💡To find the value of (frac{15!}{6!9!}), I start by recognizing that (15!) can be expressed as (15 times 14 times 13 times 12 times 11 times 10 times 9!). This allows me to cancel out the (9!) in both the numerator and the denominator.Next, I simplify the remaining fraction by dividing both the numerator and the denominator by common factors. Breaking down the division step-by-step helps manage the calculations more effectively.After simplifying each pair of terms, I multiply the results together to arrive at the final value. This method ensures accuracy and makes the computation more straightforward.

💡First, I need to determine the speed of the train. When the train crosses a signal pole, it covers its own length in 30 seconds. So, the speed of the train is:[text{Speed} = frac{text{Distance}}{text{Time}} = frac{500 text{ meters}}{30 text{ seconds}} = 16.overline{6} text{ meters per second}]Next, when the train crosses the platform, it covers the combined length of the train and the platform in 75 seconds. Let ( P ) be the length of the platform.The total distance covered while crossing the platform is:[500 + P = 16.overline{6} times 75]Calculating the right side:[16.overline{6} times 75 = 1250]Now, solve for ( P ):[500 + P = 1250][P = 1250 - 500][P = 750]Therefore, the length of the platform is 750 meters.

💡Okay, so I have this problem here: Consider a number x such that the ceiling of x minus the floor of x equals zero. Then, define a new number y as x plus one half. I need to find the value of the ceiling of y minus y. Hmm, let me try to break this down step by step.First, I remember that the ceiling function, denoted by ⎡x⎤, gives the smallest integer greater than or equal to x. On the other hand, the floor function, denoted by ⎣x⎦, gives the largest integer less than or equal to x. So, if I subtract the floor of x from the ceiling of x and get zero, that must mean something specific about x.Let me write that down: ⎡x⎤ - ⎣x⎦ = 0. If I rearrange this, it implies that ⎡x⎤ = ⎣x⎦. Now, when does the ceiling of x equal the floor of x? Well, if x is an integer, both the ceiling and the floor functions would just give x itself. So, if x is an integer, then ⎡x⎤ = ⎣x⎦ = x, which means their difference is zero. That makes sense.So, from this, I can conclude that x must be an integer. Okay, that's a key point. Now, moving on to the next part: defining y as x + 1/2. Since x is an integer, adding 1/2 to it would make y a number that's halfway between two integers. For example, if x is 2, then y would be 2.5; if x is -3, y would be -2.5, and so on.Now, I need to find the value of ⎡y⎤ - y. Let's think about what the ceiling of y would be. Since y is x + 1/2 and x is an integer, y is a half-integer. The ceiling function of a half-integer would round it up to the next integer. For instance, if y is 2.5, then ⎡y⎤ would be 3; if y is -2.5, then ⎡y⎤ would be -2 because -2 is greater than -2.5.Wait, hold on, let me make sure I got that right. If y is negative, like -2.5, the ceiling function would give me -2 because -2 is the smallest integer greater than or equal to -2.5. That seems correct. So, regardless of whether x is positive or negative, adding 1/2 to it makes y a half-integer, and the ceiling of y would be the next integer in the positive direction.So, if I denote x as an integer, then y = x + 1/2. Therefore, ⎡y⎤ would be x + 1 because it's the next integer after y. Now, subtracting y from this ceiling value: ⎡y⎤ - y = (x + 1) - (x + 1/2). Let's compute that.Expanding the subtraction: (x + 1) - x - 1/2. The x terms cancel out, leaving 1 - 1/2, which is 1/2. So, regardless of the value of x, as long as it's an integer, the difference between the ceiling of y and y itself is always 1/2.Let me test this with an example to make sure. Suppose x is 3. Then y would be 3 + 1/2 = 3.5. The ceiling of 3.5 is 4. So, 4 - 3.5 = 0.5, which is 1/2. Another example: x is -1. Then y is -1 + 1/2 = -0.5. The ceiling of -0.5 is 0. So, 0 - (-0.5) = 0.5, which is again 1/2. Hmm, that works.Wait, let me try another one. If x is 0, then y is 0 + 1/2 = 0.5. The ceiling of 0.5 is 1. So, 1 - 0.5 = 0.5. Yep, still 1/2. It seems consistent across different integers, both positive and negative, including zero.Is there any case where this wouldn't hold? Well, if x isn't an integer, but the problem states that ⎡x⎤ - ⎣x⎦ = 0, which only happens when x is an integer. So, we don't have to consider non-integer x here.Therefore, I can confidently say that the value of ⎡y⎤ - y is always 1/2 when x is an integer. So, the answer should be 1/2.**Final Answer**The value of lceil y rceil - y is boxed{dfrac{1}{2}}.

💡Okay, so I have this geometry problem here. It says: Given chords AB and CD of a circle S and a point J on chord CD, find a point X on the circle such that chords AX and BX cut a segment EF on chord CD, and EF is bisected at point J. Hmm, that sounds a bit complicated, but let me try to break it down.First, let me visualize the problem. There's a circle S with two chords, AB and CD. Point J is somewhere on CD. I need to find a point X on the circle such that when I draw chords AX and BX, they intersect CD at points E and F respectively, forming a segment EF. The key here is that J is the midpoint of EF. So, EJ = JF.Alright, so maybe I should start by drawing a diagram. I imagine a circle, draw two chords AB and CD intersecting somewhere inside the circle. Then, mark point J on CD. Now, I need to find X such that AX and BX intersect CD at E and F, with J being the midpoint.Hmm, reflections might come into play here. If J is the midpoint of EF, then reflecting points across J could help. Let me think. If I reflect point A over J, I get a new point A'. Similarly, reflecting X over J would give me X'. Maybe there's a relationship between these reflected points and the chords.Wait, if AX intersects CD at E, then reflecting AX over J would make it pass through F. So, the reflected line A'X' would pass through F. That seems useful. So, A' is the reflection of A over J, and X' is the reflection of X over J.Since AX and A'X' are reflections, they should be symmetric with respect to J. That means the angles they make with CD should be related. Maybe the angles are equal or supplementary?Let me recall some circle theorems. The angle subtended by a chord at the center is twice the angle subtended at the circumference. So, if I have point X on the circle, the angle AXB is half the measure of the arc AB. Similarly, angle A'XB would be related to the arc A'B.Wait, but A' is the reflection of A over J, so A' lies on the line AJ extended beyond J. So, A' is as far from J as A is, but on the opposite side. Similarly, X' is the reflection of X over J, so it's on the line XJ extended beyond J.Hmm, maybe I can use the properties of cyclic quadrilaterals here. If AX and BX are chords intersecting CD at E and F, then points A, X, B, and some other points might form a cyclic quadrilateral. But I'm not sure yet.Alternatively, maybe using power of a point could help. The power of point E with respect to circle S should be equal for both chords EA and EX. Similarly, for point F, the power should be equal for FB and FX. But since E and F are related through J being the midpoint, maybe there's a way to set up equations based on that.Let me denote some variables. Let’s say the length of EJ is equal to JF, so let's call that length 'k'. Then, EJ = JF = k. So, the total length of EF is 2k. Now, if I can express the lengths of EA and FB in terms of k, maybe I can find a relationship.Wait, but without knowing the exact positions, this might be tricky. Maybe coordinate geometry could help here. Let me assign coordinates to the points. Let's set the center of the circle at the origin (0,0) for simplicity. Let’s assume the circle has radius 'r'. Then, points A, B, C, D can be placed somewhere on the circumference.But this might get too algebraic and messy. Maybe there's a better synthetic approach.Going back to reflections. If I reflect A over J to get A', then the line A'X' is the reflection of AX over J. Since AX passes through E, A'X' passes through F. So, the points E and F are related through this reflection. Therefore, the angles formed by AX and A'X' with CD should be related.Let me think about the angles. If AX makes an angle θ with CD at E, then A'X' should make the same angle θ with CD at F because of the reflection. So, angle between AX and CD is equal to the angle between A'X' and CD.But since AX and A'X' are reflections, the angles they make with CD should be equal. So, maybe the angles AXJ and A'X'J are equal?Wait, but J is the midpoint, so maybe triangle EJX and FJX' are congruent? Hmm, not sure.Alternatively, maybe the arcs subtended by AX and A'X' are related. Since reflections preserve angles, the arcs should be congruent.Wait, if AX and A'X' are reflections, then the arcs they subtend from A to X and from A' to X' should be equal. So, arc AX is equal to arc A'X'.But since A' is the reflection of A over J, and X' is the reflection of X over J, maybe there's a rotational symmetry here.Alternatively, maybe using harmonic division or projective geometry concepts could help, but I'm not too familiar with those.Let me try another approach. Since J is the midpoint of EF, and E and F lie on CD, maybe I can use the concept of midpoints in intersecting chords.I remember that if two chords intersect, the products of the segments are equal. So, if AX and CD intersect at E, then EA * EX = EC * ED. Similarly, for BX and CD intersecting at F, FB * FX = FC * FD.But since J is the midpoint of EF, we have EJ = JF. Let me denote EJ = JF = k. So, EF = 2k.Therefore, E is at a distance k from J towards one end, and F is at a distance k from J towards the other end.So, if I can express EA and FB in terms of k, maybe I can relate them.Wait, but without knowing the lengths of EC and ED, or FC and FD, it's hard to proceed.Alternatively, maybe using similar triangles. If I can find triangles that are similar involving E, F, J, A, B, X.Wait, if I consider triangles AEJ and BFJ. Since EJ = JF, if I can show that these triangles are similar, then maybe some proportions can be set up.But I don't know if they are similar. Maybe the angles are equal?Alternatively, since AX and BX are chords passing through E and F, and J is the midpoint, maybe there's a spiral similarity or something.Wait, another idea: maybe inversion. If I invert the figure with respect to a circle centered at J, maybe the problem becomes simpler. But inversion is a bit advanced for me right now.Wait, going back to the reflection idea. If I reflect A over J to get A', then line A'X' is the reflection of AX over J. So, if AX passes through E, A'X' passes through F.Therefore, points E and F are related through reflection over J. So, E and F are symmetric with respect to J.Therefore, if I can find a point X such that AX and A'X' intersect CD at E and F, which are symmetric over J, then J will be the midpoint.So, maybe I can construct A' as the reflection of A over J, then find X such that A'X' passes through F, which is the reflection of E.But how do I find such an X?Wait, since X lies on the circle, and A'X' is the reflection of AX, which passes through E, which is the intersection of AX and CD.So, if I can find X such that both AX and A'X' intersect CD at points E and F, which are symmetric over J, then that X would satisfy the condition.Alternatively, maybe I can construct the reflection of A over J, get A', then find the intersection of A'B with CD, which would give me F, and then E would be the reflection of F over J.Wait, let me try that. So, if I reflect A over J to get A', then draw line A'B. Let this line intersect CD at F. Then, E would be the reflection of F over J. Then, AX would be the reflection of A'X' over J, so AX would pass through E.But does this ensure that X lies on the circle?Hmm, maybe not directly. Because X is the intersection of AX and the circle, but we need to ensure that AX passes through E, which is the reflection of F over J.Wait, perhaps the key is to realize that X must lie on the circle and also lie on the reflection of A'X' over J, which passes through E.Alternatively, maybe using power of a point. For point E, power with respect to circle S is EA * EX = EC * ED. Similarly, for point F, FB * FX = FC * FD.But since E and F are symmetric over J, maybe EC * ED = FC * FD? Hmm, not necessarily, unless CD is symmetric with respect to J, which it isn't unless J is the midpoint of CD.But J is just a point on CD, not necessarily the midpoint.Wait, so EC * ED might not equal FC * FD. So, that approach might not work.Alternatively, maybe using Menelaus' theorem or Ceva's theorem.Wait, Menelaus' theorem relates the ratios of lengths when a transversal crosses the sides of a triangle. Maybe if I consider triangle ABD or something, but I'm not sure.Alternatively, Ceva's theorem involves concurrent lines, but I don't see concurrent lines here.Wait, another idea: since J is the midpoint of EF, maybe the polar of J with respect to the circle passes through some significant points.But I'm not too familiar with poles and polars.Wait, maybe using harmonic conjugates. If EF is bisected by J, then J is the midpoint, so perhaps it's the harmonic conjugate of some point.But I'm not sure.Wait, going back to reflections. If I reflect A over J to get A', then line A'B intersects CD at F. Then, E is the reflection of F over J. Then, line AE would be the reflection of line A'F over J.But since X lies on both AX and the circle, maybe X is the intersection of the reflection of A'X' over J with the circle.Wait, this is getting a bit convoluted. Maybe I should try to formalize it.Let me denote:- Let’s reflect point A over J to get point A'.- Let’s reflect point X over J to get point X'.- Then, line AX reflects to line A'X'.- Since AX intersects CD at E, A'X' intersects CD at F, which is the reflection of E over J.Therefore, if I can construct A' and then find X such that A'X' passes through F, which is the reflection of E, then X would be the desired point.But how do I find X?Wait, since X lies on the circle, and X' is the reflection of X over J, then X' lies on the reflection of the circle over J.So, if I reflect the entire circle S over point J, I get another circle S'. Then, X' lies on S'.Therefore, point X' is the intersection of line A'B and circle S'.Wait, that might be a way to construct it.So, steps:1. Reflect point A over J to get A'.2. Reflect circle S over J to get circle S'.3. Draw line A'B.4. The intersection of line A'B with circle S' is point X'.5. Reflect X' over J to get X.6. Then, X is the desired point on circle S.Does this make sense?Let me check:- Reflecting A over J gives A'.- Reflecting circle S over J gives S', which is another circle with the same radius as S, but center reflected over J.- Line A'B intersects S' at X'.- Reflecting X' over J gives X on S.- Then, line AX passes through E, which is the reflection of F over J.- Since A'X' passes through F, AX passes through E.- Therefore, EF is bisected at J.Yes, that seems to work.So, the construction is:1. Reflect A over J to get A'.2. Reflect the circle S over J to get S'.3. Draw line A'B and find its intersection with S' to get X'.4. Reflect X' over J to get X on S.Therefore, X is the desired point.But wait, reflecting the entire circle over J might be a bit involved. How exactly do I reflect a circle over a point?Reflecting a circle over a point J means that every point P on the circle is reflected over J to a point P' such that J is the midpoint of PP'. So, the reflected circle S' will have its center at the reflection of the original center over J, and the same radius.So, if the original circle S has center O, then the reflected circle S' has center O', which is the reflection of O over J.Therefore, to construct S', I need to:- Find the center O of circle S.- Reflect O over J to get O'.- Draw circle S' with center O' and same radius as S.Then, line A'B intersects S' at X'.So, putting it all together, the steps are:1. Find the center O of circle S.2. Reflect O over J to get O'.3. Draw circle S' with center O' and same radius as S.4. Reflect point A over J to get A'.5. Draw line A'B.6. Find the intersection point X' of line A'B with circle S'.7. Reflect X' over J to get X on circle S.8. Then, X is the desired point such that chords AX and BX intersect CD at E and F, with J being the midpoint of EF.I think this should work. Let me verify with a simple case.Suppose circle S is centered at O, and J is the midpoint of CD. Then, reflecting O over J would give O', which is such that OJ = JO'. Then, circle S' would be symmetric to S with respect to J.Reflecting A over J gives A', and line A'B would intersect S' at X'. Reflecting X' over J gives X on S.In this case, since J is the midpoint, the construction should hold.Another test case: suppose J is very close to C on CD. Then, reflecting A over J would place A' far from A. Drawing line A'B might intersect S' at a point X' near C, and reflecting back would give X near C on S. Then, chords AX and BX would intersect CD near C and D, with J being the midpoint.Yes, that seems plausible.So, in conclusion, the method involves reflecting key points and the circle itself over J, then finding the intersection to determine X.I think this is a valid approach. It uses reflection properties and circle geometry to construct the desired point X.

💡Okay, so I have this problem where I need to find the minimum distance between a point P on the curve y = e^x and a point Q on the line y = x. Hmm, that sounds like a calculus problem where I have to minimize the distance between two points on different curves. Let me start by visualizing the problem. The curve y = e^x is an exponential curve that increases rapidly as x increases, and it's always above the x-axis. The line y = x is a straight line passing through the origin at a 45-degree angle. I need to find the shortest distance between any point on the exponential curve and any point on this line.I remember that the distance between two points (x1, y1) and (x2, y2) is given by the formula sqrt[(x2 - x1)^2 + (y2 - y1)^2]. So, if I let P be (x, e^x) on the curve and Q be (a, a) on the line y = x, then the distance PQ would be sqrt[(a - x)^2 + (a - e^x)^2]. But minimizing this distance directly might be complicated because it involves two variables, x and a. Maybe there's a way to express a in terms of x or vice versa. Alternatively, I recall that the shortest distance from a point to a line can be found using the formula |Ax + By + C| / sqrt(A^2 + B^2) if the line is given by Ax + By + C = 0. Wait, so if I fix a point P on the curve y = e^x, the shortest distance from P to the line y = x would be |x - e^x| / sqrt(2). But then I need to minimize this expression over all x. That seems more manageable because it reduces the problem to a single variable calculus problem.Let me write that down. The distance from P(x, e^x) to the line y = x is |x - e^x| / sqrt(2). So, to find the minimum distance, I can minimize the function f(x) = |x - e^x| / sqrt(2). Since sqrt(2) is a positive constant, minimizing f(x) is equivalent to minimizing |x - e^x|.But dealing with absolute values can be tricky, so maybe I can square the function to make it easier. The function to minimize becomes (x - e^x)^2 / 2. Let's call this g(x) = (x - e^x)^2 / 2. Now, I can take the derivative of g(x) with respect to x, set it equal to zero, and solve for x to find the critical points.Calculating the derivative: g'(x) = [2(x - e^x)(1 - e^x)] / 2 = (x - e^x)(1 - e^x). Setting this equal to zero gives (x - e^x)(1 - e^x) = 0. So, either x - e^x = 0 or 1 - e^x = 0.Let's solve these equations. 1. x - e^x = 0: This implies x = e^x. Hmm, solving this equation. I know that for x = 0, e^0 = 1, which is greater than 0. For x = 1, e^1 ≈ 2.718, which is greater than 1. For x = -1, e^{-1} ≈ 0.368, which is greater than -1. It seems that x = e^x doesn't have a real solution because e^x is always positive and grows faster than x. So, this case doesn't give us a real solution.2. 1 - e^x = 0: This implies e^x = 1, so x = 0. That's a valid solution. So, the critical point is at x = 0.Now, I should check if this critical point is a minimum. I can do this by using the second derivative test or analyzing the behavior of the first derivative around x = 0.Let's compute the second derivative. First, g'(x) = (x - e^x)(1 - e^x). Let's expand this: (x - e^x)(1 - e^x) = x(1 - e^x) - e^x(1 - e^x) = x - x e^x - e^x + e^{2x}.So, g'(x) = x - x e^x - e^x + e^{2x}. Now, taking the derivative again, g''(x) = 1 - e^x - x e^x - e^x + 2 e^{2x}.Simplify: g''(x) = 1 - 2 e^x - x e^x + 2 e^{2x}.Now, evaluate g''(0): 1 - 2 e^0 - 0 * e^0 + 2 e^{0} = 1 - 2(1) - 0 + 2(1) = 1 - 2 + 2 = 1. Since g''(0) = 1 > 0, the function is concave upward at x = 0, which means it's a local minimum. Therefore, the minimum distance occurs at x = 0. Let's compute the distance at this point. The point P is (0, e^0) = (0, 1). The distance from P to the line y = x is |0 - 1| / sqrt(2) = 1 / sqrt(2) = sqrt(2)/2.Wait a second, but is this the actual minimum distance between the two curves? Because I minimized the distance from a point on the curve to the line, but does that correspond to the minimum distance between the two curves? I think it does because the closest point on the line to P would be the perpendicular distance. So, yes, this should give the minimum distance.But just to be thorough, let me think about another approach. Maybe parametrizing both points and using calculus to minimize the distance. Let's say P is (x, e^x) and Q is (a, a). Then the distance squared between P and Q is (a - x)^2 + (a - e^x)^2. Let's call this D = (a - x)^2 + (a - e^x)^2.To minimize D, we can take partial derivatives with respect to a and x and set them to zero.First, partial derivative with respect to a: 2(a - x) + 2(a - e^x) = 0. Simplify: 2a - 2x + 2a - 2 e^x = 0 => 4a - 2x - 2 e^x = 0 => 2a = x + e^x => a = (x + e^x)/2.Now, partial derivative with respect to x: -2(a - x) + 2(a - e^x)(-e^x) = 0. Let's substitute a from above.First, compute -2(a - x): -2[( (x + e^x)/2 ) - x] = -2[ (x + e^x - 2x)/2 ] = -2[ ( -x + e^x ) / 2 ] = -(-x + e^x) = x - e^x.Next, compute 2(a - e^x)(-e^x): 2[ ( (x + e^x)/2 - e^x ) ](-e^x) = 2[ (x + e^x - 2 e^x)/2 ](-e^x) = 2[ (x - e^x)/2 ](-e^x) = (x - e^x)(-e^x).So, putting it together: x - e^x + (x - e^x)(-e^x) = 0.Factor out (x - e^x): (x - e^x)(1 - e^x) = 0.So, either x - e^x = 0 or 1 - e^x = 0.As before, x - e^x = 0 has no real solution, and 1 - e^x = 0 gives x = 0.So, x = 0, then a = (0 + e^0)/2 = (0 + 1)/2 = 1/2.Therefore, the point Q is (1/2, 1/2), and the point P is (0, 1). The distance between them is sqrt[(1/2 - 0)^2 + (1/2 - 1)^2] = sqrt[(1/2)^2 + (-1/2)^2] = sqrt[1/4 + 1/4] = sqrt[1/2] = sqrt(2)/2.So, both methods give the same result, which is reassuring. Therefore, the minimum distance |PQ| is sqrt(2)/2.I think this makes sense because the closest point on the line y = x to the curve y = e^x occurs where the perpendicular from the curve meets the line. Since the line y = x has a slope of 1, the perpendicular has a slope of -1. The point on the curve where the tangent line has a slope of -1 would be the point where the distance is minimized.Wait, let me check that. The derivative of y = e^x is y' = e^x. So, the slope of the tangent at any point x is e^x. For the tangent to be perpendicular to y = x, which has slope 1, the product of their slopes should be -1. So, e^x * 1 = -1? That would imply e^x = -1, which is impossible because e^x is always positive. Hmm, that seems contradictory.Wait, maybe I made a mistake here. If the line connecting P and Q is perpendicular to y = x, then its slope should be -1. So, the line PQ has slope -1. Let me see.The slope of PQ is (a - e^x)/(a - x). We want this slope to be -1. So, (a - e^x)/(a - x) = -1 => a - e^x = - (a - x) => a - e^x = -a + x => 2a = x + e^x => a = (x + e^x)/2, which is exactly what we found earlier. So, that's consistent.So, even though the tangent to the curve doesn't have a slope of -1, the line connecting P and Q does, which is why the distance is minimized there. That makes sense because the minimal distance occurs when PQ is perpendicular to the line y = x.Therefore, after checking through different approaches and verifying the results, I'm confident that the minimum distance |PQ| is sqrt(2)/2.

💡First, I need to determine which of the given fractions is the largest. To do this, I'll convert each fraction to have a common denominator, which will make it easier to compare them directly.The denominators of the fractions are 5, 7, 9, 11, and 8. The least common multiple (LCM) of these numbers is 2520. By converting each fraction to have this common denominator, I can easily compare their numerators.For each fraction:- ( frac{2}{5} ) becomes ( frac{1008}{2520} )- ( frac{3}{7} ) becomes ( frac{1080}{2520} )- ( frac{5}{9} ) becomes ( frac{1400}{2520} )- ( frac{4}{11} ) becomes ( frac{916}{2520} )- ( frac{3}{8} ) becomes ( frac{945}{2520} )Now, I'll compare the numerators:- 1008 (A)- 1080 (B)- 1400 (C)- 916 (D)- 945 (E)The largest numerator is 1400, which corresponds to the fraction ( frac{5}{9} ). Therefore, the largest fraction among the given options is ( frac{5}{9} ).