💡First, I need to establish the relationships between the skipping speeds of Bruce, Tony, Brandon, Colin, and Daniel.1. **Tony's Speed**: Tony skips at twice Bruce's speed. Since Bruce skips at 1 mile per hour, Tony's speed is 2 * 1 = 2 miles per hour.2. **Brandon's Speed**: Brandon skips at one-third of Tony's speed. Therefore, Brandon's speed is 2 / 3 ≈ 0.6667 miles per hour.3. **Colin's Speed**: Colin skips at six times Brandon's speed. Thus, Colin's speed is 6 * 0.6667 ≈ 4 miles per hour.By following these steps, I determine that Colin can skip at a speed of 4 miles per hour.

💡Okay, so Jennifer has a square with side length 1, and she wants to turn it into a regular octagon by cutting off the corners. Hmm, I remember that a regular octagon has all sides equal and all angles equal. So, if she cuts off the corners of the square, each corner will be a right triangle, right? Because the square has right angles, and cutting off a corner would involve removing a triangle.Let me visualize this. If I have a square, and I cut off a small triangle from each corner, the remaining figure should be an octagon. Since it's a regular octagon, all the sides should be equal, and all the angles should be equal too. That means the triangles we cut off must be identical and also right-angled because the corners of the square are right angles.So, let's denote the side length of each triangle that we're cutting off as 's'. Since each corner is a right triangle, both legs of the triangle are equal in length, which is 's'. When we cut off these triangles, the original side of the square, which was length 1, will be reduced by twice the length 's' because we're cutting off a triangle from both ends of each side. Therefore, the new side length of the octagon would be 1 - 2s.But wait, the octagon has sides that are not just the remaining parts of the square's sides; the sides of the octagon also include the hypotenuses of the triangles we've cut off. So, each side of the octagon is actually the hypotenuse of one of these right triangles. Since the legs of each triangle are both 's', the hypotenuse would be s√2, right?So now, we have two expressions for the side length of the octagon: one is 1 - 2s, and the other is s√2. Since the octagon is regular, all its sides are equal, so these two expressions must be equal. Therefore, we can set up the equation:1 - 2s = s√2Now, let's solve for 's'. First, I'll bring all the terms involving 's' to one side:1 = s√2 + 2sFactor out 's' from the right side:1 = s(√2 + 2)Now, solve for 's' by dividing both sides by (√2 + 2):s = 1 / (√2 + 2)Hmm, this denominator has a radical, so I should rationalize it. To rationalize the denominator, I'll multiply the numerator and the denominator by the conjugate of the denominator, which is (√2 - 2):s = [1 * (√2 - 2)] / [(√2 + 2)(√2 - 2)]Let's compute the denominator:(√2 + 2)(√2 - 2) = (√2)^2 - (2)^2 = 2 - 4 = -2So, the denominator is -2. Therefore:s = (√2 - 2) / (-2) = (2 - √2) / 2Simplify that:s = (2 - √2) / 2 = 1 - (√2)/2So, the side length of each triangle we're cutting off is 1 - (√2)/2. But wait, that doesn't seem right because if s is 1 - (√2)/2, then 2s would be 2 - √2, and 1 - 2s would be 1 - (2 - √2) = √2 - 1, which is positive because √2 is approximately 1.414, so √2 - 1 is about 0.414. That seems plausible.But let me double-check my steps. I set up the equation 1 - 2s = s√2, which makes sense because the remaining side of the square after cutting off two triangles of length 's' should equal the hypotenuse of the triangle, which is s√2. Solving for 's', I got s = 1 / (√2 + 2), which I rationalized to (2 - √2)/2. That seems correct.But wait, the problem asks for the side length of the octagon, not the side length of the triangles. The side length of the octagon is s√2, which is equal to 1 - 2s. So, let's compute that.We have s = (2 - √2)/2. So, s√2 = [(2 - √2)/2] * √2 = [2√2 - (√2 * √2)] / 2 = [2√2 - 2] / 2 = √2 - 1.Alternatively, since 1 - 2s = s√2, and we know s = (2 - √2)/2, then 1 - 2s = 1 - 2*(2 - √2)/2 = 1 - (2 - √2) = √2 - 1.So, the side length of the octagon is √2 - 1. Let me check if that makes sense numerically. √2 is approximately 1.414, so √2 - 1 is approximately 0.414, which is positive and less than 1, which makes sense because the octagon is smaller than the original square.Alternatively, I could have approached this problem by considering the relationship between the side length of the square and the side length of the octagon. In a regular octagon formed by cutting the corners of a square, the side length of the octagon is related to the amount cut off from each corner. The formula for the side length of the octagon is indeed √2 - 1 when the original square has a side length of 1.So, after carefully going through the steps and verifying my calculations, I'm confident that the side length of the octagon Jennifer obtains is √2 - 1.

💡Okay, so I have this problem where Alyosha was adding two natural numbers, but he accidentally added an extra digit, which changed the result from 23456 to 55555. I need to figure out which digit he accidentally added. Hmm, let's break this down step by step.First, let's assume that Alyosha was adding two numbers, let's call them x and y. So, the correct sum should be x + y = 23456. But instead, he added an extra digit, let's say digit 'a', to one of the numbers. Let's assume he added it to the number x. So, instead of x, he used 10x + a (since adding a digit to the end of a number is like multiplying by 10 and then adding the digit). So, the incorrect sum becomes (10x + a) + y = 55555.Now, we have two equations:1. x + y = 234562. (10x + a) + y = 55555If I subtract the first equation from the second, I can eliminate y and solve for x and a. Let's do that:(10x + a + y) - (x + y) = 55555 - 23456Simplifying this, we get:9x + a = 32099So, 9x + a = 32099. Now, since 'a' is a single digit, it must be between 0 and 9. That means 32099 - 9x must be between 0 and 9. Let's write that as inequalities:0 ≤ 32099 - 9x ≤ 9Let's solve these inequalities for x.First, the left inequality:32099 - 9x ≥ 032099 ≥ 9xDivide both sides by 9:32099 / 9 ≥ xCalculating that, 32099 divided by 9 is approximately 3566.555...So, x ≤ 3566.555...Now, the right inequality:32099 - 9x ≤ 932099 - 9 ≤ 9x32090 ≤ 9xDivide both sides by 9:32090 / 9 ≤ xCalculating that, 32090 divided by 9 is approximately 3565.555...So, x ≥ 3565.555...Since x must be an integer (because we're dealing with natural numbers), the only possible integer value for x is 3566.Now, let's plug x = 3566 back into the equation 9x + a = 32099 to find 'a':9 * 3566 + a = 32099Calculate 9 * 3566:3566 * 9 = 32094So, 32094 + a = 32099Subtract 32094 from both sides:a = 32099 - 32094a = 5So, the digit Alyosha accidentally added is 5.Wait, let me double-check that. If x is 3566, then y would be 23456 - 3566, which is 19890. Then, adding the extra digit 5 to x would make it 35665. Adding 35665 and 19890 gives 55555, which matches the incorrect sum. So, yes, that checks out.Alright, so I think I've got it. The digit Alyosha added by mistake is 5.Now, moving on to the second problem. There's a flask filled with a 96% hydrochloric acid solution. 12 liters of acid were poured out and the flask was refilled with water. Then, 18 more liters were poured out and the flask was refilled with water again, after which the concentration of the acid in the flask was 32%. I need to find the volume of the flask.Okay, let's denote the volume of the flask as V liters. Initially, it's filled with 96% acid, so the amount of acid is 0.96V liters.First, 12 liters are poured out. Since it's a uniform solution, the concentration remains 96%, so the amount of acid poured out is 0.96 * 12 liters. Let me calculate that: 0.96 * 12 = 11.52 liters. So, after pouring out 12 liters, the remaining acid is 0.96V - 11.52 liters.Then, the flask is refilled with water, so the total volume is back to V liters. Now, the concentration of acid is (0.96V - 11.52) / V. Let me compute that: (0.96V - 11.52)/V = 0.96 - 11.52/V.Then, 18 liters are poured out again. Now, the concentration is 0.96 - 11.52/V, so the amount of acid poured out is (0.96 - 11.52/V) * 18 liters.After pouring out 18 liters, the remaining acid is (0.96V - 11.52) - (0.96 - 11.52/V)*18.Then, the flask is refilled with water again, bringing the total volume back to V liters. The concentration is now 32%, which is 0.32. So, the amount of acid is 0.32V liters.So, let's set up the equation:Remaining acid after second pouring = 0.32VWhich is:(0.96V - 11.52) - (0.96 - 11.52/V)*18 = 0.32VLet me expand this equation step by step.First, compute (0.96 - 11.52/V)*18:= 0.96*18 - (11.52/V)*18= 17.28 - (207.36)/VSo, the remaining acid is:(0.96V - 11.52) - (17.28 - 207.36/V) = 0.32VSimplify the left side:0.96V - 11.52 - 17.28 + 207.36/V = 0.32VCombine like terms:0.96V - 28.8 + 207.36/V = 0.32VBring all terms to one side:0.96V - 0.32V - 28.8 + 207.36/V = 0Simplify:0.64V - 28.8 + 207.36/V = 0Multiply both sides by V to eliminate the denominator:0.64V^2 - 28.8V + 207.36 = 0Now, we have a quadratic equation in terms of V:0.64V^2 - 28.8V + 207.36 = 0Let me simplify this equation by multiplying all terms by 100 to eliminate decimals:64V^2 - 2880V + 20736 = 0Now, let's see if we can simplify this equation further. All coefficients are divisible by 16:64 ÷ 16 = 42880 ÷ 16 = 18020736 ÷ 16 = 1296So, the equation becomes:4V^2 - 180V + 1296 = 0Let's divide all terms by 4 to simplify further:V^2 - 45V + 324 = 0Now, we have a quadratic equation:V^2 - 45V + 324 = 0Let's solve for V using the quadratic formula:V = [45 ± sqrt(45^2 - 4*1*324)] / 2Compute discriminant:45^2 = 20254*1*324 = 1296Discriminant = 2025 - 1296 = 729Square root of 729 is 27.So,V = [45 ± 27] / 2This gives two solutions:V = (45 + 27)/2 = 72/2 = 36V = (45 - 27)/2 = 18/2 = 9Now, let's check which of these solutions makes sense in the context of the problem.If V = 9 liters, then initially, the amount of acid is 0.96*9 = 8.64 liters. After pouring out 12 liters, which is more than the total volume, that doesn't make sense because you can't pour out more than what's in the flask. So, V = 9 liters is not feasible.Therefore, the volume of the flask must be 36 liters.Let me verify this.Initial volume: 36 liters, 96% acid = 34.56 liters acid.First, pour out 12 liters: acid poured out = 0.96*12 = 11.52 liters. Remaining acid = 34.56 - 11.52 = 23.04 liters. Refill with water, so total volume back to 36 liters. Concentration now is 23.04 / 36 = 0.64 or 64%.Then, pour out 18 liters. The concentration is 64%, so acid poured out = 0.64*18 = 11.52 liters. Remaining acid = 23.04 - 11.52 = 11.52 liters. Refill with water, total volume back to 36 liters. Concentration now is 11.52 / 36 = 0.32 or 32%, which matches the given condition.So, yes, V = 36 liters is correct.**Final Answer**The digit Alyosha accidentally added is boxed{5}, and the volume of the flask is boxed{36} liters.

💡First, I recognize that the time Maria takes to climb each flight forms an arithmetic sequence. The first term ( a ) is 15 seconds, and the common difference ( d ) is 8 seconds.To find the total time for the first seven flights, I need to calculate the sum of the first seven terms of this sequence. The formula for the sum of the first ( n ) terms of an arithmetic series is:[ S_n = frac{n}{2} times (2a + (n - 1)d) ]Plugging in the values:[ S_7 = frac{7}{2} times (2 times 15 + (7 - 1) times 8) ][ S_7 = frac{7}{2} times (30 + 48) ][ S_7 = frac{7}{2} times 78 ][ S_7 = 7 times 39 ][ S_7 = 273 ]Therefore, the total time taken to climb the seven flights of stairs is 273 seconds.

💡Okay, so I've got this math expression to compute: (left(frac{1}{3}right)^{-1} + |1 - sqrt{3}| - 2sin 60^{circ} + (pi - 2016)^{0} - sqrt[3]{8}). Hmm, that looks a bit intimidating at first glance, but I think I can break it down step by step. Let me try to tackle each part one by one.First, I see (left(frac{1}{3}right)^{-1}). I remember that a negative exponent means the reciprocal, so (left(frac{1}{3}right)^{-1}) should be the same as (3). Yeah, that makes sense because flipping the fraction (frac{1}{3}) gives me 3. So, that part is straightforward.Next up is the absolute value part: (|1 - sqrt{3}|). I know that absolute value makes sure the result is positive, regardless of what's inside. Since (sqrt{3}) is approximately 1.732, which is greater than 1, (1 - sqrt{3}) would be negative. So, the absolute value would flip the sign, making it (sqrt{3} - 1). Okay, got that.Then, there's (-2sin 60^{circ}). I recall that (sin 60^{circ}) is (frac{sqrt{3}}{2}). So, multiplying that by 2 gives me (sqrt{3}), and since there's a negative sign in front, it becomes (-sqrt{3}). Wait, no, hold on. It's (-2 times frac{sqrt{3}}{2}), which simplifies to (-sqrt{3}). Yeah, that's right.Moving on, there's ((pi - 2016)^{0}). Any number raised to the power of 0 is 1, as long as the base isn't zero. Here, (pi - 2016) is definitely not zero, so this part is just 1.Lastly, there's (-sqrt[3]{8}). The cube root of 8 is 2 because (2 times 2 times 2 = 8). So, this simplifies to (-2).Now, let me put all these simplified parts back into the expression:[3 + (sqrt{3} - 1) - sqrt{3} + 1 - 2]Hmm, let's simplify this step by step. Starting from the left:1. (3 + (sqrt{3} - 1)): That would be (3 + sqrt{3} - 1), which simplifies to (2 + sqrt{3}).2. Then, subtract (sqrt{3}): (2 + sqrt{3} - sqrt{3}). The (sqrt{3}) terms cancel out, leaving me with 2.3. Next, add 1: (2 + 1 = 3).4. Finally, subtract 2: (3 - 2 = 1).Wait, so after simplifying all those terms, everything cancels out except for 1? That seems too simple. Let me double-check each step to make sure I didn't make a mistake.Starting again:- (left(frac{1}{3}right)^{-1} = 3). Correct.- (|1 - sqrt{3}| = sqrt{3} - 1). Correct, since (sqrt{3} > 1).- (-2sin 60^{circ} = -2 times frac{sqrt{3}}{2} = -sqrt{3}). Correct.- ((pi - 2016)^{0} = 1). Correct.- (-sqrt[3]{8} = -2). Correct.Putting them together:[3 + (sqrt{3} - 1) - sqrt{3} + 1 - 2]Simplify term by term:- (3 + sqrt{3} - 1 = 2 + sqrt{3})- (2 + sqrt{3} - sqrt{3} = 2)- (2 + 1 = 3)- (3 - 2 = 1)Yep, it all checks out. So, despite the expression looking complicated with all those operations, it simplifies neatly to 1. I guess that's the beauty of math sometimes—complex-looking expressions can simplify to something very simple!I was a bit worried when I saw the absolute value and the trigonometric function, but breaking them down step by step made it manageable. It's also good to remember properties like negative exponents and the zero exponent, which can simplify things quickly. And of course, knowing that the cube root of 8 is 2 was helpful in the end.So, after carefully going through each part and simplifying, I'm confident that the final answer is 1.

💡First, I need to convert the train's speed from kilometers per hour to meters per second to match the units of the train's length and the time given.To convert km/hr to m/s, I'll use the conversion factor: 1 km/hr = 1000 meters / 3600 seconds.So, the train's speed in m/s is:45 km/hr * (1000 m / 3600 s) = 12.5 m/s.Next, I'll calculate the total distance the train travels in 30 seconds by multiplying the speed by the time:Distance = Speed * TimeDistance = 12.5 m/s * 30 s = 375 meters.This total distance includes both the length of the train and the length of the bridge. Since the train is 160 meters long, I'll subtract the train's length from the total distance to find the length of the bridge:Length of the bridge = Total distance - Length of the trainLength of the bridge = 375 m - 160 m = 215 meters.Therefore, the length of the bridge is 215 meters.