answer：Okay, so I have this problem here with points A(3,0) and B(-1,0), and point C is moving. The condition given is that the dot product of vectors AC and BC is equal to 5. I need to find the equation of the trajectory of point C, which is part (1). Then, in part (2), there's a line l: x - y + 3 = 0 that intersects this curve at points M and N, and I have to find the length of |MN|.Starting with part (1). Let me recall that the dot product of two vectors is given by the product of their magnitudes times the cosine of the angle between them. But in coordinate form, if I have vectors AC and BC, I can express them in terms of coordinates and then compute the dot product.Let me denote point C as (x, y). Then vector AC would be from A to C, which is (x - 3, y - 0) = (x - 3, y). Similarly, vector BC is from B to C, which is (x - (-1), y - 0) = (x + 1, y).The dot product of AC and BC is then (x - 3)(x + 1) + y * y. According to the problem, this equals 5.So, let's compute that:(x - 3)(x + 1) + y^2 = 5.Expanding (x - 3)(x + 1):x^2 + x - 3x - 3 = x^2 - 2x - 3.So, the equation becomes:x^2 - 2x - 3 + y^2 = 5.Let me bring the constants to the right side:x^2 - 2x + y^2 = 5 + 3,which simplifies to:x^2 - 2x + y^2 = 8.Hmm, this looks like the equation of a circle, but it's not in the standard form. Let me complete the square for the x terms.The x terms are x^2 - 2x. To complete the square, take half of the coefficient of x, which is -2, so half is -1, square it to get 1. So, add and subtract 1:(x^2 - 2x + 1) - 1 + y^2 = 8.This becomes:(x - 1)^2 + y^2 - 1 = 8.Adding 1 to both sides:(x - 1)^2 + y^2 = 9.So, the equation is a circle with center at (1, 0) and radius 3. That makes sense because the dot product condition often leads to a circle or some conic section.So, part (1) is done, the equation is (x - 1)^2 + y^2 = 9.Moving on to part (2). The line l is given by x - y + 3 = 0, which I can rewrite as x = y - 3. I need to find the points M and N where this line intersects the circle from part (1), and then compute the distance between M and N.First, let's substitute x from the line equation into the circle equation.Given x = y - 3, substitute into (x - 1)^2 + y^2 = 9:(y - 3 - 1)^2 + y^2 = 9.Simplify inside the first square:(y - 4)^2 + y^2 = 9.Expanding (y - 4)^2:y^2 - 8y + 16 + y^2 = 9.Combine like terms:2y^2 - 8y + 16 = 9.Subtract 9 from both sides:2y^2 - 8y + 7 = 0.So, we have a quadratic equation in y: 2y^2 - 8y + 7 = 0.Let me solve this using the quadratic formula. The quadratic is of the form ay^2 + by + c = 0, so:y = [8 ± sqrt(64 - 4*2*7)] / (2*2).Compute discriminant D:D = 64 - 56 = 8.So, sqrt(D) = sqrt(8) = 2*sqrt(2).Thus, y = [8 ± 2*sqrt(2)] / 4.Simplify:y = [8/4] ± [2*sqrt(2)/4] = 2 ± (sqrt(2)/2).So, the y-coordinates of M and N are 2 + sqrt(2)/2 and 2 - sqrt(2)/2.Now, since x = y - 3, let's find the corresponding x-coordinates.For y = 2 + sqrt(2)/2:x = (2 + sqrt(2)/2) - 3 = (-1) + sqrt(2)/2.Similarly, for y = 2 - sqrt(2)/2:x = (2 - sqrt(2)/2) - 3 = (-1) - sqrt(2)/2.So, the points M and N are:M: ( -1 + sqrt(2)/2 , 2 + sqrt(2)/2 )N: ( -1 - sqrt(2)/2 , 2 - sqrt(2)/2 )Now, to find |MN|, the distance between M and N.Let me denote M as (x1, y1) and N as (x2, y2).So, x1 = -1 + sqrt(2)/2, y1 = 2 + sqrt(2)/2x2 = -1 - sqrt(2)/2, y2 = 2 - sqrt(2)/2The distance formula is sqrt[(x2 - x1)^2 + (y2 - y1)^2].Compute x2 - x1:(-1 - sqrt(2)/2) - (-1 + sqrt(2)/2) = (-1 - sqrt(2)/2 +1 - sqrt(2)/2) = (-sqrt(2)/2 - sqrt(2)/2) = -sqrt(2).Similarly, y2 - y1:(2 - sqrt(2)/2) - (2 + sqrt(2)/2) = (2 - sqrt(2)/2 -2 - sqrt(2)/2) = (-sqrt(2)/2 - sqrt(2)/2) = -sqrt(2).So, both differences are -sqrt(2). But since we square them, the sign doesn't matter.Compute (x2 - x1)^2 = (-sqrt(2))^2 = 2.Similarly, (y2 - y1)^2 = (-sqrt(2))^2 = 2.So, the distance squared is 2 + 2 = 4.Therefore, the distance |MN| is sqrt(4) = 2.Wait, that seems straightforward. Let me verify.Alternatively, since the line intersects the circle, the distance between the two intersection points is the length of the chord. There is a formula for the length of a chord given the distance from the center to the line.Let me recall that formula. The length of the chord is 2*sqrt(r^2 - d^2), where r is the radius and d is the distance from the center to the line.Given that, let me compute d, the distance from the center of the circle to the line l: x - y + 3 = 0.The center of the circle is (1, 0). The distance from a point (x0, y0) to the line ax + by + c = 0 is |ax0 + by0 + c| / sqrt(a^2 + b^2).Here, the line is x - y + 3 = 0, so a = 1, b = -1, c = 3.So, distance d = |1*1 + (-1)*0 + 3| / sqrt(1 + 1) = |1 + 0 + 3| / sqrt(2) = |4| / sqrt(2) = 4 / sqrt(2) = 2*sqrt(2).Wait, but the radius is 3. So, plugging into the chord length formula:Length = 2*sqrt(r^2 - d^2) = 2*sqrt(9 - (8)) = 2*sqrt(1) = 2.So, that confirms the length is 2. So, that's consistent with my earlier calculation.Therefore, the length |MN| is 2.So, summarizing:1) The equation of the trajectory is (x - 1)^2 + y^2 = 9.2) The length of |MN| is 2.**Final Answer**(1) The equation of the trajectory is boxed{(x - 1)^2 + y^2 = 9}.(2) The length of |MN| is boxed{2}.

answer：Okay, so I need to figure out what the result of this program is. Let me read through it carefully.The program starts by setting i equal to 11 and S equal to 1. Then there's a do-while loop. Inside the loop, it multiplies S by i and then decreases i by 1. The loop continues as long as i is greater than or equal to 9. After the loop finishes, it prints S.Hmm, let me break this down step by step. Maybe I should write out each iteration to see what's happening.First, initial values:i = 11S = 1Now, the loop starts. Since it's a do-while loop, it will execute at least once before checking the condition.First iteration:S = S * i = 1 * 11 = 11i = i - 1 = 11 - 1 = 10Now, check if i >= 9. 10 is greater than 9, so loop again.Second iteration:S = 11 * 10 = 110i = 10 - 1 = 9Check condition: 9 >= 9, so loop again.Third iteration:S = 110 * 9 = 990i = 9 - 1 = 8Now, check condition: 8 >= 9? No, it's not. So the loop stops.Then, the program prints S, which is 990.Wait, let me double-check that. So the loop runs when i is 11, 10, and 9. Each time, it multiplies S by i. So S starts at 1, then becomes 11, then 110, then 990. Yep, that seems right.I think that's correct. So the final value of S is 990.

answer：To find the secant of 120 degrees, I start by recalling that secant is the reciprocal of cosine. So, I need to determine the cosine of 120 degrees first.120 degrees is located in the second quadrant of the unit circle, where cosine values are negative. I can express 120 degrees as 180 degrees minus 60 degrees. Using the cosine of a difference identity, I know that cos(180° - θ) equals -cos(θ). Therefore, cos(120°) is equal to -cos(60°).I remember that cos(60°) is 0.5. Substituting this value in, I find that cos(120°) is -0.5.Finally, to find the secant, I take the reciprocal of cosine. So, sec(120°) is 1 divided by -0.5, which equals -2.

answer：Alright, so I have this problem here where I need to find the probability ( p ) that the product of three distinct integers selected randomly from 1 to 2016 is odd. Then, I have to figure out which of the given options is correct regarding this probability. Let me try to break this down step by step.First, I know that for a product of numbers to be odd, all the numbers involved must be odd themselves. If even one of them is even, the entire product becomes even. So, in this case, I need all three selected integers to be odd for their product to be odd.Next, I should figure out how many odd numbers are there between 1 and 2016. Since 2016 is an even number, the count of odd numbers would be exactly half of 2016. Let me calculate that:Total numbers from 1 to 2016: 2016Number of odd numbers: ( frac{2016}{2} = 1008 )So, there are 1008 odd numbers and 1008 even numbers in this range.Now, the problem is about selecting three distinct integers. So, the total number of ways to select any three integers from 2016 is the combination ( C(2016, 3) ). Similarly, the number of ways to select three odd integers is ( C(1008, 3) ).Therefore, the probability ( p ) that all three selected integers are odd is:( p = frac{C(1008, 3)}{C(2016, 3)} )I need to compute this probability and see where it falls in the given options.Let me recall the formula for combinations:( C(n, k) = frac{n!}{k!(n - k)!} )So, plugging in the numbers:( C(1008, 3) = frac{1008 times 1007 times 1006}{3 times 2 times 1} )Similarly,( C(2016, 3) = frac{2016 times 2015 times 2014}{3 times 2 times 1} )So, the probability ( p ) becomes:( p = frac{1008 times 1007 times 1006}{2016 times 2015 times 2014} )Hmm, this looks a bit complicated, but maybe I can simplify it.First, notice that 1008 is exactly half of 2016. So, 1008 = 2016 / 2.Similarly, 1007 is just one less than 1008, and 1006 is two less than 1008.Similarly, 2015 is one less than 2016, and 2014 is two less than 2016.So, maybe I can write each term in the numerator as a fraction of the denominator.Let me write each term:1008 / 2016 = 1/21007 / 2015 = (2015 - 1008) / 2015 = Wait, maybe another approach.Alternatively, let me express 1007 as (2014)/2 + something? Hmm, maybe not.Wait, perhaps factor out 1008 from numerator and 2016 from denominator.Wait, 1008 = 2016 / 2, so let's substitute that:Numerator: (2016 / 2) * (2016 / 2 - 1) * (2016 / 2 - 2)Denominator: 2016 * (2016 - 1) * (2016 - 2)So, substituting:Numerator: (2016/2) * (2016/2 - 1) * (2016/2 - 2) = (1008) * (1007) * (1006)Denominator: 2016 * 2015 * 2014So, the probability becomes:( p = frac{(2016/2) times (2016/2 - 1) times (2016/2 - 2)}{2016 times (2016 - 1) times (2016 - 2)} )Which simplifies to:( p = frac{(1/2) times (1/2 - 1/2016) times (1/2 - 2/2016)}{1 times (1 - 1/2016) times (1 - 2/2016)} )Wait, maybe that's not the best way to look at it. Alternatively, let's factor out 1/2 from each term in the numerator.Wait, actually, let me write each fraction separately:( frac{1008}{2016} = frac{1}{2} )( frac{1007}{2015} = frac{1007}{2015} )( frac{1006}{2014} = frac{1006}{2014} )So, the probability ( p ) is:( p = frac{1}{2} times frac{1007}{2015} times frac{1006}{2014} )Hmm, that might be easier to compute.Let me compute each fraction:First, ( frac{1}{2} ) is straightforward.Second, ( frac{1007}{2015} ). Let's see, 2015 divided by 1007 is approximately 2, but let me compute 1007/2015.Similarly, ( frac{1006}{2014} ). 2014 is 2*1007, so 1006 is 1007 -1, so 1006/2014 = (1007 -1)/(2*1007) = (1007/2*1007) - (1/2*1007) = 1/2 - 1/(2*1007). So, approximately 1/2 - a small fraction.Similarly, ( frac{1007}{2015} ). 2015 is 2*1007 + 1, so 1007/(2*1007 +1) = approximately 1/2 - a small fraction.So, both ( frac{1007}{2015} ) and ( frac{1006}{2014} ) are slightly less than 1/2.Therefore, the product ( frac{1}{2} times frac{1007}{2015} times frac{1006}{2014} ) is going to be a bit less than ( frac{1}{2} times frac{1}{2} times frac{1}{2} = frac{1}{8} ).Wait, but actually, let me compute it more accurately.Let me compute each fraction:First, ( frac{1007}{2015} ).2015 divided by 1007 is approximately 2, as 1007*2=2014, so 2015 is 2014 +1, so 2015 = 2*1007 +1.Therefore, ( frac{1007}{2015} = frac{1007}{2*1007 +1} = frac{1}{2 + 1/1007} approx frac{1}{2.000993} approx 0.4996 )Similarly, ( frac{1006}{2014} ).2014 is 2*1007, so 1006 is 1007 -1, so:( frac{1006}{2014} = frac{1007 -1}{2*1007} = frac{1007}{2*1007} - frac{1}{2*1007} = frac{1}{2} - frac{1}{2014} approx 0.5 - 0.000496 approx 0.4995 )So, both ( frac{1007}{2015} ) and ( frac{1006}{2014} ) are approximately 0.4996 and 0.4995, respectively.Therefore, multiplying these together with 1/2:( p approx frac{1}{2} times 0.4996 times 0.4995 )First, compute 0.4996 * 0.4995:Let me compute 0.5 * 0.5 = 0.25But since both are slightly less than 0.5, the product will be slightly less than 0.25.Compute 0.4996 * 0.4995:= (0.5 - 0.0004) * (0.5 - 0.0005)= 0.5*0.5 - 0.5*0.0005 - 0.5*0.0004 + 0.0004*0.0005= 0.25 - 0.00025 - 0.0002 + 0.0000002= 0.25 - 0.00045 + 0.0000002≈ 0.24955So, approximately 0.24955Then, multiply by 1/2:0.24955 * 0.5 ≈ 0.124775So, approximately 0.124775, which is roughly 0.125, which is 1/8.But wait, the exact value is slightly less than 0.125 because 0.24955 is slightly less than 0.25, so when multiplied by 0.5, it's slightly less than 0.125.Therefore, the probability ( p ) is slightly less than 1/8.Wait, but hold on, let me verify this because I might have made an approximation error.Alternatively, maybe I can compute the exact value.Let me compute ( frac{1008}{2016} times frac{1007}{2015} times frac{1006}{2014} ).Simplify each fraction:( frac{1008}{2016} = frac{1}{2} )( frac{1007}{2015} = frac{1007}{2015} )( frac{1006}{2014} = frac{1006}{2014} = frac{503}{1007} ) (divided numerator and denominator by 2)So, now, the probability is:( frac{1}{2} times frac{1007}{2015} times frac{503}{1007} )Notice that 1007 cancels out:( frac{1}{2} times frac{503}{2015} )So, ( p = frac{503}{2 times 2015} = frac{503}{4030} )Now, let's compute this fraction.Divide 503 by 4030.First, note that 4030 divided by 503 is approximately 8, because 503*8=4024, which is close to 4030.So, 503/4030 = 1/(8.008) approximately.Because 503*8=4024, so 503*8.008 ≈ 4030.Therefore, 503/4030 ≈ 1/8.008 ≈ 0.1248So, approximately 0.1248, which is just slightly less than 1/8 (which is 0.125).Therefore, the probability ( p ) is approximately 0.1248, which is just a tiny bit less than 1/8.Wait, but hold on, let me check my calculation again.Wait, 503*8=4024, so 4030-4024=6, so 503*8 +6=4030, so 503*8.008 is not exactly 4030, but 503*(8 + 6/503)=4030.So, 503/4030=1/(8 + 6/503)=1/(8.0119)≈0.1248.Yes, so approximately 0.1248, which is about 0.125 - 0.0002, so slightly less than 1/8.Therefore, the probability ( p ) is just a little less than 1/8.Wait, but hold on, the answer choices are:(A) p < 1/8(B) p = 1/8(C) 1/8 < p < 1/3(D) p = 1/3(E) p > 1/3So, if p is approximately 0.1248, which is just slightly less than 1/8 (0.125), then p < 1/8, so option A.But hold on, let me think again.Wait, is my calculation correct?I had:( p = frac{C(1008, 3)}{C(2016, 3)} = frac{1008 times 1007 times 1006}{2016 times 2015 times 2014} )Then, I simplified each term:1008/2016 = 1/21007/2015 = 1007/20151006/2014 = 503/1007So, multiplying together:1/2 * 1007/2015 * 503/1007 = 1/2 * 503/2015 = 503/4030 ≈ 0.1248Yes, that seems correct.Alternatively, maybe I can compute the exact decimal.Compute 503 divided by 4030.Let me do this division:4030 divided by 503.Compute how many times 503 goes into 4030.503*8=4024, as above.So, 503 goes into 4030 eight times with a remainder of 6.So, 503/4030 = 8 + 6/503 in the denominator, which is 1/(8 + 6/503).Compute 6/503 ≈ 0.0119So, 8 + 0.0119 ≈ 8.0119Therefore, 1/8.0119 ≈ 0.1248So, yes, approximately 0.1248, which is just under 0.125.Therefore, p ≈ 0.1248 < 1/8.So, the correct answer should be (A) p < 1/8.But wait, hold on, let me think again.Wait, 1008 is exactly half of 2016, so the first term is exactly 1/2.Then, 1007/2015 is slightly less than 1/2, and 1006/2014 is slightly less than 1/2.So, multiplying three terms: 1/2 * slightly less than 1/2 * slightly less than 1/2.So, the result is slightly less than 1/8.Therefore, p is slightly less than 1/8, so p < 1/8, which is option A.Wait, but hold on, another way to think about it is that the probability of selecting three odd numbers is (number of odd numbers / total numbers) * (number of odd numbers -1 / total numbers -1) * (number of odd numbers -2 / total numbers -2).So, in this case:( p = frac{1008}{2016} times frac{1007}{2015} times frac{1006}{2014} )Which is exactly what I computed earlier.So, 1008/2016 = 1/21007/2015 ≈ 0.49961006/2014 ≈ 0.4995Multiplying these together: 0.5 * 0.4996 * 0.4995 ≈ 0.1248, which is just under 0.125.Therefore, p is approximately 0.1248, which is less than 1/8.Hence, the correct answer is (A) p < 1/8.But wait, just to make sure, let me compute the exact value of 503/4030.Compute 503 divided by 4030.Let me do this division step by step.4030 divided by 503:503 * 8 = 4024So, 503 goes into 4030 eight times with a remainder of 6.So, 503/4030 = 8 + 6/503 in the denominator, which is 1/(8 + 6/503).Compute 6/503:503 goes into 6 zero times. So, 6/503 ≈ 0.0119Therefore, 8 + 0.0119 ≈ 8.0119So, 1/8.0119 ≈ 0.1248Therefore, 503/4030 ≈ 0.1248, which is approximately 0.1248, which is 0.125 - 0.0002, so indeed, p is just a little less than 1/8.Therefore, the correct answer is (A) p < 1/8.Wait, but hold on, the answer choices are:(A) p < 1/8(B) p = 1/8(C) 1/8 < p < 1/3(D) p = 1/3(E) p > 1/3So, since p ≈ 0.1248 < 0.125, which is 1/8, so (A) is correct.But wait, hold on, I think I made a mistake in the initial calculation.Wait, 503/4030 is approximately 0.1248, which is 12.48%.But 1/8 is 0.125, which is 12.5%.So, 0.1248 is indeed less than 0.125, so p < 1/8.Therefore, the correct answer is (A).But wait, hold on, let me think again.Wait, 1008 is exactly half of 2016, so the first term is 1/2.Then, 1007 is one less than 1008, and 2015 is one less than 2016.So, 1007/2015 is slightly less than 1/2.Similarly, 1006 is two less than 1008, and 2014 is two less than 2016.So, 1006/2014 is slightly less than 1/2.Therefore, the product is 1/2 * slightly less than 1/2 * slightly less than 1/2, which is slightly less than 1/8.Therefore, p is slightly less than 1/8, so (A) is correct.Alternatively, maybe I can compute the exact value without approximating.Let me compute 503/4030.Compute 503 divided by 4030.Let me write it as:503 ÷ 4030= (503 ÷ 503) ÷ (4030 ÷ 503)= 1 ÷ (8 + 6/503)= 1 ÷ (8.011928)≈ 0.1248So, 0.1248 is approximately 12.48%, which is less than 12.5%, which is 1/8.Therefore, p is less than 1/8, so (A) is correct.Wait, but let me think about this another way.Suppose instead of computing the exact probability, I consider the probability step by step.First, the probability that the first number selected is odd is 1008/2016 = 1/2.Then, given that the first number was odd, the probability that the second number is also odd is 1007/2015.Similarly, given that the first two numbers were odd, the probability that the third number is odd is 1006/2014.So, the combined probability is 1/2 * 1007/2015 * 1006/2014.As we saw earlier, this is equal to 503/4030 ≈ 0.1248, which is less than 1/8.Therefore, the probability is less than 1/8, so option (A) is correct.Wait, but hold on, I'm a bit confused because sometimes when dealing with probabilities, especially with dependent events, the exact calculation might be a bit different, but in this case, since we're dealing with combinations, it's correct to compute it as the ratio of combinations.Yes, so I think my calculation is correct.Therefore, the probability is slightly less than 1/8, so the correct answer is (A).But wait, hold on, let me check once more.Compute 503 divided by 4030.Let me compute 503 * 8 = 4024So, 503 * 8 = 40244030 - 4024 = 6So, 503/4030 = 8 + 6/503 in the denominator, which is 1/(8 + 6/503) ≈ 1/8.0119 ≈ 0.1248.Yes, that's correct.Therefore, p ≈ 0.1248 < 0.125, so p < 1/8.Hence, the correct answer is (A).**Final Answer**boxed{A}

answer：Okay, so I have this problem about stratified sampling. Let me try to understand it step by step. The swimming team has 32 male athletes and 24 female athletes. So, in total, there are 32 + 24 = 56 athletes. They want to draw a sample of 14 athletes using stratified sampling. I need to find out how many male athletes will be drawn in this sample.Hmm, stratified sampling. I remember that stratified sampling is a method where the population is divided into subgroups, called strata, and then a random sample is taken from each stratum. The idea is to ensure that each subgroup is proportionally represented in the sample. So, in this case, the two strata are male and female athletes.First, I should figure out the proportion of male and female athletes in the entire team. There are 32 males out of 56 total athletes. So, the proportion of males is 32/56, and the proportion of females is 24/56. Let me calculate those.32 divided by 56. Let me simplify that fraction. Both 32 and 56 are divisible by 8. 32 ÷ 8 = 4, and 56 ÷ 8 = 7. So, 32/56 simplifies to 4/7. Similarly, 24/56 simplifies to 3/7 because 24 ÷ 8 = 3 and 56 ÷ 8 = 7.So, the proportion of males is 4/7, and females is 3/7. Now, since we're using stratified sampling, the sample should reflect these proportions. The total sample size is 14 athletes. Therefore, the number of males in the sample should be 4/7 of 14, and the number of females should be 3/7 of 14.Let me compute that. For males: 4/7 multiplied by 14. 14 divided by 7 is 2, and 2 multiplied by 4 is 8. So, 4/7 * 14 = 8. Similarly, for females: 3/7 * 14. 14 divided by 7 is 2, and 2 multiplied by 3 is 6. So, 3/7 * 14 = 6.Wait, let me double-check that. If I take 8 males and 6 females, that adds up to 14, which is the sample size. So, that seems correct. But just to make sure, let me think about it another way.The ratio of males to females is 32:24, which simplifies to 4:3. So, for every 7 athletes, 4 are male and 3 are female. If we have a sample of 14, which is double 7, then the number of males should be double 4, which is 8, and females double 3, which is 6. Yep, that matches what I got earlier.Alternatively, I can think in terms of percentages. The proportion of males is 4/7, which is approximately 57.14%, and females is 3/7, approximately 42.86%. So, 57.14% of 14 is roughly 8, and 42.86% is roughly 6. That also makes sense.Is there another way to approach this? Maybe using equations. Let me denote the number of males in the sample as M and females as F. We know that M + F = 14. Also, since it's stratified sampling, the ratio of M to F should be the same as the ratio of males to females in the population, which is 32:24 or 4:3.So, M/F = 4/3. Therefore, M = (4/3)F. Plugging this into the first equation: (4/3)F + F = 14. Combining the terms: (4/3 + 3/3)F = 14 => (7/3)F = 14. Solving for F: F = 14 * (3/7) = 6. Then, M = 14 - 6 = 8. So, that also gives M = 8.Wait, so all these different methods are giving me the same answer. That gives me more confidence that 8 is the correct number of male athletes in the sample.Just to recap, the key steps are:1. Determine the total population: 32 males + 24 females = 56 athletes.2. Calculate the proportion of males and females: 32/56 = 4/7 and 24/56 = 3/7.3. Apply these proportions to the sample size of 14: (4/7)*14 = 8 males and (3/7)*14 = 6 females.4. Verify using alternative methods like ratios and equations to ensure consistency.I think I've covered all the bases here. I don't see any mistakes in my reasoning, and all the calculations add up correctly. So, I feel confident that the number of male athletes drawn in the sample is 8.**Final Answer**The number of male athletes drawn is boxed{8}.

answer：Okay, so I have this problem here: Each day, Jenny eats 20% of the jellybeans that were in her jar at the beginning of that day. After the second day, 32 jellybeans remain. I need to find out how many jellybeans were in the jar originally. Hmm, let me think about how to approach this.First, let's parse the problem. Jenny starts with a certain number of jellybeans in her jar. Each day, she eats 20% of the jellybeans that are there at the beginning of the day. So, every day, she's reducing the number of jellybeans by 20%. After two days, there are 32 left. I need to figure out how many she started with.I think the key here is to work backwards. Since we know the number of jellybeans at the end of the second day, maybe we can figure out how many were there at the beginning of the second day, and then work our way back to the original number.Let me denote the number of jellybeans at the beginning of the first day as J. Then, at the end of the first day, she would have eaten 20% of J, so she would have 80% of J left. So, the number of jellybeans at the end of the first day is 0.8 * J.Now, moving on to the second day. At the beginning of the second day, she has 0.8 * J jellybeans. Then, she eats 20% of that amount, so she eats 0.2 * (0.8 * J) jellybeans. Therefore, the number of jellybeans left at the end of the second day would be 0.8 * (0.8 * J) = (0.8)^2 * J.According to the problem, at the end of the second day, there are 32 jellybeans remaining. So, we can set up the equation:(0.8)^2 * J = 32Let me compute (0.8)^2 first. 0.8 multiplied by 0.8 is 0.64. So, the equation becomes:0.64 * J = 32To find J, we can divide both sides by 0.64:J = 32 / 0.64Hmm, let me calculate that. 32 divided by 0.64. Hmm, 0.64 goes into 32 how many times? Let me think. 0.64 times 50 is 32, because 0.64 * 50 = 32. So, J = 50.Wait, let me double-check that. If she starts with 50 jellybeans, on the first day she eats 20% of 50, which is 10 jellybeans. So, she has 40 left. On the second day, she eats 20% of 40, which is 8 jellybeans. So, she has 32 left. Yes, that matches the problem statement.So, the original number of jellybeans was 50.But just to make sure I didn't make any mistakes, let me go through the calculations again.Starting with J jellybeans.End of day 1: J - 0.2J = 0.8JEnd of day 2: 0.8J - 0.2*(0.8J) = 0.8*(0.8J) = 0.64JGiven that 0.64J = 32, so J = 32 / 0.64 = 50.Yes, that seems consistent.Alternatively, I can think of it as each day the number of jellybeans is multiplied by 0.8. So, after two days, it's multiplied by 0.8^2, which is 0.64. So, 32 is 64% of the original amount. Therefore, original amount is 32 / 0.64 = 50.Another way to think about it is to reverse the operations. If at the end of the second day, there are 32 jellybeans, that means that 32 is 80% of the amount at the beginning of the second day. So, let me denote the amount at the beginning of the second day as A.So, 0.8 * A = 32Therefore, A = 32 / 0.8 = 40So, at the beginning of the second day, there were 40 jellybeans. Now, that amount is 80% of the original amount, because Jenny ate 20% on the first day. So, let me denote the original amount as J.0.8 * J = 40Therefore, J = 40 / 0.8 = 50Yes, same result. So, that confirms that the original number was 50.I think that's solid. I've approached it both by working forward and backward, and both methods give me the same answer. So, I feel confident that the original number of jellybeans was 50.**Final Answer**The jar originally contained boxed{50} jellybeans.