ D + B + 10 = 8 + 6 + 10 = 24 ]Therefore, the price of the book was **24 euros**.

💡To simplify the expression (sin (x-y) cos y + cos (x-y) sin y), we can recognize that it resembles the sine addition formula. The sine addition formula is:[sin(a + b) = sin a cos b + cos a sin b]By comparing this with our expression, we can let (a = x - y) and (b = y). Applying the formula:[sin((x - y) + y) = sin(x - y) cos y + cos(x - y) sin y]Simplifying the left side:[sin(x - y + y) = sin x]Therefore, the expression simplifies to (sin x).

💡Okay, so I've got this problem here: A positive number (x) satisfies the condition that (x%) of (x) equals 9. I need to find what (x) is, and the options are (A) 25, (B) 30, (C) 35, (D) 40, and (E) 45. Hmm, let me try to figure this out step by step.First, I need to understand what the problem is saying. It mentions that (x%) of (x) equals 9. So, in mathematical terms, when they say "(x%) of (x)", that translates to (x) percent of (x). I remember that "percent" means per hundred, so (x%) is the same as (x/100). So, if I write that out, "(x%) of (x)" would be ((x/100) times x). Let me write that down:[frac{x}{100} times x = 9]Simplifying that, I get:[frac{x^2}{100} = 9]Now, I need to solve for (x). To do that, I can multiply both sides of the equation by 100 to get rid of the denominator:[x^2 = 9 times 100][x^2 = 900]Okay, so now I have (x^2 = 900). To find (x), I take the square root of both sides. Since (x) is a positive number, I only need to consider the positive square root:[x = sqrt{900}][x = 30]Wait, let me double-check that. If (x = 30), then (30%) of 30 should be 9. Let me calculate that:[30% text{ of } 30 = frac{30}{100} times 30 = 0.3 times 30 = 9]Yes, that works out. So, (x = 30) is the correct answer. Looking back at the options, (B) is 30, so that must be the right choice.Just to make sure I didn't make any mistakes, let me try plugging in another option, say (A) 25, and see what happens:[25% text{ of } 25 = frac{25}{100} times 25 = 0.25 times 25 = 6.25]That's not 9, so (A) is incorrect. How about (D) 40?[40% text{ of } 40 = frac{40}{100} times 40 = 0.4 times 40 = 16]That's too high. And (E) 45:[45% text{ of } 45 = frac{45}{100} times 45 = 0.45 times 45 = 20.25]Also way too high. So, yeah, (B) 30 is definitely the correct answer. I think I've got it.**Final Answer**boxed{B}

💡First, I need to determine the number of women on the airplane. Since the number of men and women is equal, and there are 30 men, there must also be 30 women.Next, I'll calculate the total number of men and women combined. Adding the number of men and women gives me 30 + 30 = 60.Finally, to find out how many children are on the airplane, I'll subtract the total number of men and women from the total number of passengers. Subtracting 60 from 80 gives me 20 children.

💡First, I recognize that the expression (601^2 - 599^2) is a difference of squares, which can be factored using the identity (a^2 - b^2 = (a + b)(a - b)).Applying this identity, I rewrite the expression as:[601^2 - 599^2 = (601 + 599)(601 - 599)]Next, I calculate the sum and difference of the numbers:[601 + 599 = 1200][601 - 599 = 2]Finally, I multiply these results to find the value:[1200 times 2 = 2400]Therefore, the value of (601^2 - 599^2) is 2400.