💡First, solve the first inequality ( x - 2 > 1 ) by adding 2 to both sides, resulting in ( x > 3 ).Next, consider the second inequality ( x < 4 ).The solution set must satisfy both conditions, which means ( x ) must be greater than 3 and less than 4.Therefore, the solution set is ( 3 < x < 4 ).

💡Okay, so I have this problem where I need to prove that the greatest common divisor (gcd) of any two elements in the sequence {a_n} is 1. The sequence is defined by a_1 = 3 and a_{n+1} = a_n^2 - 2. Hmm, interesting. I remember that sequences defined by quadratic recursions can sometimes have properties related to coprimality, but I'm not exactly sure how to approach this. Let me try to break it down step by step.First, let me write out the first few terms of the sequence to get a better feel for it. Starting with a_1 = 3:- a_1 = 3- a_2 = a_1^2 - 2 = 3^2 - 2 = 9 - 2 = 7- a_3 = a_2^2 - 2 = 7^2 - 2 = 49 - 2 = 47- a_4 = a_3^2 - 2 = 47^2 - 2 = 2209 - 2 = 2207- a_5 = a_4^2 - 2 = 2207^2 - 2 = ... Hmm, that's a big number, but I don't think I need to compute it exactly right now.Looking at these terms: 3, 7, 47, 2207, etc. They seem to be growing very rapidly. Also, all of them are odd numbers. That might be useful because if all terms are odd, then they don't share a common factor of 2. But I need to show that any two terms are coprime, not just that they are all odd.So, maybe I can use induction to prove that for any n < k, gcd(a_n, a_k) = 1. Let me try that approach.**Base Case:** Let's take k = n + 1. So, I need to show that gcd(a_n, a_{n+1}) = 1. From the definition, a_{n+1} = a_n^2 - 2. Let's suppose that there is a prime p that divides both a_n and a_{n+1}. Then p divides a_n, so a_n ≡ 0 mod p. Therefore, a_{n+1} = a_n^2 - 2 ≡ 0 - 2 ≡ -2 mod p. So p divides -2, which means p divides 2. But since all terms are odd, p cannot be 2. Therefore, the only possible common divisor is 1. So, gcd(a_n, a_{n+1}) = 1. That takes care of the base case.**Inductive Step:** Now, suppose that for some k > n, gcd(a_n, a_k) = 1. I need to show that gcd(a_n, a_{k+1}) = 1. From the recursion, a_{k+1} = a_k^2 - 2. Suppose there is a prime p that divides both a_n and a_{k+1}. Then p divides a_n and p divides a_{k+1} = a_k^2 - 2. Since p divides a_n, and by the induction hypothesis, p does not divide a_k (because gcd(a_n, a_k) = 1). Therefore, a_k is not congruent to 0 mod p. So, from a_{k+1} ≡ 0 mod p, we have a_k^2 ≡ 2 mod p. That means that 2 is a quadratic residue modulo p. Wait, I'm not sure if that helps directly. Maybe I need another approach. Let me think about the properties of the sequence. I recall that sequences defined by a_{n+1} = a_n^2 - c can sometimes be related to solutions of Pell's equation or other Diophantine equations, but I'm not sure if that's applicable here. Alternatively, perhaps I can use properties of modular arithmetic or look for patterns in the sequence.Another idea: Maybe I can express a_{k} in terms of a_n. Since each term is defined based on the previous term, perhaps there's a way to write a_{k} in terms of a_n and show that any common divisor would have to divide some constant, which would be 1.Wait, let me try to formalize the induction step. Suppose that gcd(a_n, a_k) = 1. Then, if p divides a_n and a_{k+1}, then p divides a_k^2 - 2. But since p does not divide a_k (because gcd(a_n, a_k) = 1), we can say that a_k has an inverse modulo p. Therefore, from a_k^2 ≡ 2 mod p, we can write (a_k^{-1})^2 * a_k^2 ≡ 2 * (a_k^{-1})^2 mod p, which simplifies to 1 ≡ 2 * (a_k^{-1})^2 mod p. Therefore, 2 is a quadratic residue modulo p, which implies that p ≡ ±1 mod 8 or something like that. But I'm not sure if that's helpful.Alternatively, maybe I can use the fact that if p divides both a_n and a_{k+1}, then p divides a_{k+1} + 2 = a_k^2. But p doesn't divide a_k, so p must divide 2, which is impossible since p is odd. Wait, that seems promising.Let me rephrase that. Suppose p is a prime that divides both a_n and a_{k+1}. Then p divides a_{k+1} = a_k^2 - 2, so p divides a_k^2 - 2. Also, p divides a_n, and by the induction hypothesis, p does not divide a_k. Therefore, a_k is invertible modulo p. So, from a_k^2 ≡ 2 mod p, we can write (a_k^{-1})^2 * a_k^2 ≡ 2 * (a_k^{-1})^2 mod p, which simplifies to 1 ≡ 2 * (a_k^{-1})^2 mod p. Therefore, 2 is a quadratic residue modulo p, which implies that p ≡ 1 mod 8 or p ≡ -1 mod 8, but I'm not sure if that helps.Wait, maybe I'm overcomplicating this. Let me think differently. If p divides both a_n and a_{k+1}, then p divides a_{k+1} + 2 = a_k^2. But p doesn't divide a_k (since gcd(a_n, a_k) = 1), so p must divide 2. But p is odd, so p cannot divide 2. Therefore, the only possible common divisor is 1. So, gcd(a_n, a_{k+1}) = 1.Wait, that seems to work! Let me make sure I'm not missing anything. If p divides a_{k+1}, then p divides a_k^2 - 2. If p also divides a_n, and since gcd(a_n, a_k) = 1, p does not divide a_k. Therefore, a_k is invertible modulo p, and from a_k^2 ≡ 2 mod p, we can manipulate this to find that 2 is a quadratic residue modulo p. But actually, I don't need to go that far. The key point is that if p divides a_{k+1} and a_n, then p divides a_k^2 - 2 and a_n. Since p doesn't divide a_k, it must divide 2, which is impossible because p is odd. Therefore, no such p exists, so gcd(a_n, a_{k+1}) = 1.Okay, that seems solid. So, by induction, for any n < k, gcd(a_n, a_k) = 1. Therefore, any two elements of the sequence are coprime.Wait, but I should also check if the induction step holds for all k > n, not just k = n + 1. Let me think about that. The base case was k = n + 1, and the induction step assumes that for some k, gcd(a_n, a_k) = 1, and then shows that gcd(a_n, a_{k+1}) = 1. So, by induction, this holds for all k > n. Therefore, the proof is complete.I think that covers it. The key idea is that if a prime divides two terms, it would have to divide 2, which is impossible because all terms are odd. Therefore, the gcd must be 1.

💡First, I need to determine the number of intra-division games. Each team plays every other team in its own division three times. Since there are 8 teams in a division, each team plays 7 other teams three times, resulting in 21 intra-division games per team.Next, I calculate the total intra-division games for one division by multiplying the number of teams by the number of games each team plays within the division and then dividing by 2 to avoid double-counting:(8 teams × 21 games) / 2 = 84 games per division.Since there are two divisions, the total intra-division games for both divisions are:84 games × 2 = 168 games.Now, I determine the number of inter-division games. Each team plays every team in the other division twice. With 8 teams in the other division, each team plays 8 games twice, resulting in 16 inter-division games per team.To find the total inter-division games, I multiply the number of teams by the number of inter-division games each team plays:16 games × 8 teams = 128 games.Finally, I add the total intra-division and inter-division games to find the total number of games in the season:168 games + 128 games = 296 games.

💡Okay, so I have this problem about a soccer round-robin tournament. Let me try to understand what it's asking. There are teams participating, and each team plays every other team exactly once. That's what a round-robin tournament is, right? So, if there are, say, k teams, each team plays k-1 matches.The scores for the top four teams are given: 11, 9, 7, and 5 points respectively. Also, the fifth place team has a different score from the fourth place team. In soccer, a win gives 2 points, a draw gives 1 point to each team, and a loss gives 0 points. So, points are accumulated based on wins and draws.The question is asking two things: how many teams participated in the tournament, and what's the total points scored by all teams.Alright, let me break this down step by step.First, I need to figure out how many teams there are. Let's denote the number of teams as k. In a round-robin tournament, each team plays every other team once, so the total number of matches is C(k,2), which is k(k-1)/2. Each match contributes either 2 points (if one team wins and the other loses) or 2 points in total (1 point each if it's a draw). So, regardless of the outcome, each match contributes 2 points to the total points pool. Therefore, the total points scored by all teams combined should be 2 * C(k,2) = k(k-1).So, total points = k(k-1). That's a key formula.Now, the top four teams have scores of 11, 9, 7, and 5. Let's add those up: 11 + 9 + 7 + 5 = 32 points. So, the top four teams have 32 points in total.The remaining teams (which would be k - 4 teams) must account for the rest of the points. Let's denote the total points as T = k(k-1). So, the remaining points would be T - 32.Now, the fifth place team has a different score from the fourth place team, which is 5 points. So, the fifth place team must have a score different from 5. It could be higher or lower, but since it's fifth place, it's likely lower than 5, but not necessarily. Wait, no, because in a tournament, the fifth place could have the same or different points as the fourth, but here it's specified that it's different. So, the fifth place team has a different score from the fourth place team, which is 5. So, the fifth place team has a score not equal to 5.But how does that help us? Maybe it's just a clue to ensure that the fifth place isn't also 5, which might affect the distribution of points among the remaining teams.So, moving on. The total points are k(k-1). The top four have 32 points, so the remaining points are k(k-1) - 32. These remaining points are distributed among the remaining k - 4 teams.Now, in a tournament, the maximum number of points a team can have is 2*(k-1), which would be if they won all their matches. The minimum is 0, but realistically, teams will have varying points based on their performance.Given that the top four have 11, 9, 7, and 5, which are all odd numbers, except 5 is also odd. Wait, 11, 9, 7, 5 are all odd. That might be significant because points are accumulated in increments of 1 or 2, so having an odd number of points suggests an odd number of draws or something.But maybe that's not directly relevant. Let's think about the possible number of teams.We need to find k such that the total points T = k(k-1) is greater than or equal to 32 plus the points from the remaining teams. But we also need to ensure that the points can be distributed in a way that makes sense with the given constraints.Let me try plugging in some numbers for k.If k = 5, then total points = 5*4 = 20. But the top four teams already have 32 points, which is more than 20. So, k can't be 5.If k = 6, total points = 6*5 = 30. Again, the top four have 32, which is more than 30. So, k can't be 6.If k = 7, total points = 7*6 = 42. The top four have 32, so remaining points = 42 - 32 = 10. These 10 points need to be distributed among the remaining 3 teams.Now, the fifth place team must have a different score from the fourth place team, which is 5. So, the fifth place team can't have 5 points. The remaining teams can have points less than or equal to 5, but not equal to 5 for the fifth place.Wait, but if the fifth place team has a different score from the fourth place, which is 5, then the fifth place team must have a score different from 5. It could be higher or lower, but in reality, since it's fifth place, it's likely lower. But let's see.We have 10 points to distribute among 3 teams. Let's see if it's possible.Possible distributions:- 4, 3, 3: Total = 10. So, the fifth place team could have 4, and the others 3 and 3.- 4, 4, 2: Total = 10. So, fifth place could have 4, and the others 4 and 2.- 5, 3, 2: But fifth place can't have 5, as per the problem statement.- 6, 2, 2: But 6 is higher than 5, which is the fourth place, which might not make sense because fifth place shouldn't have more points than fourth place.Wait, actually, in a tournament, teams are ranked by points, so fifth place should have less than or equal to fourth place. But since the fifth place has a different score, it must have less than 5.So, 6 is higher than 5, which would make it fourth place, which contradicts. So, 6 is not possible for fifth place.Similarly, 5 is not allowed for fifth place.So, the fifth place must have less than 5 points. So, possible points for fifth place are 4, 3, 2, 1, or 0.But let's see.If we have 10 points to distribute among 3 teams, and the fifth place must have less than 5, let's see:- 4, 3, 3: Fifth place has 4, which is less than 5. That works.- 4, 4, 2: Fifth place has 4, which is less than 5. That works.- 3, 3, 4: Same as above.- 3, 4, 3: Same.- 2, 4, 4: Same.- 3, 2, 5: But fifth place can't have 5.- 2, 3, 5: Same issue.- 1, 4, 5: No.- 0, 5, 5: No.So, the only valid distributions are 4, 3, 3 and 4, 4, 2.So, for k=7, it's possible.Let me check if k=8 works.Total points = 8*7=56.Top four have 32, so remaining points = 56 - 32 = 24.These 24 points need to be distributed among 4 teams.Now, the fifth place team must have a different score from the fourth place team, which is 5. So, fifth place can't have 5.But with 24 points among 4 teams, the possible distributions could be various.But let's see if it's possible to have the fifth place team with, say, 4 points, and the rest distributed accordingly.But wait, if we have 24 points among 4 teams, the maximum a team can have is 14 (if they won all their matches, but in reality, they have to have less because they lost to the top teams). But let's not get bogged down.The key is that for k=7, it's possible, and for k=8, it's also possible, but the problem is asking for the number of teams, and likely the minimal number that satisfies the conditions.But let's think again.Wait, in the problem, it's stated that the fifth place team has a different score from the fourth place team. So, the fourth place is 5, fifth place is not 5.But if k=7, the remaining teams are 3, and their points are 4,3,3 or 4,4,2.So, in both cases, the fifth place team has 4 points, which is different from 5.If k=8, the remaining 4 teams have 24 points. Let's see if it's possible to have the fifth place team with, say, 4 points, and the rest distributed.But 24 points among 4 teams, with fifth place having 4, and the others having, say, 6,6,6, but that would make the fifth place team have 4, which is less than 6, which is okay.But wait, the fifth place team is the fifth highest, so it's possible that the fifth place team has 4, and the others have higher.But actually, the fifth place team must be the fifth highest, so if the top four are 11,9,7,5, then the fifth place must be less than or equal to 5, but not equal to 5.Wait, no, that's not necessarily true. In a tournament, teams can have the same points but be ranked by other factors like goal difference, but in this problem, it's just based on points.So, if two teams have the same points, they could be ranked by other criteria, but the problem doesn't specify that. So, for simplicity, let's assume that the fifth place team has a different score from the fourth place team, which is 5, so fifth place must have less than 5 or more than 5, but since it's fifth place, it's likely less than 5.But in reality, in a tournament, the fifth place could have more points than the fourth place if they have the same points but lost on other criteria, but the problem says the fifth place has a different score, so it must have a different number of points.So, if the fourth place has 5, the fifth place must have something else, like 4, 3, etc.But in the case of k=8, the remaining 4 teams have 24 points. Let's see if it's possible to have the fifth place team with 4 points, and the rest distributed.But 24 points among 4 teams, with fifth place having 4, then the remaining 3 teams have 20 points. That seems possible, but we need to ensure that the points are distributed in a way that makes sense.But actually, the problem is that the fifth place team must have a different score from the fourth place, which is 5. So, in k=7, the fifth place team has 4, which is different from 5, and it's feasible.In k=8, the fifth place team could have 4, but then the remaining teams could have higher points, but that might not make sense because the fifth place should have less than or equal to the fourth place.Wait, no, in reality, the fifth place team could have more points than the fourth place team if they have the same points but lost on other criteria, but the problem says the fifth place has a different score, so it must have a different number of points.But if the fifth place team has more points than the fourth place, that would mean the fourth place team is actually fifth, which contradicts the ranking.Therefore, the fifth place team must have less than or equal to the fourth place team's points, but since it's different, it must have less than 5.So, in k=7, the fifth place team has 4, which is less than 5, and it's feasible.In k=8, the fifth place team would have to have less than 5, but with 24 points to distribute among 4 teams, it's possible that the fifth place team has 4, and the others have higher, but that would mean the fifth place team is actually higher than some teams, which contradicts the ranking.Wait, no, because the fifth place team is the fifth highest, so it must have less than or equal to the fourth place team's points, which is 5. So, in k=8, the fifth place team must have less than 5, but the remaining 4 teams have 24 points. That would mean that the fifth place team has, say, 4, and the others have 6,6,6, but that would mean the fifth place team has 4, which is less than 6, which is okay, but the fifth place team is fifth, so it's okay to have less than the teams above it.But wait, the fifth place team is fifth, so it's possible that the fifth place team has 4, and the others have higher, but that would mean that the fifth place team is actually below the fourth place team, which is 5, so it's okay.But the problem is that the fifth place team must have a different score from the fourth place team, which is 5, so fifth place can't have 5, but can have less.So, in k=8, it's possible, but we need to check if the total points make sense.But let's think about the minimal number of teams. Since k=7 works, and k=8 also works, but the problem is likely expecting the minimal number, which is 7.Wait, but let's check if k=7 is possible.Total points = 42.Top four have 32, so remaining 10 points among 3 teams.Possible distributions: 4,3,3 or 4,4,2.So, the fifth place team has 4, which is different from 5, satisfying the condition.Therefore, k=7 is possible.If we try k=6, total points=30, but top four have 32, which is more than 30, so k=6 is impossible.k=5 is even worse.k=7 is the minimal number that satisfies the conditions.Therefore, the number of teams is 7, and the total points scored by all teams is 42.

💡First, I need to determine the additional area the calf can graze when the rope length is increased from 9 meters to 23 meters.The area that the calf can graze is a circle with the rope as the radius. The formula for the area of a circle is A = πr², where A is the area and r is the radius (length of the rope).First, I'll calculate the area with the original rope length of 9 meters:A₁ = π * (9 m)²A₁ = π * 81 m²A₁ = 81π m²Next, I'll calculate the area with the increased rope length of 23 meters:A₂ = π * (23 m)²A₂ = π * 529 m²A₂ = 529π m²Now, I'll find the additional area the calf can graze by subtracting the original area from the increased area:Additional area = A₂ - A₁Additional area = 529π m² - 81π m²Additional area = (529 - 81)π m²Additional area = 448π m²So, the calf can graze an additional 448π square meters of grassy ground with the increased rope length.

💡Okay, so I have this geometry problem here, and I need to figure out the value of (frac{a}{p} + frac{b}{q} + frac{c}{r}). Let me try to understand what's given and what I need to find.First, the problem mentions the origin (O) and a fixed point ((2a, 2b, 2c)). There's a plane passing through this fixed point and intersecting the x-axis, y-axis, and z-axis at points (A), (B), and (C), respectively. None of these points coincide with the origin, which is good because otherwise, the sphere might not be well-defined.Then, there's a sphere passing through points (A), (B), (C), and (O), with its center at ((p, q, r)). I need to find the value of (frac{a}{p} + frac{b}{q} + frac{c}{r}).Hmm, okay. Let me break this down step by step.**Understanding the Plane and Points A, B, C:**Since the plane intersects the x-axis at (A), y-axis at (B), and z-axis at (C), these points must be of the form:- (A = (k, 0, 0))- (B = (0, m, 0))- (C = (0, 0, n))Where (k), (m), and (n) are some constants. But the problem says the plane passes through the fixed point ((2a, 2b, 2c)), so this point must satisfy the equation of the plane.**Equation of the Plane:**The general equation of a plane intercepting the axes at (A), (B), and (C) is:[frac{x}{k} + frac{y}{m} + frac{z}{n} = 1]Since the plane passes through ((2a, 2b, 2c)), substituting this point into the equation gives:[frac{2a}{k} + frac{2b}{m} + frac{2c}{n} = 1]So that's one equation relating (k), (m), and (n).**Sphere Passing Through O, A, B, C:**Now, the sphere passes through the origin (O = (0, 0, 0)) and the points (A), (B), and (C). The center of the sphere is given as ((p, q, r)). Since all these points lie on the sphere, the distance from the center to each of these points must be equal (the radius).So, the distance from ((p, q, r)) to (O) is:[sqrt{(p - 0)^2 + (q - 0)^2 + (r - 0)^2} = sqrt{p^2 + q^2 + r^2}]Similarly, the distance from ((p, q, r)) to (A = (k, 0, 0)) is:[sqrt{(p - k)^2 + (q - 0)^2 + (r - 0)^2} = sqrt{(p - k)^2 + q^2 + r^2}]Since these distances must be equal, we have:[sqrt{p^2 + q^2 + r^2} = sqrt{(p - k)^2 + q^2 + r^2}]Squaring both sides to eliminate the square roots:[p^2 + q^2 + r^2 = (p - k)^2 + q^2 + r^2]Simplifying:[p^2 = p^2 - 2pk + k^2]Subtracting (p^2) from both sides:[0 = -2pk + k^2]Which simplifies to:[2pk = k^2 implies 2p = k quad text{(assuming (k neq 0))}]So, (k = 2p). Similarly, by applying the same logic for points (B) and (C), we can find expressions for (m) and (n).For point (B = (0, m, 0)), the distance from the center is:[sqrt{(p - 0)^2 + (q - m)^2 + (r - 0)^2} = sqrt{p^2 + (q - m)^2 + r^2}]Setting this equal to the radius:[sqrt{p^2 + q^2 + r^2} = sqrt{p^2 + (q - m)^2 + r^2}]Squaring both sides:[p^2 + q^2 + r^2 = p^2 + (q - m)^2 + r^2]Simplifying:[q^2 = q^2 - 2qm + m^2]Which gives:[0 = -2qm + m^2 implies 2qm = m^2 implies 2q = m quad text{(assuming (m neq 0))}]So, (m = 2q).Similarly, for point (C = (0, 0, n)), the distance from the center is:[sqrt{(p - 0)^2 + (q - 0)^2 + (r - n)^2} = sqrt{p^2 + q^2 + (r - n)^2}]Setting this equal to the radius:[sqrt{p^2 + q^2 + r^2} = sqrt{p^2 + q^2 + (r - n)^2}]Squaring both sides:[p^2 + q^2 + r^2 = p^2 + q^2 + (r - n)^2]Simplifying:[r^2 = r^2 - 2rn + n^2]Which gives:[0 = -2rn + n^2 implies 2rn = n^2 implies 2r = n quad text{(assuming (n neq 0))}]So, (n = 2r).**Relating Back to the Plane Equation:**Earlier, we had the equation of the plane passing through ((2a, 2b, 2c)):[frac{2a}{k} + frac{2b}{m} + frac{2c}{n} = 1]But we found that (k = 2p), (m = 2q), and (n = 2r). Substituting these into the equation:[frac{2a}{2p} + frac{2b}{2q} + frac{2c}{2r} = 1]Simplifying each term:[frac{a}{p} + frac{b}{q} + frac{c}{r} = 1]So, that's exactly the expression we needed to find!**Wait, Let Me Double-Check:**Hmm, let me make sure I didn't skip any steps or make any mistakes. So, we found that the intercepts (k), (m), and (n) are twice the coordinates of the center of the sphere. Then, substituting these into the plane equation, we ended up with the desired expression equal to 1. That seems consistent.Is there another way to approach this problem? Maybe using the concept of the sphere's equation directly?**Alternative Approach Using Sphere Equation:**The general equation of a sphere is:[x^2 + y^2 + z^2 + Dx + Ey + Fz + G = 0]Since the sphere passes through the origin (O = (0, 0, 0)), substituting this point gives:[0 + 0 + 0 + 0 + 0 + 0 + G = 0 implies G = 0]So, the equation simplifies to:[x^2 + y^2 + z^2 + Dx + Ey + Fz = 0]The center of the sphere is at (left(-frac{D}{2}, -frac{E}{2}, -frac{F}{2}right)), which is given as ((p, q, r)). Therefore:[p = -frac{D}{2} implies D = -2p][q = -frac{E}{2} implies E = -2q][r = -frac{F}{2} implies F = -2r]So, the equation of the sphere becomes:[x^2 + y^2 + z^2 - 2px - 2qy - 2rz = 0]Now, since the sphere passes through points (A = (k, 0, 0)), (B = (0, m, 0)), and (C = (0, 0, n)), let's substitute these into the sphere's equation.For point (A = (k, 0, 0)):[k^2 + 0 + 0 - 2pk - 0 - 0 = 0 implies k^2 - 2pk = 0 implies k(k - 2p) = 0]Since (k neq 0) (as (A) is not the origin), we have (k = 2p).Similarly, for point (B = (0, m, 0)):[0 + m^2 + 0 - 0 - 2qm - 0 = 0 implies m^2 - 2qm = 0 implies m(m - 2q) = 0]Again, (m neq 0), so (m = 2q).And for point (C = (0, 0, n)):[0 + 0 + n^2 - 0 - 0 - 2rn = 0 implies n^2 - 2rn = 0 implies n(n - 2r) = 0]Thus, (n = 2r).So, this method also leads us to (k = 2p), (m = 2q), and (n = 2r), which is consistent with what I found earlier.**Substituting Back into the Plane Equation:**As before, the plane equation is:[frac{x}{k} + frac{y}{m} + frac{z}{n} = 1]Substituting (k = 2p), (m = 2q), (n = 2r), we get:[frac{x}{2p} + frac{y}{2q} + frac{z}{2r} = 1]Since the plane passes through ((2a, 2b, 2c)), substituting this point into the plane equation:[frac{2a}{2p} + frac{2b}{2q} + frac{2c}{2r} = 1]Simplifying each term:[frac{a}{p} + frac{b}{q} + frac{c}{r} = 1]So, this confirms the result again.**Is There a Geometric Interpretation?**Thinking about this, the sphere passing through the origin and the intercepts of the plane seems to relate the center of the sphere to the intercepts in a linear way. The fact that each intercept is twice the corresponding coordinate of the center suggests a kind of symmetry or scaling.Also, the plane equation being satisfied by the fixed point ((2a, 2b, 2c)) ties together the coefficients of the plane (which are related to the sphere's center) with the fixed point. It's interesting how the algebra worked out neatly to give the sum (frac{a}{p} + frac{b}{q} + frac{c}{r} = 1).**Potential Pitfalls:**I should consider if there are any special cases or if I made any assumptions that might not hold. For instance, I assumed that (k), (m), and (n) are non-zero, which is given in the problem statement since none of the points (A), (B), or (C) coincide with the origin. So, that's safe.Another thing is that the sphere must pass through four non-coplanar points (O, A, B, C), which should uniquely define the sphere. Since A, B, C are on the axes, they are not coplanar with O unless the plane is the coordinate plane, but in this case, the plane is arbitrary except for passing through ((2a, 2b, 2c)). So, the sphere should be uniquely defined, and the center should be uniquely determined, which is given as ((p, q, r)).**Conclusion:**After working through both the geometric approach with distances and the algebraic approach using the sphere's equation, I consistently arrive at the conclusion that (frac{a}{p} + frac{b}{q} + frac{c}{r} = 1). I don't see any errors in my reasoning, and both methods corroborate each other. So, I feel confident that this is the correct answer.**Final Answer**boxed{1}