answer：Okay, so I have this problem where X is at the center of a circular pond, and Y is at the edge. Y can't swim but can run around the edge at a speed of 4v. X can swim at a speed of v and can run faster than 4v. The question is, can X escape?Hmm, let's break this down. First, the pond is circular, so it's symmetric. X is at the center, which is the farthest point from the edge in terms of swimming. Y is at the edge, but can't swim, so Y can only move along the circumference. Y's speed is 4v, which is pretty fast.X can swim at speed v, which is slower than Y's running speed. But X can run faster than 4v once on land. So, if X can get out of the pond, X can outrun Y. The key is whether X can reach the edge before Y can get to that point.Let me visualize this. The pond has radius r. So, the distance from the center to the edge is r. X needs to swim this distance. The time it takes X to swim to the edge is t = r / v.Meanwhile, Y is at the edge. If X swims straight towards a point on the edge, Y can start running towards that point. Since Y is at the edge, the distance Y needs to cover is along the circumference. The circumference is 2πr. If X swims to a point directly opposite Y, the distance Y needs to run is half the circumference, which is πr. But if X swims to a point that's not directly opposite, Y might have a shorter distance to run.Wait, actually, if X swims in a straight line towards a point on the edge, Y can choose the shorter arc to reach that point. So, depending on the angle, Y might have to run less than πr. Hmm, so maybe X should not swim straight towards the opposite point but at some angle to maximize the distance Y has to run.Alternatively, maybe X can swim in a circle or some other path to confuse Y or make Y have to run more. But since Y is on the edge, maybe it's better for X to swim in a straight line towards a point, but not directly opposite.Let me think about the time it takes for X to swim to the edge. If X swims a distance r at speed v, time is t = r / v.In that same time, Y can run a distance of 4v * t = 4v * (r / v) = 4r. So, Y can cover 4r distance in the time it takes X to swim to the edge.But the circumference is 2πr, which is approximately 6.28r. So, 4r is less than the full circumference. That means Y can't make a full loop around the pond in the time X swims to the edge. So, if X swims towards a point, Y can run 4r along the circumference. If the distance Y needs to run is more than 4r, then X can escape. If it's less, Y can intercept.But wait, the maximum distance Y needs to run to reach the point where X is heading is πr, which is about 3.14r. Since 4r is greater than 3.14r, Y can actually reach any point on the circumference faster than X can swim to it. That seems bad for X.But hold on, maybe X can choose a different point. If X swims towards a point that's not directly opposite, the distance Y has to run is less. So, if X swims towards a point that's θ radians away from Y's starting position, the distance Y has to run is rθ (along the circumference). The time for X to swim to that point is still r / v. In that time, Y can run 4v * (r / v) = 4r. So, Y can cover 4r distance, which is equivalent to 4r / r = 4 radians. Since 4 radians is about 229 degrees, which is more than half the circle (π radians ≈ 3.14 radians or 180 degrees). So, if X swims towards a point that's more than 4 radians away, Y can't reach it in time. But since the pond is circular, the maximum angle between two points is π radians (half the circle). So, 4 radians is more than π, which is about 3.14. So, actually, Y can reach any point on the circumference in less time than X can swim to it because 4r is greater than the maximum distance Y needs to run, which is πr.Wait, that can't be right. Because if X swims towards a point that's θ radians away, the distance Y needs to run is min(θ, 2π - θ) * r. So, the maximum distance Y needs to run is πr. Since 4r is greater than πr (since 4 > π ≈ 3.14), Y can always reach any point on the circumference before X can swim there. That would mean Y can always intercept X, so X can't escape.But that contradicts the initial thought that X can swim to the edge and then run away. Maybe I'm missing something.Wait, perhaps X doesn't have to swim straight to the edge. Maybe X can swim in a spiral or some other path to confuse Y or make Y have to run more. But since Y is on the edge, any point X swims towards, Y can just run along the circumference to intercept. Since Y is faster, it seems Y can always get there first.Alternatively, maybe X can swim towards a point, and when Y starts running, X changes direction. But since Y is on the edge, X can't really outmaneuver Y because Y can always adjust to the shortest path.Wait, another thought: if X swims towards a point, Y will start running towards that point. If X swims at an angle, maybe X can reach a different point before Y can adjust. But since Y is on the edge, the distance Y needs to cover is along the circumference, which is proportional to the angle. If X swims at an angle θ, the distance X swims is r / cos(θ/2) or something? Wait, no, if X swims at an angle, the distance is still r, but the direction is different.Wait, maybe I should model this mathematically. Let's denote the radius of the pond as r. X is at the center, Y is at a point on the circumference. Let's say Y is at angle 0, and X swims towards angle θ. The distance X swims is r, so time is t = r / v.In that time, Y can run a distance of 4v * t = 4r. The distance Y needs to run to reach angle θ is the minimum of θ*r and (2π - θ)*r. So, Y will take the shorter arc, which is min(θ, 2π - θ)*r.So, for X to escape, the time it takes for Y to reach the point must be greater than t. So, min(θ, 2π - θ)*r / (4v) > t = r / v.Simplify: min(θ, 2π - θ) / 4 > 1.So, min(θ, 2π - θ) > 4.But θ is an angle in radians, and the maximum min(θ, 2π - θ) can be is π (when θ = π). Since π ≈ 3.14 < 4, this inequality can never be satisfied. Therefore, Y can always reach the point before X can swim there.Wait, that means X can never escape? But that seems counterintuitive because X can run faster than Y once on land. Maybe the key is that once X reaches the edge, X can start running away from Y, who is on the circumference.So, perhaps when X reaches the edge, Y is not exactly at that point, but somewhere else. Then, X can start running along the circumference away from Y, using the fact that X can run faster than Y.Let me think about this. Suppose X swims to a point P on the edge. The time taken is t = r / v. In that time, Y has run a distance of 4v * t = 4r. So, Y has moved 4r along the circumference. The circumference is 2πr, so 4r is 4/(2π) = 2/π ≈ 0.6366 of the circumference. So, Y has moved about 63.66% of the circumference in the time X swims to the edge.Now, when X reaches point P, Y is at a point Q, which is 4r distance away from the starting point along the circumference. The distance between P and Q along the circumference is |4r - rθ|, but I need to think carefully.Wait, actually, when X swims to point P, which is θ radians away from Y's starting point, Y would have moved 4r distance, which is 4r / r = 4 radians. So, Y would have moved 4 radians from the starting point. The angle between P and Q is |θ - 4| radians, but since angles wrap around, it's actually the minimum of |θ - 4| and 2π - |θ - 4|.But since θ is the angle X chose to swim towards, and Y started at 0, after time t, Y is at angle 4 radians. So, the angle between P and Q is |θ - 4|.Now, when X reaches P, X can start running along the circumference away from Y. X's running speed is greater than 4v, let's say it's k*v where k > 4. So, X can run faster than Y.The distance between P and Q along the circumference is |θ - 4| * r. But since angles are periodic, the actual distance is the minimum of |θ - 4| * r and (2π - |θ - 4|) * r.Wait, but if X starts running away from Y, the distance between them can be either clockwise or counterclockwise. X will choose the direction where the distance is shorter.But actually, since Y is moving towards P, when X reaches P, Y is at Q, which is 4 radians away from the starting point. So, the angle between P and Q is |θ - 4|.If X starts running in the direction away from Y, the distance X needs to cover to get away is the shorter arc between P and Q.Wait, maybe I need to model this as a pursuit problem. Once X is on the edge, both X and Y are on the circumference, with X at P and Y at Q, separated by some angle. Then, X can run away from Y, and Y can run towards X. Since X is faster, X can increase the distance between them until it's safe.But how much time does X have to get away before Y can catch up?Alternatively, maybe X can swim to a point where, after reaching the edge, X can run away before Y can catch up.Let me try to calculate the maximum angle θ that X can swim towards such that, after reaching the edge, X can run away before Y can intercept.When X swims to P, which is θ radians from Y's starting point, Y has moved 4 radians in that time. So, the angle between P and Q is |θ - 4|.Now, when X is at P, Y is at Q, which is |θ - 4| radians away from P along the circumference. The distance between P and Q is min(|θ - 4|, 2π - |θ - 4|) * r.X can start running away from Y at speed k*v, and Y can run towards X at speed 4v.The relative speed between X and Y is (k*v - 4v) = v(k - 4). Since k > 4, X is moving away faster.The initial distance between them is d = min(|θ - 4|, 2π - |θ - 4|) * r.The time it takes for X to get far enough away is when the distance between them is increasing. But actually, since they are on a circle, if X runs in the opposite direction of Y, the distance will increase.Wait, no. If X runs in the direction away from Y, the distance between them increases. If X runs towards Y, the distance decreases.But X wants to get away, so X will run in the direction that increases the distance. So, the initial distance is d = min(|θ - 4|, 2π - |θ - 4|) * r.But if X runs away, the distance will increase at a rate of (k*v - 4v) = v(k - 4). So, the time it takes for X to get to a safe distance is when the distance becomes large enough that Y can't catch up.But actually, since the pond is circular, X can just keep running away, maintaining the distance. Wait, no, because both are on the circumference, so if X runs away, Y can also run towards X, but since X is faster, the distance will increase.Wait, actually, on a circular track, if two people are moving in the same direction, the relative speed is the difference. So, if X is running away from Y at speed k*v, and Y is running towards X at speed 4v, but in the opposite direction, the relative speed is k*v + 4v. Wait, no, if they are moving in opposite directions, their relative speed is the sum. If they are moving in the same direction, it's the difference.Wait, let's clarify. If X is running clockwise and Y is running counterclockwise, their relative speed is k*v + 4v. If they are running in the same direction, it's k*v - 4v.But in this case, when X reaches P, Y is at Q, which is |θ - 4| radians away. If X runs in the direction away from Y, which would be the direction that increases the distance, then Y has to run in the opposite direction to catch up. So, their relative speed is k*v + 4v.Wait, no, if X is running away from Y in one direction, Y can choose to run in the opposite direction to intercept. So, the relative speed is k*v + 4v, because they are moving towards each other in opposite directions.Wait, no, if X is running clockwise away from Y, and Y is running counterclockwise towards X, their relative speed is k*v + 4v. So, the time it takes for them to meet again is the initial distance divided by their relative speed.But X doesn't want to meet Y again. X wants to get far enough away that Y can't catch up. Wait, but since they are on a circular track, unless X can get to a point where Y can't reach, but since Y is on the edge, it's a closed loop.Wait, maybe I'm overcomplicating. Let's think about it differently. Once X is on the edge, X can start running away from Y. Since X is faster, X can maintain a distance from Y. But Y is also on the edge, so Y can run around the pond to intercept.But since X is faster, X can always stay ahead of Y. Wait, but Y is already on the edge, so if X is on the edge, Y can just run around to intercept. But since X is faster, X can choose a direction to run where Y can't catch up.Wait, actually, if X is faster, once X is on the edge, X can run in the opposite direction of Y and increase the distance between them. Since X is faster, the distance will keep increasing until X can get to a point where Y is far enough away, and then X can run off the pond.But wait, the pond is circular, so X can't really "run off" unless X reaches a point where Y can't intercept. But since Y is on the edge, Y can always run around to intercept.Wait, maybe the key is that once X is on the edge, X can run in a direction such that Y has to run more than half the circumference to intercept, which would take more time than X needs to get away.But since X is faster, maybe X can reach a safe point before Y can intercept.Wait, let's try to calculate this. Suppose X swims to point P, which is θ radians from Y's starting point. The time taken is t1 = r / v. In that time, Y has run 4v * t1 = 4r, which is 4 radians around the circumference.So, Y is now at point Q, which is 4 radians from the starting point. The angle between P and Q is |θ - 4| radians.Now, X is at P and can start running away from Y. Let's say X runs in the direction that increases the distance from Y. The initial distance between P and Q is min(|θ - 4|, 2π - |θ - 4|) * r.Let's denote this distance as d = min(|θ - 4|, 2π - |θ - 4|) * r.Now, X runs away from Y at speed k*v, and Y runs towards X at speed 4v. Since they are moving in opposite directions, their relative speed is k*v + 4v.The time it takes for them to meet again is t2 = d / (k*v + 4v).But X doesn't want to meet Y again. Instead, X wants to reach a point where Y can't intercept. Wait, but on a circular track, they will meet again after time t2.But if X can reach a point where Y is far enough away before Y can intercept, maybe X can escape.Wait, perhaps instead of trying to outrun Y on the circumference, X can immediately start running tangentially away from the pond. But the problem says X can run faster than 4v, but it doesn't specify whether X can run on land or only in the water. Wait, the problem says X can swim at speed v and can run faster than 4v. So, once X is out of the pond, X can run on land at speed greater than 4v.So, maybe the strategy is for X to swim to the edge, then run tangentially away from the pond at speed greater than 4v, while Y is constrained to run along the circumference at 4v.In that case, once X is on the edge, X can start running tangentially, and Y can only run along the circumference to intercept. Since X is running tangentially, the distance X needs to cover to get away is the straight line from the edge, but Y has to run along the circumference.Wait, but once X is on the edge, if X runs tangentially, Y can run along the circumference towards the point where X is heading. But since X is running faster, maybe X can get far enough away before Y can intercept.Wait, let's model this. Suppose X swims to point P on the edge. Time taken: t1 = r / v. In that time, Y has run 4r along the circumference, ending up at point Q, which is 4 radians from the starting point.Now, X starts running tangentially from P at speed k*v (k > 4). Y starts running from Q towards P at speed 4v.The distance X needs to cover to get away is the straight line from P, but Y has to run along the circumference to reach P. Wait, no, Y is at Q, which is 4 radians from the starting point. The distance between Q and P along the circumference is |θ - 4| * r, but since θ is the angle X chose to swim towards, which we can choose.Wait, maybe X should swim towards a point P such that after reaching P, the distance Y has to run to intercept is maximized.But since Y is already at Q, which is 4 radians from the starting point, the angle between P and Q is |θ - 4|.If X chooses θ such that |θ - 4| is maximized, that would be θ = 0, making |0 - 4| = 4 radians, but since angles wrap around, the actual distance is min(4, 2π - 4) ≈ min(4, 2.28) ≈ 2.28 radians.Wait, so if X swims towards θ = 0, which is directly opposite Y's starting point, then after time t1, Y has moved 4 radians, ending up at Q = 4 radians. The distance between P (θ=0) and Q (4 radians) is 4 radians, but since the circumference is 2π ≈ 6.28 radians, the shorter distance is 2π - 4 ≈ 2.28 radians.So, the distance Y has to run to reach P is 2.28r.Now, X starts running tangentially from P at speed k*v, and Y starts running from Q towards P at speed 4v.The time it takes for Y to reach P is t2 = (2.28r) / (4v) ≈ (2.28 / 4) * (r / v) ≈ 0.57 * (r / v).In that time, X has run a distance of k*v * t2 = k*v * 0.57 * (r / v) = 0.57k * r.To escape, X needs to have run far enough that Y can't catch up. But since X is running tangentially, the distance X needs to cover to get away is the straight line from P, but Y is constrained to the circumference.Wait, actually, once X is running tangentially, the distance between X and Y is increasing because X is moving away in a straight line while Y is moving along the circumference.But how much time does X have before Y can reach the point where X is heading?Wait, maybe I need to calculate the time it takes for Y to reach the point where X is heading, and see if X can get there first.But X is running tangentially, so the point where X is heading is moving away. Wait, no, X is running in a straight line tangent to the pond at P. So, the direction is fixed.Wait, perhaps a better approach is to consider the angular speed. Once X is on the edge, X can run tangentially at speed k*v, which translates to an angular speed of (k*v) / r (since tangential speed = r * angular speed). Y is running along the circumference at speed 4v, which is an angular speed of 4v / r.So, the angular speed of X is k*v / r, and Y's is 4v / r. Since k > 4, X's angular speed is greater than Y's.Wait, but X is running tangentially, so X's angular position relative to the center is changing. Wait, no, X is running away from the pond, so X's angular position isn't changing relative to the center. Instead, X is moving in a straight line, so the angle between X and the center is fixed.Wait, maybe I'm overcomplicating. Let's think about the distance between X and Y as functions of time.When X starts running tangentially from P at speed k*v, the position of X at time t after reaching P is (r, 0) + (k*v * t, 0) in polar coordinates, but actually, it's better to use Cartesian coordinates.Let me set up a coordinate system where the center of the pond is at (0,0), Y starts at (r,0), and X swims to point P at (r,0) as well, but that's where Y is. Wait, no, if X swims to (r,0), Y is already there. So, X should swim to a different point.Wait, let's say X swims to point P at (r, θ). Then, Y is at (r, 0) initially. After time t1 = r / v, Y has moved to (r, 4t1 / r * r) = (r, 4t1). Wait, no, Y's angular position is 4v * t1 / r, since angular speed is linear speed divided by radius.Wait, Y's linear speed is 4v, so angular speed is 4v / r. So, in time t1 = r / v, Y moves θ_Y = (4v / r) * t1 = (4v / r) * (r / v) = 4 radians.So, Y ends up at angle 4 radians from the starting point. So, if X swims to point P at angle θ, then after time t1, Y is at angle 4 radians.The distance between P and Y's new position along the circumference is |θ - 4| radians, but the shorter arc is min(|θ - 4|, 2π - |θ - 4|).Now, X starts running tangentially from P at speed k*v. The direction of tangential run is fixed, so X's position at time t after reaching P is (r + k*v * t * cos(θ + π/2), k*v * t * sin(θ + π/2)), assuming θ is the angle from the x-axis to P.Wait, maybe it's better to set θ = 0 for simplicity. Let me assume X swims to point P at (r,0). Then, Y starts at (r,0) and moves to angle 4 radians, ending up at (r cos 4, r sin 4).Now, X starts running tangentially from (r,0) at speed k*v. The direction of tangential run is along the positive y-axis, so X's position at time t after reaching P is (r, k*v * t).Y is at (r cos 4, r sin 4) and can run along the circumference at speed 4v. The angular speed of Y is 4v / r.Now, the distance between X and Y as functions of time needs to be calculated. But this might get complicated.Alternatively, maybe we can consider the time it takes for Y to reach the point where X is heading. If X is running tangentially, Y has to run along the circumference to reach the point where X is heading. But since X is moving away, the point where X is heading is moving away as well.Wait, perhaps a better approach is to consider the relative speeds. Once X is on the edge, X can run tangentially at speed k*v, while Y can run along the circumference at 4v. The key is whether X can get far enough away before Y can intercept.But since X is running tangentially, the distance X needs to cover to get away is the straight line from P, while Y has to run along the circumference to reach P. Wait, no, Y is already at Q, which is 4 radians away from P.Wait, maybe I need to calculate the time it takes for Y to reach P, and see if X can get far enough away in that time.The distance Y needs to run to reach P is the shorter arc between Q and P, which is min(|θ - 4|, 2π - |θ - 4|) * r. Since θ = 0, it's min(4, 2π - 4) * r ≈ min(4, 2.28) * r = 2.28r.So, time for Y to reach P is t2 = 2.28r / (4v) ≈ 0.57r / v.In that time, X has run a distance of k*v * t2 = k*v * 0.57r / v = 0.57k r.To escape, X needs to have run far enough that Y can't catch up. But since X is running tangentially, the distance between X and Y is increasing. The distance between X and Y after time t2 is sqrt((r + 0.57k r)^2 + (r sin 4)^2), but this might not be necessary.Wait, actually, once X is running tangentially, the distance between X and Y is the straight line from X's position to Y's position. But Y is moving along the circumference, so the distance is changing over time.This is getting too complicated. Maybe I should look for a simpler approach.I recall that in similar problems, the key is whether the swimmer can reach a point on the edge such that the runner can't intercept in time. The formula often involves the ratio of speeds.In this case, the swimmer's speed is v, and the runner's speed is 4v. The ratio is 1:4.The critical angle for escape is when the swimmer can reach a point where the runner can't intercept. The formula for the critical angle is θ = arcsin(v / V), where V is the runner's speed. But in this case, V = 4v, so θ = arcsin(1/4) ≈ 14.47 degrees.Wait, but that might be for a different setup. Let me think.Actually, the standard problem is when the pursuer is on the edge and the pursued is in the water. The critical angle is when the pursued can reach the edge before the pursuer can run to that point.In our case, the pursuer is Y on the edge, and the pursued is X in the water. So, the critical angle is when X can swim to a point such that Y can't run to that point in time.The formula for the critical angle θ is given by θ = arcsin(v / V), where v is the swimmer's speed and V is the runner's speed.So, θ = arcsin(v / 4v) = arcsin(1/4) ≈ 14.47 degrees.This means that if X swims towards a point within 14.47 degrees of Y's starting position, Y can intercept. But if X swims beyond that angle, Y can't intercept, and X can escape.Wait, that seems promising. So, if X swims towards a point more than 14.47 degrees away from Y's starting position, X can reach the edge before Y can run to that point.But in our case, the pond is circular, so the maximum angle X can swim towards is 180 degrees (directly opposite). So, if X swims towards a point more than 14.47 degrees away, X can escape.But wait, in the standard problem, the pursuer is on the edge, and the pursued is in the water. The critical angle is the angle beyond which the pursued can escape. So, in our case, if X swims towards a point beyond the critical angle, X can escape.So, the critical angle is θ_c = arcsin(v / V) = arcsin(1/4) ≈ 0.2527 radians ≈ 14.47 degrees.Therefore, if X swims towards a point more than θ_c radians away from Y's starting position, X can reach the edge before Y can run to that point.In our case, since the pond is circular, X can choose any point on the edge. So, X can choose a point that's more than θ_c radians away from Y's starting position, ensuring that Y can't intercept.Once X reaches the edge, X can start running tangentially at speed k*v > 4v, and Y can't catch up because X is faster.Therefore, X can escape.Wait, but earlier I thought that Y can always intercept because Y can run 4r in the time X swims r. But that was under the assumption that X swims straight towards a point, and Y runs along the circumference. But if X swims towards a point beyond the critical angle, Y can't intercept.So, the key is that X doesn't have to swim directly opposite but can choose a point that's beyond the critical angle, allowing X to reach the edge before Y can run to that point.Therefore, the answer is yes, X can escape.

answer：First, I need to determine the cost price of the toy car. Let's denote the cost price as ( C ) yuan.On the first day, the car was priced with a 100% profit, which means the initial selling price was ( 2C ) yuan.On the second day, the price was reduced by 10%, so the new price became ( 0.9 times 2C = 1.8C ) yuan.On the third day, the price was further reduced by 360 yuan, making the final selling price ( 1.8C - 360 ) yuan. According to the problem, this final price is 1.44 times the cost price, so we have the equation:[1.8C - 360 = 1.44C]Solving for ( C ):[1.8C - 1.44C = 360 0.36C = 360 C = frac{360}{0.36} C = 1000]Therefore, the cost price of the toy car is 1000 yuan.

answer：First, I'll identify the number of people who attended class each day. On Monday, there were 10 attendees, and on Tuesday, there were 15. From Wednesday through Friday, there were 10 attendees each day.Next, I'll calculate the total number of attendees over the five days. Adding Monday and Tuesday gives 10 + 15 = 25. For Wednesday, Thursday, and Friday, there are 3 days with 10 attendees each, totaling 30. Adding these together, the total attendance is 25 + 30 = 55.Finally, to find the average daily attendance, I'll divide the total number of attendees by the number of days. So, 55 divided by 5 equals 11. Therefore, the average number of people who attended class each day is 11.

answer：Okay, so I have this problem here about converting a polar equation to Cartesian and finding the parametric equation of a line, then figuring out the sum of distances from a point to the intersection points. Hmm, let me try to break it down step by step.First, part (1): Find the Cartesian equation of curve C and the parametric equation of line l.Alright, starting with curve C. The polar equation is given as ρ = 4 cos(θ - π/3). I remember that converting polar equations to Cartesian involves using the relationships x = ρ cos θ and y = ρ sin θ. Also, ρ² = x² + y².So, let me recall the general formula for converting polar equations. If I have ρ = a cos(θ - α), this is a circle in polar coordinates. The general form is a circle with radius a/2, centered at (a/2 cos α, a/2 sin α). Is that right? Let me verify.Yes, because in polar coordinates, ρ = 2a cos θ is a circle with radius a centered at (a, 0). So, if it's cos(θ - α), it's just shifted by α. So, the center would be at (a cos α, a sin α), right? Wait, no, actually, if it's ρ = 2a cos(θ - α), then the center is at (a cos α, a sin α). So, in this case, our equation is ρ = 4 cos(θ - π/3). So, comparing to ρ = 2a cos(θ - α), we have 2a = 4, so a = 2, and α = π/3.Therefore, the center should be at (a cos α, a sin α) = (2 cos π/3, 2 sin π/3). Calculating that, cos π/3 is 0.5, and sin π/3 is (√3)/2. So, 2 * 0.5 = 1, and 2 * (√3)/2 = √3. So, the center is at (1, √3). And the radius is a, which is 2. So, the Cartesian equation should be a circle with center (1, √3) and radius 2.Let me write that out: (x - 1)² + (y - √3)² = 4. That seems right.Alternatively, I can derive it step by step to make sure. Starting from ρ = 4 cos(θ - π/3). Let me expand that using the cosine difference identity: cos(θ - π/3) = cos θ cos π/3 + sin θ sin π/3.So, ρ = 4 [cos θ * 0.5 + sin θ * (√3)/2] = 4*(0.5 cos θ + (√3)/2 sin θ) = 2 cos θ + 2√3 sin θ.Now, multiply both sides by ρ: ρ² = 2 ρ cos θ + 2√3 ρ sin θ.But ρ² is x² + y², ρ cos θ is x, and ρ sin θ is y. So, substituting, we get:x² + y² = 2x + 2√3 y.Bring all terms to one side: x² - 2x + y² - 2√3 y = 0.Now, complete the square for x and y.For x: x² - 2x = (x - 1)² - 1.For y: y² - 2√3 y = (y - √3)² - (√3)² = (y - √3)² - 3.So, putting it all together:(x - 1)² - 1 + (y - √3)² - 3 = 0Simplify: (x - 1)² + (y - √3)² - 4 = 0 => (x - 1)² + (y - √3)² = 4.Yep, that's the same as before. So, the Cartesian equation is correct.Okay, moving on to the parametric equation of line l. The line passes through point P(0, -√3) and has an inclination angle of π/3. Hmm, inclination angle is the angle the line makes with the positive x-axis, measured counterclockwise. So, the slope of the line is tan(π/3) = √3.So, the slope m = √3, and it passes through (0, -√3). So, the equation in point-slope form is y - (-√3) = √3 (x - 0), which simplifies to y + √3 = √3 x, so y = √3 x - √3.But the question asks for the parametric equation. So, parametric equations can be written as:x = x0 + aty = y0 + btWhere (x0, y0) is a point on the line, and (a, b) is the direction vector. Since the slope is √3, the direction vector can be (1, √3), because rise over run is √3/1.So, starting from point P(0, -√3), the parametric equations would be:x = 0 + 1*t = ty = -√3 + √3*tSo, parametric equations:x = ty = -√3 + √3 tAlternatively, we can write it as:x = ty = √3 t - √3I think that's correct. Let me check: when t = 0, x = 0, y = -√3, which is point P. When t = 1, x = 1, y = √3(1) - √3 = 0. So, that point is (1, 0). The slope between (0, -√3) and (1, 0) is (0 - (-√3))/(1 - 0) = √3, which matches. So, that seems right.So, part (1) is done. The Cartesian equation of curve C is (x - 1)² + (y - √3)² = 4, and the parametric equation of line l is x = t, y = √3 t - √3.Now, part (2): Suppose line l intersects curve C at two points A and B, find the value of |PA| + |PB|.Hmm, so we need to find points A and B where line l intersects curve C, then compute the distances from P to A and P to B, and add them together.Alternatively, maybe there's a smarter way without finding A and B explicitly. Let me think.Wait, point P is (0, -√3), which is on line l. So, line l passes through P, and intersects curve C at A and B. So, P is one of the points on line l, but not necessarily on curve C. So, we have points A and B on both l and C, and P is another point on l.So, |PA| + |PB| is the sum of distances from P to A and P to B.Alternatively, since A and B are points on the line l, which is parametrized as x = t, y = √3 t - √3, we can substitute this parametric equation into the Cartesian equation of curve C and solve for t. Then, the values of t will correspond to points A and B. Then, compute the distances from P to A and P to B, which are just the differences in t multiplied by the direction vector's magnitude, but maybe it's easier to compute using coordinates.Let me try substituting the parametric equations into the circle equation.So, curve C: (x - 1)² + (y - √3)² = 4.Substitute x = t, y = √3 t - √3.So, (t - 1)² + (√3 t - √3 - √3)² = 4.Simplify each term:First term: (t - 1)² = t² - 2t + 1.Second term: (√3 t - 2√3)² = (√3(t - 2))² = 3(t - 2)² = 3(t² - 4t + 4).So, putting it together:(t² - 2t + 1) + 3(t² - 4t + 4) = 4.Expand:t² - 2t + 1 + 3t² - 12t + 12 = 4.Combine like terms:(1t² + 3t²) + (-2t -12t) + (1 + 12) = 44t² -14t +13 = 4.Subtract 4 from both sides:4t² -14t +9 = 0.So, quadratic equation: 4t² -14t +9 = 0.Let me solve for t.Using quadratic formula: t = [14 ± √(14² - 4*4*9)] / (2*4)Compute discriminant: 196 - 144 = 52.So, t = [14 ± √52]/8.Simplify √52: √(4*13) = 2√13.So, t = [14 ± 2√13]/8 = [7 ± √13]/4.So, t1 = (7 + √13)/4, t2 = (7 - √13)/4.So, the points A and B correspond to these t values.Therefore, coordinates of A and B:For t1: x = (7 + √13)/4, y = √3*(7 + √13)/4 - √3.Similarly, for t2: x = (7 - √13)/4, y = √3*(7 - √13)/4 - √3.Simplify y-coordinate:For t1: y = (7√3 + √3*√13)/4 - √3 = (7√3 + √39)/4 - √3.Express √3 as 4√3/4: so, y = (7√3 + √39 - 4√3)/4 = (3√3 + √39)/4.Similarly, for t2: y = (7√3 - √39)/4 - √3 = (7√3 - √39 - 4√3)/4 = (3√3 - √39)/4.So, points A and B are:A: ((7 + √13)/4, (3√3 + √39)/4)B: ((7 - √13)/4, (3√3 - √39)/4)Now, we need to compute |PA| and |PB|, then add them.Point P is (0, -√3). Let's compute the distance from P to A.Using distance formula: |PA| = sqrt[(x_A - 0)^2 + (y_A - (-√3))^2] = sqrt[x_A² + (y_A + √3)^2].Similarly for |PB|.Let me compute |PA| first.Compute x_A = (7 + √13)/4, so x_A² = [(7 + √13)/4]^2 = (49 + 14√13 + 13)/16 = (62 + 14√13)/16.Compute y_A + √3 = (3√3 + √39)/4 + √3 = (3√3 + √39 + 4√3)/4 = (7√3 + √39)/4.So, (y_A + √3)^2 = [(7√3 + √39)/4]^2.Let me compute that:(7√3 + √39)^2 = (7√3)^2 + 2*7√3*√39 + (√39)^2 = 49*3 + 14√(117) + 39 = 147 + 14√(9*13) + 39 = 147 + 14*3√13 + 39 = 147 + 42√13 + 39 = 186 + 42√13.So, (y_A + √3)^2 = (186 + 42√13)/16.Therefore, |PA| = sqrt[(62 + 14√13)/16 + (186 + 42√13)/16] = sqrt[(62 + 14√13 + 186 + 42√13)/16] = sqrt[(248 + 56√13)/16].Simplify numerator: 248 + 56√13 = 8*(31) + 8*(7√13) = 8*(31 + 7√13). Wait, 248 divided by 8 is 31, and 56 divided by 8 is 7. So, 248 +56√13 =8*(31 +7√13).So, sqrt[(8*(31 +7√13))/16] = sqrt[(31 +7√13)/2].Wait, 8/16 is 1/2, so sqrt[(31 +7√13)/2].Hmm, that's |PA|.Similarly, let's compute |PB|.Point B: x_B = (7 - √13)/4, y_B = (3√3 - √39)/4.Compute |PB| = sqrt[x_B² + (y_B + √3)^2].x_B² = [(7 - √13)/4]^2 = (49 -14√13 +13)/16 = (62 -14√13)/16.Compute y_B + √3 = (3√3 - √39)/4 + √3 = (3√3 - √39 +4√3)/4 = (7√3 - √39)/4.So, (y_B + √3)^2 = [(7√3 - √39)/4]^2.Compute numerator: (7√3 - √39)^2 = (7√3)^2 - 2*7√3*√39 + (√39)^2 = 147 - 14√(117) + 39 = same as before, 147 +39 = 186, and -14√(117) = -14*3√13 = -42√13. So, total is 186 -42√13.Thus, (y_B + √3)^2 = (186 -42√13)/16.Therefore, |PB| = sqrt[(62 -14√13)/16 + (186 -42√13)/16] = sqrt[(62 +186 -14√13 -42√13)/16] = sqrt[(248 -56√13)/16].Again, 248 -56√13 =8*(31 -7√13), so sqrt[(8*(31 -7√13))/16] = sqrt[(31 -7√13)/2].So, |PA| = sqrt[(31 +7√13)/2], |PB| = sqrt[(31 -7√13)/2].Therefore, |PA| + |PB| = sqrt[(31 +7√13)/2] + sqrt[(31 -7√13)/2].Hmm, that looks a bit complicated. Maybe we can simplify this expression.Let me denote A = sqrt[(31 +7√13)/2] and B = sqrt[(31 -7√13)/2]. So, we need to compute A + B.Let me compute (A + B)^2 = A² + 2AB + B².Compute A² = (31 +7√13)/2, B² = (31 -7√13)/2.So, A² + B² = (31 +7√13 +31 -7√13)/2 = 62/2 =31.Compute AB: sqrt[(31 +7√13)/2] * sqrt[(31 -7√13)/2] = sqrt[ (31 +7√13)(31 -7√13) /4 ].Compute numerator: (31)^2 - (7√13)^2 = 961 - 49*13 = 961 - 637 = 324.So, AB = sqrt[324/4] = sqrt[81] =9.Therefore, (A + B)^2 = 31 + 2*9 =31 +18=49.Thus, A + B = sqrt(49)=7.So, |PA| + |PB| =7.Wow, that's neat. So, the sum of the distances is 7.Alternatively, maybe there's a geometric interpretation that could have led us here without computing all the coordinates. Since P is on line l, and l intersects the circle C at A and B, then |PA| + |PB| is the sum of the lengths from P to A and P to B. In circle geometry, if P is outside the circle, then |PA| + |PB| can be related to the power of the point or something else, but I'm not sure. Alternatively, since we have a quadratic equation in t, and the sum of the roots is related to the coefficients, but in this case, we had t1 and t2, and the distances are related to |t1| and |t2| scaled by the direction vector's magnitude.Wait, actually, in parametric terms, the parameter t corresponds to a point moving along the line l. The distance from P to a general point on l is |t| times the length of the direction vector. Since the direction vector is (1, √3), its length is sqrt(1 + 3)=2. So, the distance from P to a point on l is 2|t|. So, |PA| = 2|t1|, |PB|=2|t2|.But wait, in our earlier substitution, t1 and t2 were the parameter values where the line intersects the circle. So, t1 and t2 are the parameter values for points A and B. So, the distances from P to A and P to B would be |t1 - 0| * 2 and |t2 - 0| * 2, since P corresponds to t=0.But wait, in our parametrization, when t=0, we are at P(0, -√3). So, the distance from P to A is |t1| * 2, and from P to B is |t2| * 2.But in our case, t1 = (7 + √13)/4 and t2 = (7 - √13)/4. Both are positive because √13 ≈3.605, so 7 - √13 ≈3.395, which is positive. So, both t1 and t2 are positive, so |PA| = 2t1, |PB|=2t2.Therefore, |PA| + |PB| = 2(t1 + t2).From quadratic equation, t1 + t2 = [7 + √13 +7 - √13]/4 =14/4=7/2.Therefore, |PA| + |PB| =2*(7/2)=7.Ah, that's a much simpler way! So, instead of computing the distances separately, recognizing that the parameter t corresponds to a scaled distance from P, and since the sum of the roots t1 + t2 is 7/2, multiplying by 2 gives the total distance.So, that confirms the earlier result. Therefore, the value of |PA| + |PB| is 7.**Final Answer**The value of (|PA| + |PB|) is boxed{7}).

answer：Okay, so I have this problem here: Given the expression (left( dfrac{a}{x} - sqrt{ dfrac{x}{2}} right)^9), the coefficient of (x^3) in its expansion is (dfrac{9}{4}). I need to find the value of (a). Hmm, let me think about how to approach this.First, I remember that when dealing with binomial expansions, the general term can be found using the binomial theorem. The binomial theorem states that ((x + y)^n = sum_{k=0}^{n} dbinom{n}{k} x^{n - k} y^{k}). So, in this case, my expression is (left( dfrac{a}{x} - sqrt{ dfrac{x}{2}} right)^9), which is similar to ((x + y)^n) where (x = dfrac{a}{x}) and (y = -sqrt{ dfrac{x}{2}}), and (n = 9).So, the general term in the expansion will be (dbinom{9}{k} left( dfrac{a}{x} right)^{9 - k} left( -sqrt{ dfrac{x}{2}} right)^k). I need to find the term where the power of (x) is 3 because the coefficient of (x^3) is given as (dfrac{9}{4}).Let me write out the general term more clearly:[T_k = dbinom{9}{k} left( dfrac{a}{x} right)^{9 - k} left( -sqrt{ dfrac{x}{2}} right)^k]Simplify each part step by step.First, let's simplify (left( dfrac{a}{x} right)^{9 - k}). That would be (dfrac{a^{9 - k}}{x^{9 - k}}).Next, let's simplify (left( -sqrt{ dfrac{x}{2}} right)^k). The negative sign can be written as ((-1)^k), and the square root can be written as (left( dfrac{x}{2} right)^{k/2}). So, that becomes ((-1)^k cdot left( dfrac{x}{2} right)^{k/2}).Putting it all together, the term (T_k) becomes:[T_k = dbinom{9}{k} cdot dfrac{a^{9 - k}}{x^{9 - k}} cdot (-1)^k cdot left( dfrac{x}{2} right)^{k/2}]Let me simplify the exponents and constants here.First, let's handle the constants:- The (dbinom{9}{k}) is just a binomial coefficient.- The (a^{9 - k}) is straightforward.- The ((-1)^k) is also straightforward.- The (dfrac{1}{x^{9 - k}}) can be written as (x^{-(9 - k)}).- The (left( dfrac{x}{2} right)^{k/2}) can be split into (dfrac{x^{k/2}}{2^{k/2}}).So, combining all these, we have:[T_k = dbinom{9}{k} cdot a^{9 - k} cdot (-1)^k cdot dfrac{x^{-(9 - k)} cdot x^{k/2}}{2^{k/2}}]Now, let's combine the exponents of (x):The exponent of (x) in the term is (-(9 - k) + dfrac{k}{2}). Let me compute that:[-(9 - k) + dfrac{k}{2} = -9 + k + dfrac{k}{2} = -9 + dfrac{3k}{2}]So, the exponent of (x) in term (T_k) is (-9 + dfrac{3k}{2}). We are interested in the term where the exponent is 3 because we need the coefficient of (x^3). Therefore, we set up the equation:[-9 + dfrac{3k}{2} = 3]Let me solve for (k):[-9 + dfrac{3k}{2} = 3 Rightarrow dfrac{3k}{2} = 12 Rightarrow 3k = 24 Rightarrow k = 8]So, the term corresponding to (k = 8) will have the exponent of (x) equal to 3. Therefore, we can find the coefficient by plugging (k = 8) into the expression for (T_k).Let me compute each part of (T_k) when (k = 8):First, the binomial coefficient (dbinom{9}{8}). I know that (dbinom{n}{k} = dbinom{n}{n - k}), so (dbinom{9}{8} = dbinom{9}{1} = 9).Next, (a^{9 - 8} = a^{1} = a).Then, ((-1)^8 = 1) because any even power of -1 is 1.Now, the denominator term: (2^{8/2} = 2^4 = 16).So, putting it all together, the coefficient is:[dbinom{9}{8} cdot a cdot (-1)^8 cdot dfrac{1}{16} = 9 cdot a cdot 1 cdot dfrac{1}{16} = dfrac{9a}{16}]But wait, the problem states that the coefficient of (x^3) is (dfrac{9}{4}). So, we set up the equation:[dfrac{9a}{16} = dfrac{9}{4}]Now, let's solve for (a):Multiply both sides by 16:[9a = dfrac{9}{4} times 16]Simplify the right-hand side:[dfrac{9}{4} times 16 = 9 times 4 = 36]So, we have:[9a = 36 Rightarrow a = dfrac{36}{9} Rightarrow a = 4]Wait, hold on. Let me double-check my calculations because I might have made a mistake somewhere.So, starting again from the term (T_8):[T_8 = dbinom{9}{8} cdot left( dfrac{a}{x} right)^{1} cdot left( -sqrt{ dfrac{x}{2}} right)^8]Compute each part:- (dbinom{9}{8} = 9)- (left( dfrac{a}{x} right)^1 = dfrac{a}{x})- (left( -sqrt{ dfrac{x}{2}} right)^8 = (-1)^8 cdot left( sqrt{ dfrac{x}{2}} right)^8 = 1 cdot left( dfrac{x}{2} right)^{4} = dfrac{x^4}{16})So, multiplying these together:[9 cdot dfrac{a}{x} cdot dfrac{x^4}{16} = 9 cdot a cdot dfrac{x^{3}}{16}]So, the term is (dfrac{9a}{16}x^3). Therefore, the coefficient is (dfrac{9a}{16}), which is equal to (dfrac{9}{4}).So, setting (dfrac{9a}{16} = dfrac{9}{4}), we can divide both sides by 9:[dfrac{a}{16} = dfrac{1}{4}]Then, multiplying both sides by 16:[a = 4]So, that seems consistent. Therefore, (a = 4).But just to be thorough, let me check another term to see if I might have made a mistake in the exponent calculation.Wait, when I set the exponent equal to 3, I had:[-9 + dfrac{3k}{2} = 3 Rightarrow dfrac{3k}{2} = 12 Rightarrow k = 8]That seems correct. So, (k = 8) is the correct term. Therefore, the coefficient is indeed (dfrac{9a}{16}), leading to (a = 4).Alternatively, maybe I should consider if the exponent calculation is correct.Let me re-examine the exponent:From (left( dfrac{a}{x} right)^{9 - k}), the exponent of (x) is (-(9 - k)).From (left( -sqrt{ dfrac{x}{2}} right)^k), the exponent of (x) is (dfrac{k}{2}).So, total exponent is (-(9 - k) + dfrac{k}{2} = -9 + k + dfrac{k}{2} = -9 + dfrac{3k}{2}). So, setting that equal to 3:[-9 + dfrac{3k}{2} = 3 Rightarrow dfrac{3k}{2} = 12 Rightarrow 3k = 24 Rightarrow k = 8]Yes, that's correct. So, no mistake there.Alternatively, maybe I made a mistake in simplifying the constants.Wait, let's see:[left( -sqrt{ dfrac{x}{2}} right)^k = (-1)^k cdot left( dfrac{x}{2} right)^{k/2}]So, the constants would be ((-1)^k) and (dfrac{1}{2^{k/2}}). So, in the term, the constants are:[dbinom{9}{k} cdot a^{9 - k} cdot (-1)^k cdot dfrac{1}{2^{k/2}}]So, for (k = 8):[dbinom{9}{8} = 9 a^{9 - 8} = a (-1)^8 = 1 2^{8/2} = 2^4 = 16]So, the constants multiply to:[9 cdot a cdot 1 cdot dfrac{1}{16} = dfrac{9a}{16}]Which is correct. So, setting that equal to (dfrac{9}{4}), we get:[dfrac{9a}{16} = dfrac{9}{4} Rightarrow a = dfrac{9}{4} cdot dfrac{16}{9} = 4]So, that seems consistent again.Wait, another thought: Is the expansion correct? Because sometimes, when dealing with binomial expansions, especially with negative signs, it's easy to make a mistake.Wait, the original expression is (left( dfrac{a}{x} - sqrt{ dfrac{x}{2}} right)^9). So, the second term is negative, so when we take it to the power (k), it becomes ((-1)^k). So, in the term (T_k), we have a factor of ((-1)^k). So, when (k = 8), which is even, it becomes positive, as we had.So, that seems correct.Alternatively, maybe I should compute the term for (k = 8) step by step.So, (T_8 = dbinom{9}{8} cdot left( dfrac{a}{x} right)^{1} cdot left( -sqrt{ dfrac{x}{2}} right)^8).Compute each part:- (dbinom{9}{8} = 9)- (left( dfrac{a}{x} right)^1 = dfrac{a}{x})- (left( -sqrt{ dfrac{x}{2}} right)^8 = (-1)^8 cdot left( sqrt{ dfrac{x}{2}} right)^8 = 1 cdot left( dfrac{x}{2} right)^4 = dfrac{x^4}{16})So, multiplying all together:[9 cdot dfrac{a}{x} cdot dfrac{x^4}{16} = 9 cdot a cdot dfrac{x^{3}}{16}]So, the term is (dfrac{9a}{16}x^3), so the coefficient is (dfrac{9a}{16}). Since this is given as (dfrac{9}{4}), we set:[dfrac{9a}{16} = dfrac{9}{4}]Divide both sides by 9:[dfrac{a}{16} = dfrac{1}{4}]Multiply both sides by 16:[a = 4]So, that's consistent again.Wait, perhaps I should check if there's another term that could contribute to (x^3). Is (k = 8) the only term where the exponent is 3? Let me see.We had the exponent equation:[-9 + dfrac{3k}{2} = 3 Rightarrow dfrac{3k}{2} = 12 Rightarrow k = 8]So, (k) must be 8, which is an integer between 0 and 9, so it's valid. Therefore, only (k = 8) gives the exponent 3. So, no other term contributes to (x^3).Therefore, the only term contributing to (x^3) is when (k = 8), and that gives the coefficient (dfrac{9a}{16}), which we set equal to (dfrac{9}{4}) and solve for (a = 4).So, I think that's correct. Therefore, the value of (a) is 4.**Final Answer**The value of (a) is boxed{4}.

answer：Okay, so I have this problem where Oleg wrote several composite natural numbers less than 1500 on the board. The key point is that the greatest common divisor (GCD) of any two of them is 1. I need to figure out the maximum number of numbers Oleg could have written. Hmm, interesting.First, let me understand the problem better. We're dealing with composite numbers, which means each number must have more than two factors. So, numbers like 4, 6, 8, 9, etc., are composite. Prime numbers are out of the question because they only have two factors, 1 and themselves. Also, all these composite numbers must be less than 1500.The crucial condition here is that the GCD of any two numbers is 1. That means any two numbers Oleg wrote are coprime. So, they share no common factors other than 1. For example, 4 and 9 are coprime because their GCD is 1, but 4 and 6 are not because their GCD is 2.So, the task is to find as many composite numbers as possible under 1500 such that any pair is coprime. I need to maximize the count of such numbers.Let me think about how to approach this. Since all numbers must be coprime, they can't share any prime factors. So, each number must be constructed from distinct prime factors that aren't used by any other number. But since we're dealing with composite numbers, each must have at least two prime factors (they could be squares of primes, like 4=2², 9=3², etc., or products of different primes, like 6=2×3, 10=2×5, etc.).Wait, but if I use squares of primes, like 4, 9, 25, etc., each of these only uses one prime factor. So, if I use 4, which is 2², then I can't use any other number that has 2 as a factor. Similarly, if I use 9=3², I can't use any number with 3 as a factor. So, using squares of primes might be a good strategy because each only blocks one prime, allowing more numbers to be included.On the other hand, if I use composite numbers that are products of two different primes, like 6=2×3, 10=2×5, 14=2×7, etc., each of these blocks two primes. So, using such numbers would block more primes, potentially limiting the total number of numbers I can include.Therefore, maybe using squares of primes is a better strategy because each only blocks one prime, allowing more numbers. Let me explore this.First, let's list the squares of primes less than 1500. The primes themselves can be up to sqrt(1500). What's sqrt(1500)? Approximately 38.72. So, primes less than or equal to 38.Let me list all primes less than or equal to 38:2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37.So, that's 12 primes. Therefore, their squares would be:4, 9, 25, 49, 121, 169, 289, 361, 529, 841, 961, 1369.Wait, let me calculate each:2²=43²=95²=257²=4911²=12113²=16917²=28919²=36123²=52929²=84131²=96137²=1369Yes, all of these are less than 1500. So, that's 12 composite numbers, each being the square of a prime. Since each square only uses one prime, and all these primes are distinct, any two squares will have GCD 1. So, this gives me 12 numbers.But wait, maybe I can include more numbers by using other composite numbers that are products of two different primes, but ensuring that their prime factors don't overlap with the primes used in the squares.For example, if I have already used 4=2², I can't use any other number that has 2 as a factor. Similarly, if I have 9=3², I can't use any number with 3 as a factor. So, the primes 2, 3, 5, 7, etc., are already blocked by the squares.But hold on, if I don't use some squares, maybe I can use other composite numbers that use those primes but in combination with other primes not used elsewhere. Hmm, this might allow me to include more numbers.Let me think. Suppose instead of using 4=2², I use a composite number that is 2 multiplied by another prime not used in any other number. For example, 2×5=10. Then, 10 would block primes 2 and 5. But if I don't use 4, I can maybe use 10 and then 25=5²? Wait, no, because 10 already uses 5, so 25 can't be used.Alternatively, if I use 10, I can't use any other number with 2 or 5. So, perhaps it's better to use 4 and 25, which only block 2 and 5, respectively, but allow me to include two numbers instead of just one.Wait, let's see: If I use 4 and 25, that's two numbers, each blocking one prime. If I instead use 10, that's one number blocking two primes. So, using squares gives me more numbers for the same number of blocked primes.Therefore, it's more efficient to use squares of primes because each square blocks only one prime, allowing more numbers.So, if I stick with squares, I can get 12 numbers. But maybe I can include some additional composite numbers that don't interfere with these squares.Wait, let's think about higher powers. For example, 8=2³ is composite, but it's not a square. However, 8 shares the same prime factor as 4, so if I have 4 on the board, I can't include 8 because GCD(4,8)=4, which is not 1. Similarly, 16=2⁴, 27=3³, etc., would conflict with the squares.So, higher powers of primes are out because they share the same prime as their square counterparts.What about composite numbers that are products of two distinct primes, neither of which is used in the squares? For example, if I have squares of primes up to 37²=1369, the next prime is 41, but 41²=1681, which is over 1500. So, 41 is a prime whose square is over 1500, so we can't use 41².But, 41 is a prime that hasn't been used yet. So, maybe we can use composite numbers that include 41 multiplied by other primes not used in the squares.Wait, but all the primes up to 37 have been used in the squares. So, 41 is a prime that hasn't been used. So, if I use 41×43=1763, which is over 1500, so that's too big. 41×2=82, which is under 1500. But 2 is already used in the square 4. So, 82 would share a prime factor with 4, which is 2, so GCD(4,82)=2, which is not 1. So, can't use 82.Similarly, 41×3=123, but 3 is used in 9, so GCD(9,123)=3, which is not 1. So, can't use 123. 41×5=205, but 5 is used in 25, so GCD(25,205)=5, which is not 1. Same issue.Wait, so any composite number that includes 41 would have to be multiplied by a prime that's already been used in the squares, which would make it share a common factor with that square. Therefore, we can't include any composite number that includes 41 without violating the GCD condition.Similarly, primes larger than 37, like 43, 47, etc., when multiplied by any smaller prime (which are already used in the squares), would result in composite numbers that share a factor with the squares. So, we can't include those.Alternatively, could we use a composite number that is the product of two primes larger than 37? Let's see. The smallest such product would be 41×43=1763, which is over 1500. So, that's too big. Therefore, we can't include any composite numbers that are products of two primes larger than 37 because their product would exceed 1500.Therefore, it seems that we can't include any composite numbers beyond the squares of primes less than or equal to 37 because any other composite number would either share a prime factor with one of the squares or would be too large.But wait, hold on. Maybe there's another way. Instead of using all the squares, perhaps we can replace some squares with other composite numbers that might allow us to include more numbers overall.For example, suppose we don't use 4=2², but instead use 6=2×3. Then, 6 blocks both 2 and 3. But by doing so, we can't use 9=3² or 25=5², but maybe we can use other composite numbers that don't involve 2 or 3.Wait, but 6 is a composite number, and if we use 6, we can't use any other number that has 2 or 3 as a factor. So, instead of having two numbers (4 and 9), we have one number (6). So, that's worse in terms of count.Alternatively, maybe using 6 and then not using 4 and 9. But then, we can use other composite numbers that don't involve 2 or 3. For example, 25=5², 49=7², etc. So, let's see:If we use 6 instead of 4 and 9, we block 2 and 3. Then, we can still use 25, 49, 121, 169, 289, 361, 529, 841, 961, 1369. That's 10 numbers. Plus 6, so total 11. But previously, using 4,9,25,...,1369 gave us 12 numbers. So, this approach gives fewer numbers.Therefore, it's better to use squares rather than products of two small primes.Alternatively, what if we use some squares and some products? For example, use some squares and replace others with products that might allow more numbers.Wait, let me think. Suppose we use 4=2², which blocks 2. Then, instead of using 9=3², which blocks 3, maybe we can use 15=3×5, which blocks 3 and 5. Then, we can't use 25=5² because 5 is already blocked by 15. But, by doing this, we have 4 and 15, which blocks 2,3,5. Previously, 4 and 9 blocked 2 and 3, allowing 25 to be used. So, in this case, we lose the ability to use 25 but gain nothing because 15 is one number instead of two (9 and 25). So, again, this is worse.Alternatively, if we use 4=2², 9=3², 25=5², 49=7², etc., we can get 12 numbers. If we try to replace some squares with products, we end up with fewer numbers because each product blocks two primes, whereas each square blocks only one.Therefore, it seems that using squares is the optimal strategy because it allows us to block one prime per number, maximizing the count.But wait, let me check if there are composite numbers that are coprime to all the squares. For example, numbers that are products of two primes larger than 37. But as I thought earlier, the smallest such product is 41×43=1763, which is over 1500. So, we can't include any composite numbers made from two primes larger than 37 because their product would exceed 1500.Alternatively, are there composite numbers less than 1500 that are made from a single prime factor larger than 37? Well, those would be squares of primes larger than 37, but the square of 41 is 1681, which is over 1500. So, no, we can't include those either.Therefore, it seems that the maximum number of composite numbers we can have, each pair being coprime, is equal to the number of primes less than or equal to 38, which is 12. So, 12 composite numbers, each being the square of a prime.Wait, but hold on. There's another thought. Maybe we can include some composite numbers that are higher powers of primes, but not squares. For example, 8=2³, 16=2⁴, etc. But as I thought earlier, these share the same prime as their square counterparts, so they can't be included if we already have the square.Alternatively, if we don't include the square, can we include a higher power? For example, instead of 4=2², include 8=2³. But 8 is still a power of 2, so it's not coprime with 4. So, if we include 8, we can't include 4. But 8 is just another power of 2, so it doesn't help us include more numbers because it still blocks the prime 2.Similarly, 9=3², 27=3³, etc., same issue. So, higher powers don't help because they block the same prime as their square.Alternatively, maybe using semiprimes (products of two distinct primes) that don't interfere with the squares. But as I thought earlier, since all small primes are already blocked by the squares, any semiprime would have to use two primes, both of which are already blocked, so they can't be included.Wait, unless we don't use some squares to make room for semiprimes. Let me explore this.Suppose we decide not to use the square of 2, which is 4. Then, we can use semiprimes that include 2 and another prime. For example, 6=2×3, 10=2×5, 14=2×7, etc. But each of these semiprimes would block two primes. So, if we use 6, we block 2 and 3. Then, we can't use 9=3² or 25=5² because 3 and 5 are blocked.Alternatively, if we use 10=2×5, we block 2 and 5, so we can't use 4=2² or 25=5².So, each semiprime blocks two primes, which might allow us to include more numbers if we can cleverly choose semiprimes that block different sets of primes.Wait, let's think about this. If we don't use any squares, and instead use semiprimes, each semiprime blocks two primes. How many semiprimes can we include without overlapping primes?The number of primes less than 1500 is quite large, but since we're dealing with composite numbers less than 1500, the primes involved in semiprimes must be such that their product is less than 1500.But if we try to maximize the number of semiprimes, each using two unique primes, the maximum number would be limited by the number of primes available divided by 2.But wait, the number of primes less than 1500 is 249, but that's way too high. However, in our case, we need semiprimes less than 1500, so the primes involved must be such that their product is less than 1500.But if we try to pair up primes into semiprimes, each semiprime uses two primes, so the maximum number of semiprimes would be floor(249/2)=124. But that's not practical because many semiprimes would be over 1500.Wait, no, actually, the primes involved in the semiprimes must be such that their product is less than 1500. So, the primes can't be too large. For example, the largest prime p such that p×2 < 1500 is p < 750. So, primes up to 743 can be paired with 2 to make semiprimes less than 1500.But this is getting complicated. Maybe it's better to think in terms of the primes we can use.Wait, but if we don't use any squares, and instead use semiprimes, each semiprime blocks two primes. So, the number of semiprimes we can include is limited by the number of primes we have divided by 2.But in our case, if we use semiprimes, we can include more numbers because each semiprime blocks two primes, but allows us to include one number. Whereas squares block one prime and allow one number.So, if we have N primes, using squares gives us N/1 numbers, while using semiprimes gives us N/2 numbers. So, squares are better because they give more numbers.Wait, but N is the number of primes less than sqrt(1500), which is 12. So, using squares gives us 12 numbers. If we use semiprimes, we can pair these 12 primes into 6 semiprimes, which is fewer. So, squares are better.But wait, maybe we can use both squares and semiprimes, but that might not be possible because semiprimes would block primes used in squares.Alternatively, perhaps we can use squares for some primes and semiprimes for others, but I don't see how that would increase the total count beyond 12.Wait, another idea: Maybe use squares for some primes and semiprimes for others, but ensuring that the semiprimes don't share primes with the squares.For example, use squares for primes 2,3,5,7,11,13,17,19,23,29,31,37, which are 12 primes, giving 12 squares. Then, use semiprimes for primes larger than 37, but as I thought earlier, the smallest semiprime using two primes larger than 37 is 41×43=1763, which is over 1500. So, we can't include any semiprimes with primes larger than 37 because their product would exceed 1500.Therefore, it's not possible to include any semiprimes beyond the squares because they would either conflict with the squares or be too large.Wait, but what about semiprimes that include one small prime and one large prime? For example, 2×41=82, which is under 1500. But 2 is already used in the square 4, so 82 would share a factor with 4, which is 2, so GCD(4,82)=2≠1. Therefore, can't include 82.Similarly, 3×41=123, but 3 is used in 9, so GCD(9,123)=3≠1. So, can't include 123.Same with 5×41=205, which shares 5 with 25, so GCD(25,205)=5≠1. So, can't include 205.Therefore, any semiprime that includes a small prime (used in the squares) and a large prime would conflict with the squares. So, we can't include those.Alternatively, semiprimes that include two large primes, but as we saw, their product would be over 1500. So, no luck there.Therefore, it seems that the maximum number of composite numbers we can have, each pair coprime, is 12, each being the square of a prime less than or equal to 37.But wait, let me check if there are composite numbers that are coprime to all the squares but aren't squares themselves. For example, numbers like 121=11², which is already included. Or, numbers like 169=13², also included.Wait, what about numbers like 1, but 1 is neither prime nor composite, so it's excluded. So, no help there.Alternatively, what about prime powers higher than squares? For example, 8=2³, 16=2⁴, etc. But as I thought earlier, these share the same prime as their square counterparts, so they can't be included if we already have the square.Alternatively, if we don't include the square, can we include a higher power? For example, instead of 4=2², include 8=2³. But 8 is still a power of 2, so it's not coprime with 4. So, if we include 8, we can't include 4. But 8 is just another power of 2, so it doesn't help us include more numbers because it still blocks the prime 2.Therefore, higher powers don't help us include more numbers beyond what squares provide.Wait, another thought: Maybe some composite numbers are products of three or more primes, but still coprime to all the squares. For example, 30=2×3×5. But 30 shares primes with 4, 9, and 25, so GCD(4,30)=2, GCD(9,30)=3, GCD(25,30)=5. So, can't include 30.Similarly, any composite number that is a product of multiple primes would share at least one prime with one of the squares, making their GCD greater than 1. So, such numbers can't be included.Therefore, it seems that the only composite numbers we can include without violating the GCD condition are the squares of primes. Each square blocks one prime, and since all primes are distinct, any two squares are coprime.So, the number of such squares is equal to the number of primes less than or equal to sqrt(1500). As we calculated earlier, sqrt(1500) is approximately 38.72, so primes less than or equal to 38. There are 12 such primes: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37.Therefore, the maximum number of composite numbers Oleg could have written is 12.But wait, let me double-check. Is there any composite number less than 1500 that is coprime to all the squares of primes? For example, numbers like 121=11², which is already included. Or, what about 1, but 1 is not composite. Or, numbers like 169=13², also included.Wait, what about 1? No, it's not composite. So, no help.Alternatively, could we include a composite number that is a product of two primes, both larger than 37? As we saw earlier, the smallest such product is 41×43=1763, which is over 1500. So, no.Alternatively, could we include a composite number that is a power of a prime larger than 37? For example, 41²=1681, which is over 1500. So, no.Therefore, I think 12 is indeed the maximum number.Wait, but hold on. Let me think about another angle. Maybe instead of using all the squares, we can use some squares and some other composite numbers that don't interfere. For example, using 4=2², 9=3², 25=5², etc., and then also using some composite numbers that are coprime to all of these.But as we saw earlier, any composite number would have to be made from primes not used in the squares, but all primes less than 38 are already used in the squares. The next prime is 41, but any composite number involving 41 would have to be multiplied by another prime, which would either be less than 38 (and thus already blocked) or greater than 38, resulting in a product over 1500.Therefore, it's impossible to include any composite number beyond the squares without violating the GCD condition.So, I think I've considered all possibilities, and the conclusion is that the maximum number of composite numbers Oleg could have written is 12.**Final Answer**The maximum number of numbers Oleg could write is boxed{12}.