💡Okay, so Kevin the kangaroo is trying to hop from 0 to 1 on a number line. He starts by hopping 1/4 of the distance to the goal, which is 1. So, his first hop is 1/4. That makes sense because 1/4 of 1 is 0.25, right?Now, after the first hop, he's at 1/4. The remaining distance to the goal is 1 - 1/4 = 3/4. But here's the catch: each subsequent hop can only cover 1/4 of the remaining distance, and his efficiency decreases by a factor of 3/4 each time. Hmm, so not only is the remaining distance getting smaller, but his ability to cover that fraction is also decreasing. That sounds a bit tricky, but let's break it down step by step.Let me try to outline each hop:1. **First hop:** He hops 1/4 of the total distance, which is 1/4. So, total distance covered so far is 1/4.2. **Second hop:** Now, the remaining distance is 3/4. He can hop 1/4 of that, which is (1/4)*(3/4) = 3/16. So, after the second hop, he's covered an additional 3/16. Total distance now is 1/4 + 3/16.Wait, let me calculate that. 1/4 is 4/16, so 4/16 + 3/16 = 7/16. So, after two hops, he's at 7/16.3. **Third hop:** The remaining distance is 1 - 7/16 = 9/16. He can hop 1/4 of that, which is (1/4)*(9/16) = 9/64. But wait, the problem says his efficiency decreases by a factor of 3/4 each time. So, does that mean each hop is 3/4 of the previous hop's distance?Hold on, maybe I misunderstood. Let me read the problem again: "each subsequent hop can only cover 1/4 of the remaining distance to the goal, and his efficiency to cover that fraction decreases by a factor of 3/4 with each hop."So, it's two things: each hop is 1/4 of the remaining distance, and his efficiency (which I think refers to the fraction he can cover) decreases by 3/4 each time. Hmm, maybe I need to model this differently.Let me think. The first hop is 1/4 of the total distance. Then, each subsequent hop is 1/4 of the remaining distance, but each time, the fraction he can cover is multiplied by 3/4. So, the first hop is 1/4, the second hop is (1/4)*(3/4), the third hop is (1/4)*(3/4)^2, and so on.Wait, that might make more sense. So, each hop is 1/4 of the remaining distance, but the efficiency (the fraction he can cover) is decreasing by 3/4 each time. So, the first hop is 1/4, the second hop is (1/4)*(3/4), the third hop is (1/4)*(3/4)^2, etc.But then, the remaining distance after each hop would be the previous remaining distance minus the hop distance. So, maybe I need to model both the hop distance and the remaining distance.Let me try to structure this:- **Initial position:** 0- **Goal:** 1- **Hop 1:** 1/4 of the distance to the goal, so 1/4. Remaining distance: 1 - 1/4 = 3/4.- **Hop 2:** 1/4 of the remaining distance, which is 1/4 * 3/4 = 3/16. But his efficiency decreases by 3/4, so the actual hop is 3/16 * 3/4 = 9/64. Remaining distance: 3/4 - 9/64 = 48/64 - 9/64 = 39/64.- **Hop 3:** 1/4 of the remaining distance, which is 1/4 * 39/64 = 39/256. Efficiency decreases by 3/4, so hop distance is 39/256 * 3/4 = 117/1024. Remaining distance: 39/64 - 117/1024 = (624/1024 - 117/1024) = 507/1024.- **Hop 4:** 1/4 of 507/1024 = 507/4096. Efficiency decreases by 3/4, so hop distance is 507/4096 * 3/4 = 1521/16384. Remaining distance: 507/1024 - 1521/16384 = (8112/16384 - 1521/16384) = 6591/16384.- **Hop 5:** 1/4 of 6591/16384 = 6591/65536. Efficiency decreases by 3/4, so hop distance is 6591/65536 * 3/4 = 19773/262144. Remaining distance: 6591/16384 - 19773/262144 = (105456/262144 - 19773/262144) = 85683/262144.- **Hop 6:** 1/4 of 85683/262144 = 85683/1048576. Efficiency decreases by 3/4, so hop distance is 85683/1048576 * 3/4 = 257049/4194304. Remaining distance: 85683/262144 - 257049/4194304 = (1370928/4194304 - 257049/4194304) = 1113879/4194304.Wait, this seems complicated. Maybe there's a pattern or a formula I can use instead of calculating each hop step by step.Let me think about the total distance hopped after six hops. Each hop is 1/4 of the remaining distance, but each time, the efficiency is multiplied by 3/4. So, the hop distances form a geometric series where each term is (1/4)*(3/4)^(n-1), where n is the hop number.So, the total distance hopped after six hops would be the sum of the first six terms of this geometric series.The formula for the sum of the first n terms of a geometric series is S_n = a1*(1 - r^n)/(1 - r), where a1 is the first term and r is the common ratio.Here, a1 = 1/4, r = 3/4, and n = 6.So, S_6 = (1/4)*(1 - (3/4)^6)/(1 - 3/4) = (1/4)*(1 - 729/4096)/(1/4) = (1/4)*(3367/4096)/(1/4) = 3367/4096.Wait, that can't be right because 3367/4096 is approximately 0.82, but after six hops, he should be closer to 1. Maybe I made a mistake in the formula.Wait, no, because each hop is 1/4 of the remaining distance, but the remaining distance is also decreasing each time. So, the hop distances are not simply a geometric series with ratio 3/4. The remaining distance after each hop affects the next hop's distance.Hmm, maybe I need to model it differently. Let me denote the remaining distance after n hops as R_n. Then, the hop distance on the (n+1)th hop is (1/4)*R_n, and R_{n+1} = R_n - (1/4)*R_n = (3/4)*R_n.But wait, the efficiency also decreases by 3/4 each time. So, the hop distance is (1/4)*R_n*(3/4)^n?Wait, no, maybe the hop distance is (1/4)*R_n, and R_{n+1} = R_n - (1/4)*R_n = (3/4)*R_n. So, the remaining distance after each hop is multiplied by 3/4. Therefore, the hop distances are:Hop 1: (1/4)*1 = 1/4Hop 2: (1/4)*(3/4) = 3/16Hop 3: (1/4)*(3/4)^2 = 9/64Hop 4: (1/4)*(3/4)^3 = 27/256Hop 5: (1/4)*(3/4)^4 = 81/1024Hop 6: (1/4)*(3/4)^5 = 243/4096So, the total distance hopped after six hops is the sum of these six terms.Let me calculate each term:1. 1/4 = 1024/40962. 3/16 = 768/40963. 9/64 = 576/40964. 27/256 = 432/40965. 81/1024 = 324/40966. 243/4096 = 243/4096Now, adding them up:1024 + 768 = 17921792 + 576 = 23682368 + 432 = 28002800 + 324 = 31243124 + 243 = 3367So, total distance hopped is 3367/4096.Wait, but earlier I thought that was the sum, but I was confused because it seemed too low. But actually, 3367/4096 is approximately 0.82, which seems reasonable after six hops, considering each hop is getting smaller.But let me double-check. The sum of the series is S = (1/4)*(1 - (3/4)^6)/(1 - 3/4) = (1/4)*(1 - 729/4096)/(1/4) = (1/4)*(3367/4096)/(1/4) = 3367/4096.Yes, that matches. So, the total distance hopped after six hops is 3367/4096.Wait, but in the initial step-by-step calculation, I got a different result. Let me see where I went wrong.In the initial step-by-step, I was calculating each hop as (1/4)*(3/4)^(n-1), which is correct, but when I added them up, I think I made a mistake in the denominators.Wait, no, when I converted each hop to 4096 denominator, I added them correctly:1/4 = 1024/40963/16 = 768/40969/64 = 576/409627/256 = 432/409681/1024 = 324/4096243/4096 = 243/4096Adding these: 1024 + 768 = 1792; 1792 + 576 = 2368; 2368 + 432 = 2800; 2800 + 324 = 3124; 3124 + 243 = 3367. So, 3367/4096 is correct.But wait, in the initial problem, the user mentioned that the efficiency decreases by a factor of 3/4 with each hop. So, does that mean that each hop is 3/4 of the previous hop's distance? Or is it that the efficiency (the fraction of the remaining distance) decreases by 3/4 each time?I think I interpreted it correctly as each hop being 1/4 of the remaining distance, and the efficiency (the fraction) decreasing by 3/4 each time, meaning each hop is (1/4)*(3/4)^(n-1).But let me confirm with the problem statement: "each subsequent hop can only cover 1/4 of the remaining distance to the goal, and his efficiency to cover that fraction decreases by a factor of 3/4 with each hop."So, it's two separate factors: each hop is 1/4 of the remaining distance, and the efficiency (the fraction he can cover) decreases by 3/4 each time. So, the hop distance is (1/4)*(3/4)^(n-1).Therefore, the total distance is the sum of the first six terms of this series, which is 3367/4096.But wait, in the initial calculation, I thought the remaining distance after each hop is multiplied by 3/4, which is correct because he covers 1/4 of it, leaving 3/4. So, the remaining distance after n hops is (3/4)^n. Therefore, the hop distances are (1/4)*(3/4)^(n-1).So, the sum is S = (1/4)*(1 - (3/4)^6)/(1 - 3/4) = (1/4)*(1 - 729/4096)/(1/4) = 3367/4096.Yes, that seems correct. So, the total distance hopped after six hops is 3367/4096.Wait, but in the initial step-by-step, I had a different approach where I calculated each hop as (1/4)*(3/4)^(n-1) and added them up, getting 3367/4096. So, that's consistent.Therefore, the answer should be 3367/4096.But let me check if that's the simplest form. 3367 and 4096. 4096 is 2^12. 3367 divided by 7 is 481, which is a prime number. So, 3367 = 7*481. 4096 is 2^12, so no common factors. Therefore, 3367/4096 is already in simplest form.So, the final answer is 3367/4096.

💡Alright, so I have this problem here about finding the standard equation of a circle. The circle passes through two points: (4,2) and (-2,-6). Also, the sum of the four intercepts on the two coordinate axes is -2. Hmm, okay, let me try to figure this out step by step.First, I remember that the general equation of a circle is something like (x^2 + y^2 + Dx + Ey + F = 0). I think D, E, and F are constants that we need to find. Since the circle passes through the points (4,2) and (-2,-6), I can plug these points into the general equation to get two equations.Let me do that. Plugging in (4,2):(4^2 + 2^2 + D*4 + E*2 + F = 0)That simplifies to:(16 + 4 + 4D + 2E + F = 0)Which is:(20 + 4D + 2E + F = 0)Okay, so that's equation (1): (4D + 2E + F = -20)Now, plugging in (-2,-6):((-2)^2 + (-6)^2 + D*(-2) + E*(-6) + F = 0)That simplifies to:(4 + 36 - 2D - 6E + F = 0)Which is:(40 - 2D - 6E + F = 0)So that's equation (2): (-2D - 6E + F = -40)Now, I have two equations:1. (4D + 2E + F = -20)2. (-2D - 6E + F = -40)I need a third equation because there are three unknowns: D, E, and F. The problem mentions the sum of the four intercepts on the two coordinate axes is -2. Hmm, intercepts... So, the x-intercepts and y-intercepts.For the x-intercepts, we set y=0 in the circle equation:(x^2 + Dx + F = 0)Similarly, for the y-intercepts, we set x=0:(y^2 + Ey + F = 0)So, the x-intercepts are the solutions to (x^2 + Dx + F = 0), and the y-intercepts are the solutions to (y^2 + Ey + F = 0).If I let the x-intercepts be (x_1) and (x_2), then from quadratic equation properties, (x_1 + x_2 = -D).Similarly, for the y-intercepts (y_1) and (y_2), (y_1 + y_2 = -E).The sum of all four intercepts is (x_1 + x_2 + y_1 + y_2 = -D + (-E) = - (D + E)).According to the problem, this sum is -2. So:(- (D + E) = -2)Which simplifies to:(D + E = 2)That's equation (3): (D + E = 2)Now, I have three equations:1. (4D + 2E + F = -20)2. (-2D - 6E + F = -40)3. (D + E = 2)Let me try to solve these equations.From equation (3), I can express D in terms of E: (D = 2 - E)Now, substitute D into equations (1) and (2).Substituting into equation (1):(4*(2 - E) + 2E + F = -20)Simplify:(8 - 4E + 2E + F = -20)Combine like terms:(8 - 2E + F = -20)Subtract 8 from both sides:(-2E + F = -28)Let me call this equation (4): (-2E + F = -28)Now, substitute D into equation (2):(-2*(2 - E) - 6E + F = -40)Simplify:(-4 + 2E - 6E + F = -40)Combine like terms:(-4 - 4E + F = -40)Add 4 to both sides:(-4E + F = -36)Let me call this equation (5): (-4E + F = -36)Now, I have two equations:4. (-2E + F = -28)5. (-4E + F = -36)Let me subtract equation (4) from equation (5):((-4E + F) - (-2E + F) = -36 - (-28))Simplify:(-4E + F + 2E - F = -36 + 28)Combine like terms:(-2E = -8)Divide both sides by -2:(E = 4)Now, substitute E = 4 into equation (3): (D + 4 = 2), so (D = -2)Now, substitute E = 4 into equation (4): (-2*4 + F = -28)Which is:(-8 + F = -28)Add 8 to both sides:(F = -20)So, D = -2, E = 4, F = -20Therefore, the general equation of the circle is:(x^2 + y^2 - 2x + 4y - 20 = 0)To write this in standard form, I need to complete the squares for x and y.Starting with x terms: (x^2 - 2x). To complete the square, take half of -2, which is -1, square it to get 1.Similarly, for y terms: (y^2 + 4y). Half of 4 is 2, square it to get 4.So, add and subtract these squares:(x^2 - 2x + 1 - 1 + y^2 + 4y + 4 - 4 - 20 = 0)Simplify:((x - 1)^2 - 1 + (y + 2)^2 - 4 - 20 = 0)Combine constants:((x - 1)^2 + (y + 2)^2 - 25 = 0)Move the constant to the other side:((x - 1)^2 + (y + 2)^2 = 25)So, the standard equation of the circle is ((x - 1)^2 + (y + 2)^2 = 25)Let me double-check if this makes sense. The center is at (1, -2), and the radius is 5. Let's see if the points (4,2) and (-2,-6) lie on this circle.For (4,2):((4 - 1)^2 + (2 + 2)^2 = 3^2 + 4^2 = 9 + 16 = 25). Yes, that works.For (-2,-6):((-2 - 1)^2 + (-6 + 2)^2 = (-3)^2 + (-4)^2 = 9 + 16 = 25). That also works.Now, checking the intercepts. For x-intercepts, set y=0:((x - 1)^2 + (0 + 2)^2 = 25)((x - 1)^2 + 4 = 25)((x - 1)^2 = 21)So, (x - 1 = pm sqrt{21}), which gives (x = 1 pm sqrt{21})Sum of x-intercepts: (1 + sqrt{21} + 1 - sqrt{21} = 2)For y-intercepts, set x=0:((0 - 1)^2 + (y + 2)^2 = 25)(1 + (y + 2)^2 = 25)((y + 2)^2 = 24)So, (y + 2 = pm sqrt{24}), which gives (y = -2 pm 2sqrt{6})Sum of y-intercepts: (-2 + 2sqrt{6} + (-2 - 2sqrt{6}) = -4)Total sum of intercepts: 2 + (-4) = -2, which matches the given condition.Alright, everything checks out!

💡Alright, so I've got this geometry problem here about a cube and marking points to form an equilateral triangle. Let me try to wrap my head around it step by step.First, the cube is labeled ABCDA₁B₁C₁D₁. I remember that in cube notation, the vertices are labeled such that ABCD is the base face, and A₁B₁C₁D₁ is the top face, with edges connecting corresponding vertices. So, A is connected to A₁, B to B₁, and so on.Petya marked a point X on edge AB, dividing it in the ratio 1:2 from vertex A. That means AX:XB = 1:2. If I consider the edge AB to have a length of, say, 3 units for simplicity, then AX would be 1 unit and XB would be 2 units. This makes sense because 1:2 is a ratio that adds up to 3, which is the total length of AB.Now, the task is to find points Y on edge CC₁ and Z on edge A₁D₁ such that triangle XYZ is equilateral. An equilateral triangle has all sides equal and all angles equal to 60 degrees. So, I need to ensure that the distances XY, YZ, and ZX are all equal.Let me visualize the cube. Since X is on AB, Y is on CC₁, and Z is on A₁D₁, these points are on different edges, some on the base and some on the top face. I need to figure out where exactly to place Y and Z so that the triangle formed is equilateral.I think assigning coordinates might help here. Let's place the cube in a 3D coordinate system where point A is at the origin (0,0,0). Then, since it's a cube, all edges are of equal length. Let's assume each edge has length 3 units for simplicity, as I mentioned earlier.So, the coordinates of the vertices would be:- A: (0,0,0)- B: (3,0,0)- C: (3,3,0)- D: (0,3,0)- A₁: (0,0,3)- B₁: (3,0,3)- C₁: (3,3,3)- D₁: (0,3,3)Given that, point X is on AB, 1 unit from A. So, moving 1 unit along AB from A, which is along the x-axis, X would be at (1,0,0).Now, I need to find points Y on CC₁ and Z on A₁D₁ such that triangle XYZ is equilateral. Let's denote the coordinates of Y and Z.Point Y is on CC₁. Since C is at (3,3,0) and C₁ is at (3,3,3), Y must have coordinates (3,3,k) where k is between 0 and 3.Similarly, point Z is on A₁D₁. A₁ is at (0,0,3) and D₁ is at (0,3,3), so Z must have coordinates (0,m,3) where m is between 0 and 3.So, our points are:- X: (1,0,0)- Y: (3,3,k)- Z: (0,m,3)We need to find k and m such that the distances XY, YZ, and ZX are equal.Let's compute the distances using the distance formula in 3D.First, distance XY:XY = sqrt[(3-1)^2 + (3-0)^2 + (k-0)^2] = sqrt[(2)^2 + (3)^2 + k^2] = sqrt[4 + 9 + k^2] = sqrt[13 + k^2]Next, distance YZ:YZ = sqrt[(0-3)^2 + (m-3)^2 + (3 - k)^2] = sqrt[(-3)^2 + (m-3)^2 + (3 - k)^2] = sqrt[9 + (m-3)^2 + (3 - k)^2]Distance ZX:ZX = sqrt[(1-0)^2 + (0 - m)^2 + (0 - 3)^2] = sqrt[(1)^2 + (-m)^2 + (-3)^2] = sqrt[1 + m^2 + 9] = sqrt[10 + m^2]So, we have:XY = sqrt(13 + k^2)YZ = sqrt[9 + (m - 3)^2 + (3 - k)^2]ZX = sqrt(10 + m^2)We need all three distances to be equal. Therefore, we can set up equations:1. sqrt(13 + k^2) = sqrt(10 + m^2)2. sqrt(13 + k^2) = sqrt[9 + (m - 3)^2 + (3 - k)^2]Let's square both sides of the first equation to eliminate the square roots:13 + k^2 = 10 + m^2=> m^2 = 13 + k^2 - 10=> m^2 = 3 + k^2=> m = sqrt(3 + k^2) [Since m is a length, we take the positive root]Now, let's square both sides of the second equation:13 + k^2 = 9 + (m - 3)^2 + (3 - k)^2Let's expand the right-hand side:= 9 + (m^2 - 6m + 9) + (9 - 6k + k^2)= 9 + m^2 - 6m + 9 + 9 - 6k + k^2= 9 + 9 + 9 + m^2 + k^2 - 6m - 6k= 27 + m^2 + k^2 - 6m - 6kSo, the equation becomes:13 + k^2 = 27 + m^2 + k^2 - 6m - 6kSimplify by subtracting 13 + k^2 from both sides:0 = 14 + m^2 - 6m - 6kBut from the first equation, we have m^2 = 3 + k^2. Let's substitute that into this equation:0 = 14 + (3 + k^2) - 6m - 6k0 = 17 + k^2 - 6m - 6kNow, let's rearrange:k^2 - 6k - 6m + 17 = 0But we also have m = sqrt(3 + k^2). Let's substitute m into this equation:k^2 - 6k - 6*sqrt(3 + k^2) + 17 = 0This looks a bit complicated, but maybe we can solve for k numerically or look for integer solutions.Let me try to see if k is an integer. Let's test k=1:1 - 6 - 6*sqrt(4) + 17 = 1 -6 -12 +17 = 0. Hey, that works!So, k=1.Then, m = sqrt(3 + 1^2) = sqrt(4) = 2.So, k=1 and m=2.Therefore, point Y is at (3,3,1) and point Z is at (0,2,3).Let me verify the distances:XY: sqrt[(3-1)^2 + (3-0)^2 + (1-0)^2] = sqrt[4 + 9 + 1] = sqrt[14]YZ: sqrt[(0-3)^2 + (2-3)^2 + (3 - 1)^2] = sqrt[9 + 1 + 4] = sqrt[14]ZX: sqrt[(1-0)^2 + (0 - 2)^2 + (0 - 3)^2] = sqrt[1 + 4 + 9] = sqrt[14]Yes, all sides are equal to sqrt(14), so triangle XYZ is indeed equilateral.So, the points are:- X: (1,0,0)- Y: (3,3,1)- Z: (0,2,3)This satisfies the condition.**Final Answer**boxed{XYZ text{ is equilateral when } Y text{ is one-third along } CC_1 text{ and } Z text{ is two-thirds along } A_1D_1.}

💡First, define the variables:- Let ( c ) represent the number of sets of quadruplets.- The number of sets of triplets is ( 5c ) (five times as many as quadruplets).- The number of sets of twins is ( 3 times 5c = 15c ) (three times as many as triplets).Next, calculate the total number of babies:- Babies from twins: ( 2 times 15c = 30c )- Babies from triplets: ( 3 times 5c = 15c )- Babies from quadruplets: ( 4c )Set up the equation for the total number of babies:[30c + 15c + 4c = 1200][49c = 1200][c = frac{1200}{49} approx 24.49]Finally, calculate the number of babies in sets of quadruplets:[4c = 4 times 24.49 approx 98]

💡Alright, so I have this problem where I'm given that sec x + tan x equals 15/4, and I need to find csc x + cot x in the form p/q, where p and q are coprime integers, and then find p + q. Hmm, okay. Let me try to figure this out step by step.First, I remember that sec x is 1/cos x and tan x is sin x / cos x. So, maybe I can express sec x + tan x in terms of sin x and cos x. Let me write that down:sec x + tan x = 1/cos x + sin x / cos x = (1 + sin x) / cos x.So, (1 + sin x) / cos x = 15/4. That gives me an equation involving sin x and cos x. Maybe I can solve for sin x or cos x.Let me denote sin x as s and cos x as c to make it simpler. So, the equation becomes:(1 + s) / c = 15/4.Which implies that 1 + s = (15/4) c.Okay, so 1 + s = (15/4) c. I can write this as:s = (15/4) c - 1.Now, I also know that sin^2 x + cos^2 x = 1. So, s^2 + c^2 = 1. Maybe I can substitute s from the previous equation into this identity.Let me do that:[(15/4) c - 1]^2 + c^2 = 1.Expanding the square:(225/16)c^2 - (30/4)c + 1 + c^2 = 1.Simplify the terms:225/16 c^2 - 30/4 c + 1 + c^2 = 1.Combine like terms:(225/16 + 16/16) c^2 - (30/4)c + 1 = 1.So, (241/16)c^2 - (30/4)c + 1 = 1.Subtract 1 from both sides:241/16 c^2 - 30/4 c = 0.Factor out c:c (241/16 c - 30/4) = 0.So, either c = 0 or 241/16 c - 30/4 = 0.But c = cos x, and if c = 0, then sec x would be undefined, which contradicts the given equation. So, c ≠ 0. Therefore, we can solve for c:241/16 c = 30/4.Multiply both sides by 16:241 c = (30/4) * 16.Simplify 30/4 * 16: 30/4 is 15/2, and 15/2 * 16 is 15 * 8 = 120.So, 241 c = 120.Therefore, c = 120 / 241.Okay, so cos x is 120/241. Now, let's find sin x using the equation s = (15/4)c - 1.So, s = (15/4)(120/241) - 1.Calculate (15/4)(120/241):15 * 120 = 1800.4 * 241 = 964.So, 1800 / 964. Let me simplify that fraction.Divide numerator and denominator by 4: 1800 ÷ 4 = 450, 964 ÷ 4 = 241.So, 450/241.Therefore, s = 450/241 - 1.Convert 1 to 241/241:s = 450/241 - 241/241 = (450 - 241)/241 = 209/241.So, sin x = 209/241.Alright, so now I have sin x and cos x:sin x = 209/241,cos x = 120/241.Now, I need to find csc x + cot x.csc x is 1/sin x, and cot x is cos x / sin x.So, csc x + cot x = 1/sin x + cos x / sin x = (1 + cos x) / sin x.So, let's compute that:(1 + cos x) / sin x = (1 + 120/241) / (209/241).First, compute 1 + 120/241:1 is 241/241, so 241/241 + 120/241 = 361/241.So, numerator is 361/241, denominator is 209/241.Therefore, (361/241) / (209/241) = 361/209.Simplify 361/209.Let me check if 361 and 209 have any common factors.209 factors: 11 * 19.361 is 19^2.So, 361 = 19 * 19,209 = 11 * 19.So, common factor is 19.Divide numerator and denominator by 19:361 ÷ 19 = 19,209 ÷ 19 = 11.So, simplified fraction is 19/11.Therefore, csc x + cot x = 19/11.So, p = 19, q = 11. Therefore, p + q = 19 + 11 = 30.Wait, but in the initial problem statement, the user mentioned that the answer was 390, but that seems incorrect because 19 + 11 is 30. Maybe I made a mistake somewhere.Let me double-check my calculations.Starting from sec x + tan x = 15/4.Expressed as (1 + sin x)/cos x = 15/4.So, 1 + sin x = (15/4) cos x.Then, sin x = (15/4) cos x - 1.Then, sin^2 x + cos^2 x = 1.Substituting sin x:[(15/4 cos x - 1)]^2 + cos^2 x = 1.Expanding:(225/16 cos^2 x - 30/4 cos x + 1) + cos^2 x = 1.Combine like terms:(225/16 + 16/16) cos^2 x - 30/4 cos x + 1 = 1.So, 241/16 cos^2 x - 30/4 cos x = 0.Factor out cos x:cos x (241/16 cos x - 30/4) = 0.So, cos x = 0 or 241/16 cos x = 30/4.cos x = 0 is rejected, so 241/16 cos x = 30/4.Multiply both sides by 16: 241 cos x = 120.So, cos x = 120/241.Then, sin x = (15/4)(120/241) - 1.Compute (15/4)(120/241):15 * 120 = 1800,4 * 241 = 964,So, 1800/964 simplifies to 450/241.Then, sin x = 450/241 - 1 = 450/241 - 241/241 = 209/241.So, sin x = 209/241.Then, csc x + cot x = (1 + cos x)/sin x.1 + cos x = 1 + 120/241 = 361/241.Divide by sin x: (361/241)/(209/241) = 361/209.Simplify 361/209: both divisible by 19.361 ÷ 19 = 19,209 ÷ 19 = 11.So, 19/11.Therefore, p = 19, q = 11, p + q = 30.Hmm, so I think the initial thought process had an error where it concluded 390, but actually, the correct answer is 30. Maybe the initial user made a mistake in their calculation.Alternatively, perhaps I made a mistake in interpreting the problem. Let me check again.Wait, the problem says "csc x + cot x = p/q". I computed that as 19/11, which is in lowest terms, so p + q = 30.Alternatively, maybe I should have considered another identity or approach.Wait, another approach: I remember that if sec x + tan x = 15/4, then sec x - tan x is the reciprocal of that, because (sec x + tan x)(sec x - tan x) = sec^2 x - tan^2 x = 1.So, if sec x + tan x = 15/4, then sec x - tan x = 4/15.Therefore, we can solve for sec x and tan x.Let me write:sec x + tan x = 15/4,sec x - tan x = 4/15.Adding these two equations:2 sec x = 15/4 + 4/15.Compute 15/4 + 4/15:Find a common denominator, which is 60.15/4 = 225/60,4/15 = 16/60.So, 225/60 + 16/60 = 241/60.Therefore, 2 sec x = 241/60,So, sec x = 241/120.Therefore, cos x = 120/241, which matches what I found earlier.Then, subtracting the two equations:(sec x + tan x) - (sec x - tan x) = 15/4 - 4/15.So, 2 tan x = 15/4 - 4/15.Compute 15/4 - 4/15:Again, common denominator 60.15/4 = 225/60,4/15 = 16/60.So, 225/60 - 16/60 = 209/60.Therefore, 2 tan x = 209/60,So, tan x = 209/120.Which is the same as sin x / cos x = 209/120.Since cos x = 120/241, sin x = (209/120) * (120/241) = 209/241, which is consistent.So, everything checks out.Therefore, csc x + cot x = (1 + cos x)/sin x = (1 + 120/241)/(209/241) = (361/241)/(209/241) = 361/209 = 19/11.So, p = 19, q = 11, p + q = 30.Therefore, the answer is 30.**Final Answer**boxed{30}

💡Okay, let's try to figure out the range of ( a ) for which the function ( f(x) = frac{1}{3}|x^3| - frac{a}{2}x^2 + (3 - a)|x| + b ) has six different monotonic intervals. Hmm, that sounds a bit complicated, but I'll take it step by step.First, I remember that the number of monotonic intervals a function has is related to the number of critical points it has. Critical points occur where the derivative is zero or undefined. So, I think I need to find the derivative of ( f(x) ) and analyze its roots.But before that, I notice that the function has absolute value terms, specifically ( |x^3| ) and ( |x| ). Absolute value functions can complicate things because they introduce corners or points where the derivative might not exist. However, since ( |x^3| ) is actually equal to ( |x|^3 ), which is smooth everywhere except at ( x = 0 ), but since it's a higher power, maybe it's differentiable there too? Hmm, not sure. Maybe I should consider the function for ( x geq 0 ) and ( x < 0 ) separately because of the absolute values.Wait, the function is given as ( f(x) = frac{1}{3}|x^3| - frac{a}{2}x^2 + (3 - a)|x| + b ). Let me see if this function is even or odd. Plugging in ( -x ) into the function:( f(-x) = frac{1}{3}|(-x)^3| - frac{a}{2}(-x)^2 + (3 - a)|-x| + b )Simplify each term:- ( |(-x)^3| = | -x^3 | = |x^3| )- ( (-x)^2 = x^2 )- ( |-x| = |x| )So, ( f(-x) = frac{1}{3}|x^3| - frac{a}{2}x^2 + (3 - a)|x| + b = f(x) )Oh, so ( f(x) ) is an even function. That means it's symmetric about the y-axis. Therefore, the behavior of the function for ( x > 0 ) will mirror that for ( x < 0 ). So, if I can figure out the number of monotonic intervals for ( x > 0 ), I can double it (or adjust accordingly) for the entire function.The problem states there are six different monotonic intervals. Since the function is even, that suggests that for ( x > 0 ), there are three monotonic intervals, and similarly for ( x < 0 ). So, I need to focus on ( x > 0 ) and find when the derivative has two critical points, leading to three intervals.Let me compute the derivative ( f'(x) ) for ( x > 0 ). Since for ( x > 0 ), ( |x^3| = x^3 ) and ( |x| = x ). So, the function simplifies to:( f(x) = frac{1}{3}x^3 - frac{a}{2}x^2 + (3 - a)x + b )Now, take the derivative:( f'(x) = frac{d}{dx}left( frac{1}{3}x^3 - frac{a}{2}x^2 + (3 - a)x + b right) )( f'(x) = x^2 - a x + (3 - a) )So, ( f'(x) = x^2 - a x + (3 - a) ). To find critical points, set ( f'(x) = 0 ):( x^2 - a x + (3 - a) = 0 )This is a quadratic equation in ( x ). For there to be two distinct critical points (which would lead to three monotonic intervals on ( x > 0 )), the quadratic must have two distinct real roots. The condition for that is that the discriminant is positive.The discriminant ( D ) of ( x^2 - a x + (3 - a) = 0 ) is:( D = (-a)^2 - 4 times 1 times (3 - a) )( D = a^2 - 12 + 4a )( D = a^2 + 4a - 12 )For two distinct real roots, ( D > 0 ):( a^2 + 4a - 12 > 0 )Let me solve this inequality. First, find the roots of the quadratic equation ( a^2 + 4a - 12 = 0 ):Using quadratic formula:( a = frac{ -4 pm sqrt{16 + 48} }{2} )( a = frac{ -4 pm sqrt{64} }{2} )( a = frac{ -4 pm 8 }{2} )So, two solutions:1. ( a = frac{ -4 + 8 }{2} = frac{4}{2} = 2 )2. ( a = frac{ -4 - 8 }{2} = frac{ -12 }{2} = -6 )So, the quadratic ( a^2 + 4a - 12 ) factors as ( (a - 2)(a + 6) ). To determine where it's positive, we can analyze the intervals determined by the roots ( a = -6 ) and ( a = 2 ).- For ( a < -6 ), say ( a = -7 ): ( (-7 - 2)(-7 + 6) = (-9)(-1) = 9 > 0 )- For ( -6 < a < 2 ), say ( a = 0 ): ( (0 - 2)(0 + 6) = (-2)(6) = -12 < 0 )- For ( a > 2 ), say ( a = 3 ): ( (3 - 2)(3 + 6) = (1)(9) = 9 > 0 )So, ( a^2 + 4a - 12 > 0 ) when ( a < -6 ) or ( a > 2 ).But wait, in our case, ( a ) is a parameter in the original function. Let me think about the context. The function ( f(x) ) has terms with ( a ) in coefficients, but I don't see any immediate restrictions on ( a ) except those imposed by the discriminant condition.However, I should also consider whether the critical points are positive because we're looking at ( x > 0 ). So, even if the quadratic has two real roots, they need to be positive for the critical points to lie in ( x > 0 ).So, let's recall that for a quadratic equation ( x^2 - a x + (3 - a) = 0 ), the sum of the roots is ( a ) and the product is ( 3 - a ).For both roots to be positive, we need:1. Sum of roots ( > 0 ): ( a > 0 )2. Product of roots ( > 0 ): ( 3 - a > 0 ) => ( a < 3 )So, combining these, we have ( 0 < a < 3 ).But from the discriminant condition, we have ( a > 2 ) or ( a < -6 ). So, intersecting these two conditions:- From discriminant: ( a > 2 ) or ( a < -6 )- From positivity of roots: ( 0 < a < 3 )Therefore, the overlap is ( 2 < a < 3 ).Wait, but what about ( a < -6 )? If ( a < -6 ), then the sum of roots is ( a ), which would be negative, but the product is ( 3 - a ). If ( a < -6 ), then ( 3 - a > 3 - (-6) = 9 > 0 ). So, the product is positive, but the sum is negative. That would mean both roots are negative because their sum is negative and product is positive. But since we're considering ( x > 0 ), negative roots don't affect the critical points in ( x > 0 ). So, for ( x > 0 ), we don't have any critical points if ( a < -6 ), which would mean the function is monotonic on ( x > 0 ). But the problem states there are six monotonic intervals, which implies three on each side of zero. So, if ( a < -6 ), we don't get three intervals on ( x > 0 ), so that's not acceptable.Therefore, the only valid range is ( 2 < a < 3 ).Wait, let me double-check. If ( a > 2 ), then from the discriminant, we have two real roots. From the sum and product, if ( a > 2 ), then sum is positive, product is ( 3 - a ). So, if ( a < 3 ), product is positive. So, when ( 2 < a < 3 ), both roots are positive, which is what we need for ( x > 0 ). If ( a = 3 ), the product becomes zero, so one root is zero, but since we're considering ( x > 0 ), that would mean a critical point at zero, but since we're already considering ( x > 0 ), maybe it's a boundary case. Similarly, if ( a > 3 ), the product becomes negative, so one root is positive and one is negative. But since we're only considering ( x > 0 ), only one critical point there, leading to two monotonic intervals on ( x > 0 ), which would result in four total intervals (two on each side), but the problem says six, so ( a > 3 ) is also invalid.Similarly, if ( a = 2 ), discriminant is zero, so only one critical point on ( x > 0 ), leading to two intervals on ( x > 0 ), again not enough.So, putting it all together, the range of ( a ) is ( 2 < a < 3 ).**Final Answer**The range of ( a ) is boxed{(2, 3)}.