💡Okay, so I have this expression to simplify: (1 - 1/(x+3)) divided by ((x² - 9)/(x² + 6x + 9)). Hmm, that looks a bit complicated, but I think I can break it down step by step. First, let me understand what each part of the expression is.Starting with the numerator: 1 minus 1 over (x + 3). I can write that as 1 - [1/(x + 3)]. Maybe I can combine these into a single fraction to make it easier to work with. To do that, I need a common denominator. The first term is just 1, which is the same as (x + 3)/(x + 3). So, if I subtract 1/(x + 3) from that, I get:(x + 3)/(x + 3) - 1/(x + 3) = [(x + 3) - 1]/(x + 3) = (x + 2)/(x + 3).Okay, so the numerator simplifies to (x + 2)/(x + 3). That wasn't too bad.Now, looking at the denominator of the entire expression: (x² - 9)/(x² + 6x + 9). I recognize that both the numerator and the denominator here are quadratic expressions, and maybe they can be factored.Starting with the numerator: x² - 9. That's a difference of squares, right? So it factors into (x - 3)(x + 3).The denominator: x² + 6x + 9. That looks like a perfect square trinomial. It factors into (x + 3)².So, the denominator of the entire expression becomes:(x² - 9)/(x² + 6x + 9) = [(x - 3)(x + 3)] / (x + 3)².Now, putting it all together, the original expression is:[(x + 2)/(x + 3)] ÷ [(x - 3)(x + 3)/(x + 3)²].Dividing by a fraction is the same as multiplying by its reciprocal, so this becomes:[(x + 2)/(x + 3)] * [(x + 3)² / (x - 3)(x + 3)].Now, let's simplify this multiplication. I can cancel out common factors in the numerator and the denominator.First, the (x + 3) in the denominator of the first fraction cancels with one (x + 3) in the numerator of the second fraction. So, we're left with:(x + 2) * (x + 3) / (x - 3).So, the simplified expression is (x + 2)(x + 3)/(x - 3). Wait, but let me double-check that.Hold on, after canceling one (x + 3), the second fraction still has another (x + 3) in the numerator, right? So actually, it should be:(x + 2) * (x + 3) / (x - 3).Yes, that's correct. So, the simplified form is (x + 2)(x + 3)/(x - 3). Hmm, but I think I might have made a mistake here because when I canceled the (x + 3), I should have only canceled one from the denominator and one from the numerator, leaving me with (x + 3) in the numerator.Wait, let me go back. The first fraction is (x + 2)/(x + 3), and the second fraction after taking the reciprocal is (x + 3)² / [(x - 3)(x + 3)]. So, when I multiply them, I have:(x + 2)/(x + 3) * (x + 3)² / [(x - 3)(x + 3)].Now, the (x + 3) in the denominator cancels with one (x + 3) in the numerator of the second fraction, leaving me with:(x + 2) * (x + 3) / (x - 3).Yes, that's correct. So, the simplified expression is (x + 2)(x + 3)/(x - 3). But wait, can I simplify this further? Let me see.Actually, (x + 2)(x + 3) is just a quadratic, and the denominator is (x - 3). I don't think they have any common factors, so this should be the simplified form.But let me check if there are any restrictions on the values of x. The original expression had denominators of (x + 3) and (x - 3), so x cannot be -3 or 3 because that would make the denominator zero, which is undefined.Now, the problem asks me to choose a suitable number from -3, 2, 3 to substitute and evaluate. Since x cannot be -3 or 3, the only suitable number is 2.So, substituting x = 2 into the simplified expression:(2 + 2)(2 + 3)/(2 - 3) = (4)(5)/(-1) = 20/(-1) = -20.Wait, that seems a bit large. Let me double-check my simplification.Original expression: (1 - 1/(x + 3)) ÷ ((x² - 9)/(x² + 6x + 9)).Simplified to: (x + 2)(x + 3)/(x - 3).Substituting x = 2: (2 + 2)(2 + 3)/(2 - 3) = (4)(5)/(-1) = -20.Hmm, that seems correct, but let me verify by substituting x = 2 into the original expression to see if I get the same result.Original expression:Numerator: 1 - 1/(2 + 3) = 1 - 1/5 = 4/5.Denominator: (2² - 9)/(2² + 6*2 + 9) = (4 - 9)/(4 + 12 + 9) = (-5)/25 = -1/5.So, the entire expression is (4/5) ÷ (-1/5) = (4/5) * (-5/1) = -4.Wait, that's different from what I got earlier. So, there must be a mistake in my simplification.Let me go back and check my steps.Starting again:Original expression: (1 - 1/(x + 3)) ÷ ((x² - 9)/(x² + 6x + 9)).Simplify numerator: 1 - 1/(x + 3) = (x + 3 - 1)/(x + 3) = (x + 2)/(x + 3).Denominator: (x² - 9)/(x² + 6x + 9) = [(x - 3)(x + 3)] / (x + 3)².So, the entire expression becomes:[(x + 2)/(x + 3)] ÷ [(x - 3)(x + 3)/(x + 3)²] = [(x + 2)/(x + 3)] * [(x + 3)² / (x - 3)(x + 3)].Now, simplifying:(x + 2) * (x + 3)² / [(x + 3) * (x - 3)(x + 3)].Wait, I think I made a mistake in canceling. Let's see:The numerator after multiplying is (x + 2)(x + 3)².The denominator is (x + 3)(x - 3)(x + 3).So, the (x + 3) in the denominator cancels with one (x + 3) in the numerator, leaving:(x + 2)(x + 3) / (x - 3).Wait, that's what I had before. But when I substituted x = 2, I got -20, but when I substituted into the original expression, I got -4. So, there's a discrepancy.Let me check the substitution again.Simplified expression: (x + 2)(x + 3)/(x - 3).Substituting x = 2: (2 + 2)(2 + 3)/(2 - 3) = (4)(5)/(-1) = -20.Original expression:Numerator: 1 - 1/(2 + 3) = 1 - 1/5 = 4/5.Denominator: (2² - 9)/(2² + 6*2 + 9) = (4 - 9)/(4 + 12 + 9) = (-5)/25 = -1/5.So, (4/5) ÷ (-1/5) = (4/5) * (-5/1) = -4.So, the simplified expression gives -20, but the original expression gives -4 when x = 2. That means my simplification is incorrect.Where did I go wrong?Let me re-examine the simplification steps.Original expression:[(x + 2)/(x + 3)] ÷ [(x - 3)(x + 3)/(x + 3)²].This is equal to:[(x + 2)/(x + 3)] * [(x + 3)² / (x - 3)(x + 3)].Multiplying the numerators: (x + 2)(x + 3)².Multiplying the denominators: (x + 3)(x - 3)(x + 3).So, the expression becomes:(x + 2)(x + 3)² / [(x + 3)(x - 3)(x + 3)].Now, canceling one (x + 3) from numerator and denominator:(x + 2)(x + 3) / [(x - 3)(x + 3)].Wait, I think I missed that there's another (x + 3) in the denominator. So, it should be:(x + 2)(x + 3) / [(x - 3)(x + 3)].Now, the (x + 3) terms cancel out, leaving:(x + 2)/(x - 3).Ah, that's different from what I had before. So, the simplified expression is actually (x + 2)/(x - 3), not (x + 2)(x + 3)/(x - 3).That makes more sense. Let me verify this.Simplified expression: (x + 2)/(x - 3).Substituting x = 2: (2 + 2)/(2 - 3) = 4/(-1) = -4.Which matches the result from the original expression. So, my earlier mistake was not canceling all possible terms correctly.Therefore, the correct simplified expression is (x + 2)/(x - 3).Now, the problem asks to choose a suitable number from -3, 2, 3 to substitute. As before, x cannot be -3 or 3, so x = 2 is the only suitable choice.Substituting x = 2: (2 + 2)/(2 - 3) = 4/(-1) = -4.So, the final answer is -4.

💡Okay, so I'm trying to figure out this problem about a king moving on an m x n chessboard. The king starts in one of the corners, and two players take turns moving it to any unvisited square. The first player who can't move loses. I need to determine who has the winning strategy based on the dimensions of the chessboard.First, I need to understand the movement of the king. In chess, the king can move one square in any direction—horizontally, vertically, or diagonally. So, from any given square, the king can potentially move to up to eight adjacent squares, but of course, it can't move off the board or to a square that's already been visited.Since the king starts in a corner, its initial move will have fewer options. From a corner, the king can only move to three squares: one to the right, one above, and one diagonally to the right and above.Now, the game is about moving the king to unvisited squares, and the player who can't move loses. So, essentially, this is a game of visiting all possible squares, and the player who makes the last move wins.I think this might relate to something called impartial games, where both players have the same set of moves available to them. Maybe it's similar to games like Nim or Chomp, where the key is to find a winning strategy based on the game's structure.I recall that in such games, the concept of parity often comes into play. Parity refers to whether a number is odd or even. Maybe the total number of squares on the chessboard, which is m multiplied by n, could determine the winner based on whether it's odd or even.Let me think about small chessboards to see if I can spot a pattern.First, consider a 1x1 chessboard. There's only one square, and the king is already there. So, the first player can't move and loses immediately. But this is a trivial case.Next, a 1x2 chessboard. The king starts on one square, and the first player can move it to the only other square. Then, the second player has no moves and loses. So, the first player wins.Similarly, a 2x2 chessboard. The king starts in a corner. The first player can move it to any of the three adjacent squares. Suppose the first player moves it to the square diagonally opposite. Then, the second player can move it to one of the remaining two squares. Then, the first player can move it to the last square, and the second player loses. So, the first player wins.Wait, but what if the first player doesn't move diagonally? Suppose the first player moves horizontally or vertically. Then, the second player can mirror the move, perhaps leading to a different outcome. Hmm, maybe I need to think more carefully.Alternatively, maybe the key is the total number of squares. If the number of squares is odd, the second player can always mirror the first player's moves and win. If it's even, the first player can win.Wait, let's test that idea.For a 1x2 board, which has 2 squares (even), the first player wins, which fits the idea.For a 2x2 board, which has 4 squares (even), the first player wins, which also fits.What about a 3x3 board, which has 9 squares (odd). If the first player moves to the center, then the second player can mirror moves across the center. But wait, the king can't jump, so mirroring might not always be possible.Alternatively, maybe the parity of the number of squares determines the winner. If the total number of squares is even, the first player can win; if it's odd, the second player can win.But I'm not sure if that's the case. Let me think of a 1x3 board. It has 3 squares (odd). The first player moves from the first square to the second. Then, the second player can move to the third square, and the first player loses. So, in this case, the second player wins, which fits the idea.Similarly, a 2x3 board has 6 squares (even). The first player can move to any square, and then the second player can mirror the moves, but I'm not sure. Maybe the first player can always win.Wait, maybe it's more about the parity of the number of squares minus one. Because the king starts on one square, so the number of moves possible is m*n - 1.If m*n - 1 is odd, then the first player can make the last move and win. If it's even, the second player can make the last move and win.So, if m*n is even, then m*n - 1 is odd, so the first player wins.If m*n is odd, then m*n - 1 is even, so the second player wins.That seems to fit the examples I thought of.For 1x2 (m*n=2, even), m*n -1=1 (odd), first player wins.For 2x2 (m*n=4, even), m*n -1=3 (odd), first player wins.For 1x3 (m*n=3, odd), m*n -1=2 (even), second player wins.For 2x3 (m*n=6, even), m*n -1=5 (odd), first player wins.That seems consistent.So, the key is whether m*n is even or odd.If m*n is even, the first player can make the last move and win.If m*n is odd, the second player can make the last move and win.Therefore, the winning strategy depends on the parity of the product of m and n.So, the conclusion is:- If m*n is even, the first player has a winning strategy.- If m*n is odd, the second player has a winning strategy.I think that's the answer.

💡Alright, so I've got this subtraction problem here, and I need to figure out the values of X and Y, which are digits. Then, I have to find the sum of X and Y. The problem is set up like a vertical subtraction, but since I can't see the actual numbers, I'll have to work with the information given.First, let me try to visualize the subtraction. It's probably something like:``` 1 X 2- 8 Y------- 4 5```So, when we subtract 8Y from 1X2, we get 45. That means 1X2 minus 8Y equals 45. My goal is to find X and Y such that this equation holds true.Let me write this out as an equation:1X2 - 8Y = 45But since X and Y are single digits, I can represent 1X2 as 100 + 10X + 2 and 8Y as 80 + Y. So, substituting these into the equation:100 + 10X + 2 - (80 + Y) = 45Simplifying this:102 + 10X - 80 - Y = 45Which simplifies further to:22 + 10X - Y = 45Now, let's rearrange this equation to solve for Y:10X - Y = 45 - 2210X - Y = 23So, 10X - Y = 23. Since X and Y are digits (0-9), I can try different values of X to see which one makes Y a valid digit.Let's start with X = 3:10*3 - Y = 2330 - Y = 23Y = 30 - 23Y = 7Okay, Y = 7, which is a valid digit. Let's check if this works in the original subtraction problem.So, 1X2 becomes 132, and 8Y becomes 87. Subtracting 87 from 132:132 - 87 = 45Yes, that works out perfectly. So, X = 3 and Y = 7.Therefore, X + Y = 3 + 7 = 10.Wait, let me double-check to make sure I didn't make any mistakes. If X were 4:10*4 - Y = 2340 - Y = 23Y = 40 - 23Y = 17But Y has to be a single digit, so that's not possible. Similarly, if X were 2:10*2 - Y = 2320 - Y = 23Y = 20 - 23Y = -3Negative digits don't make sense here, so X can't be 2. If X were 5:10*5 - Y = 2350 - Y = 23Y = 50 - 23Y = 27Again, Y is not a single digit. So, X must be 3 and Y must be 7. Therefore, their sum is 10.I think that's solid. I don't see any errors in my reasoning, and the subtraction checks out. So, I'm confident that X + Y is 10.

💡First, I need to determine the value of ( b ) that makes the equation ( 4(3x - b) = 3(4x + 16) ) have infinitely many solutions for ( x ).To do this, I'll start by simplifying both sides of the equation. I'll distribute the 4 on the left side and the 3 on the right side.After simplifying, I'll set the equations equal to each other and solve for ( b ). If the equation has infinitely many solutions, the coefficients of ( x ) must be equal, and the constant terms must also be equal.By equating the constant terms, I'll find the value of ( b ) that satisfies this condition.

💡First, I need to determine the average speed of the boat in still water. I'll start by letting the speed of the boat in still water be ( x ) km/h. The speed of the current is given as 3 km/h.When the boat travels from port A to port B with the current, its effective speed increases by the speed of the current. Therefore, the speed during this part of the journey is ( x + 3 ) km/h. Given that this journey took 2 hours, the distance covered can be expressed as ( 2(x + 3) ).On the return journey from port B to port A against the current, the boat's effective speed decreases by the speed of the current, making it ( x - 3 ) km/h. This journey took 2.5 hours, so the distance covered can be expressed as ( 2.5(x - 3) ).Since the distance covered in both directions is the same, I can set these two expressions equal to each other to find ( x ):[2(x + 3) = 2.5(x - 3)]Expanding both sides gives:[2x + 6 = 2.5x - 7.5]Next, I'll rearrange the equation to solve for ( x ). I'll move all terms involving ( x ) to one side and constants to the other:[2x - 2.5x = -7.5 - 6]Simplifying both sides gives:[-0.5x = -13.5]Finally, I'll divide both sides by (-0.5) to solve for ( x ):[x = frac{-13.5}{-0.5} = 27]Therefore, the average speed of the boat in still water is ( boxed{27} ) km/h.

💡Okay, so I have this problem here about percentage decrease in the total production value of a factory. Hmm, let me read it again carefully."The percentage decrease in the total production value of a factory from the year before last to last year, if it increased by a% from the year before last to last year, is (　　) A: a% B: (1+a)% C: |(a+1)/(100a)| D: |a/(100+a)|"Wait, that's a bit confusing. So, the factory's production increased by a% from the year before last to last year. But the question is asking about the percentage decrease from the year before last to last year. That seems contradictory because if it increased, how can there be a decrease? Maybe I'm misunderstanding the question.Let me parse it again. It says, "the percentage decrease in the total production value of a factory from the year before last to last year, if it increased by a% from the year before last to last year." Hmm, so it's saying that the production increased by a%, but they're asking for the percentage decrease? That doesn't make sense. Maybe it's a typo? Or perhaps I'm misinterpreting the direction.Wait, maybe it's asking for the percentage decrease from last year back to the year before last? So, if last year's production was higher by a%, then going back to the year before last would be a decrease. That makes more sense. So, perhaps the question is asking, if production increased by a% from the year before last to last year, what is the percentage decrease from last year back to the year before last?Yes, that seems plausible. So, if I denote the production value of the year before last as, say, P. Then last year's production would be P multiplied by (1 + a/100), since it increased by a%. Now, if we want to find the percentage decrease from last year back to the year before last, we need to calculate how much percentage decrease is needed from last year's production to get back to the year before last's production.So, the formula for percentage decrease is [(New Value - Original Value)/Original Value] * 100%. In this case, the "New Value" is the year before last's production, which is P, and the "Original Value" is last year's production, which is P*(1 + a/100). So, plugging into the formula:Percentage Decrease = [(P - P*(1 + a/100)) / (P*(1 + a/100))] * 100%Let me compute that step by step.First, compute the numerator: P - P*(1 + a/100) = P - P - (P*a)/100 = - (P*a)/100.Then, the denominator is P*(1 + a/100).So, the entire fraction becomes (- (P*a)/100) / (P*(1 + a/100)).Simplify this: The P cancels out, so we have (-a/100) / (1 + a/100).Which is equal to (-a/100) * (100/(100 + a)) = -a/(100 + a).Since percentage decrease is a positive value, we take the absolute value, so it becomes a/(100 + a).Therefore, the percentage decrease is (a/(100 + a))%.Looking at the options, option D is |a/(100 + a)|, which is exactly what we have here. So, the correct answer should be D.Wait, let me double-check my steps to make sure I didn't make a mistake.1. Let P be the production value of the year before last.2. Last year's production is P*(1 + a/100).3. To find the percentage decrease from last year to the year before last, we use the formula: [(Last Year - Year Before Last)/Last Year] * 100%.4. Plugging in the values: [(P - P*(1 + a/100)) / (P*(1 + a/100))] * 100%.5. Simplify numerator: P - P - (P*a)/100 = - (P*a)/100.6. Denominator: P*(1 + a/100).7. So, (- (P*a)/100) / (P*(1 + a/100)) = (-a/100) / (1 + a/100) = -a/(100 + a).8. Taking absolute value gives a/(100 + a), which is the percentage decrease.Yes, that seems correct. So, the answer is D.Alternatively, let me think about it with numbers to see if it makes sense. Suppose a is 100%. So, if production increased by 100% from the year before last to last year, that means last year's production is double the year before last. So, to go back, the percentage decrease would be 50%, because you're halving it.Plugging a = 100 into option D: 100/(100 + 100) = 100/200 = 0.5, which is 50%. That matches our expectation.Another example: a = 50%. So, last year's production is 1.5 times the year before last. To go back, the percentage decrease is (0.5)/1.5 = 1/3 ≈ 33.33%. Plugging a = 50 into D: 50/(100 + 50) = 50/150 ≈ 0.3333, which is 33.33%. Correct again.So, the formula seems to hold with these examples. Therefore, I'm confident that D is the correct answer.**Final Answer**boxed{D}