💡Alright, so I've got this problem about guests at a resort. There are 2n guests, and each one knows at least n others. The goal is to prove that we can choose four guests and seat them around a round table so that each person is next to two people they know. Hmm, okay, let's break this down.First off, let's make sure I understand the problem correctly. We have 2n guests, each with at least n acquaintances. Acquaintances are mutual, so if A knows B, B knows A. We need to find four guests who can be seated around a table such that each is adjacent to two people they know. That sounds like forming a cycle of four where each person is connected to their neighbors.I think graph theory might be a good way to model this. If I represent each guest as a vertex and each acquaintance as an edge, then the problem becomes finding a cycle of length four in this graph. But wait, it's not just any cycle; it's a cycle where each vertex has degree at least two in the cycle, which is naturally satisfied since it's a cycle.But hold on, the problem is more specific. It's not just about finding a cycle of four, but about seating them around a table such that each person is next to two they know. So, in graph terms, we need a cycle of four where each vertex is connected to its two neighbors in the cycle. That is, a 4-cycle.Given that each vertex has degree at least n, and there are 2n vertices, maybe we can use some properties of graphs with high minimum degree to find such a cycle.Let me recall some graph theory concepts. In a graph with minimum degree δ, certain subgraphs must exist. For example, Dirac's theorem states that if a graph has n vertices and each vertex has degree at least n/2, then the graph is Hamiltonian, meaning it contains a cycle that visits every vertex exactly once. But we don't need a Hamiltonian cycle here, just a 4-cycle.Another thought: maybe I can use the Pigeonhole Principle. Since each guest knows at least n others, and there are 2n guests, perhaps there's some overlap in their acquaintances that can be exploited.Let me try to approach this step by step.1. **Pick any two guests who don't know each other.** Wait, but if every guest knows at least n others, and there are 2n guests, is it possible that some guests don't know each other? Yes, because knowing at least n doesn't mean they know everyone. So, there must be some pairs who don't know each other.2. **Suppose we pick two guests, A and B, who don't know each other.** Now, A knows at least n guests, and B also knows at least n guests. Since A and B don't know each other, their acquaintances are among the remaining 2n - 2 guests.3. **Let's list the acquaintances of A and B.** A has at least n acquaintances, and B has at least n acquaintances. Together, they have at least 2n acquaintances. But there are only 2n - 2 other guests besides A and B. By the Pigeonhole Principle, some guests must be acquainted with both A and B.4. **Specifically, since there are 2n acquaintances and only 2n - 2 guests, at least two guests must be common acquaintances of both A and B.** Let's call these two guests C and D.5. **Now, we have A acquainted with C and D, and B acquainted with C and D.** So, we can form a cycle A-C-B-D-A. This is a 4-cycle where each person is seated next to two people they know.Wait, but what if A and B don't have two common acquaintances? Hmm, according to the Pigeonhole Principle, since there are 2n acquaintances and only 2n - 2 guests, the overlap must be at least 2. So, there must be at least two guests who are acquainted with both A and B.But let me double-check that. If A has n acquaintances and B has n acquaintances, the total number of acquaintances is 2n. The number of possible guests is 2n - 2. So, the overlap is at least 2n - (2n - 2) = 2. Yes, that makes sense.So, this seems to work. But what if every pair of guests knows each other? That is, the graph is complete. Then, of course, any four guests will form a 4-cycle, so the condition is trivially satisfied.But in the case where not everyone knows each other, we can always find two guests who don't know each other, and then find two common acquaintances to form the cycle.Wait, but what if n is 1? Then, 2n is 2, and each guest knows at least 1 other. But with two guests, they must know each other, so it's trivial. But the problem probably assumes n > 1.Another edge case: n = 2, so 4 guests. Each guest knows at least 2 others. If the graph is such that it's a 4-cycle, then we're done. If it's a complete graph, also done. If it's something else, like a triangle plus an isolated edge, but wait, each guest must know at least 2 others, so the isolated edge isn't possible. So, in that case, it must contain a 4-cycle.Hmm, I think this approach works. So, to summarize:- If there are two guests who don't know each other, their acquaintances must overlap by at least two guests, forming a 4-cycle.- If all guests know each other, then any four guests form a 4-cycle.Therefore, in either case, we can find four guests who can be seated around a table with each adjacent to two acquaintances.I think that covers it. I should probably write this up more formally to make sure all steps are clear.

💡Okay, so I have this problem where I'm given a sequence defined by a recurrence relation. It starts with ( u_0 = frac{1}{8} ), and for each ( k geq 0 ), the next term is given by ( u_{k+1} = 2u_k - 2u_k^2 ). The question is asking for the least value of ( k ) such that the absolute difference between ( u_k ) and the limit ( L ) is less than or equal to ( frac{1}{2^{10}} ). The options are from A) 2 to E) 6.First, I need to figure out what the limit ( L ) is. Since the sequence tends to a limit, I can assume that as ( k ) approaches infinity, ( u_{k+1} ) and ( u_k ) both approach ( L ). So, plugging ( L ) into the recurrence relation gives:[L = 2L - 2L^2]Let me solve this equation for ( L ). Subtract ( 2L ) from both sides:[L - 2L = -2L^2][-L = -2L^2]Multiply both sides by -1 to make it simpler:[L = 2L^2]Now, rearrange the equation:[2L^2 - L = 0][L(2L - 1) = 0]So, the solutions are ( L = 0 ) or ( 2L - 1 = 0 ), which gives ( L = frac{1}{2} ). Since the initial term ( u_0 = frac{1}{8} ) is positive and less than ( frac{1}{2} ), and looking at the recurrence relation, each term is a function of the previous term. Let me check if the sequence is increasing or decreasing.Calculate ( u_1 ):[u_1 = 2u_0 - 2u_0^2 = 2 times frac{1}{8} - 2 times left(frac{1}{8}right)^2 = frac{1}{4} - 2 times frac{1}{64} = frac{1}{4} - frac{1}{32} = frac{8}{32} - frac{1}{32} = frac{7}{32}]So, ( u_1 = frac{7}{32} approx 0.21875 ). Comparing to ( u_0 = 0.125 ), it's larger. So the sequence is increasing.Next, calculate ( u_2 ):[u_2 = 2u_1 - 2u_1^2 = 2 times frac{7}{32} - 2 times left(frac{7}{32}right)^2 = frac{14}{32} - 2 times frac{49}{1024} = frac{7}{16} - frac{49}{512}]Convert to a common denominator:[frac{7}{16} = frac{224}{512}, quad frac{49}{512} = frac{49}{512}][u_2 = frac{224}{512} - frac{49}{512} = frac{175}{512} approx 0.3418]So, ( u_2 ) is larger than ( u_1 ). The sequence is still increasing.Calculate ( u_3 ):[u_3 = 2u_2 - 2u_2^2 = 2 times frac{175}{512} - 2 times left(frac{175}{512}right)^2][= frac{350}{512} - 2 times frac{30625}{262144}][= frac{175}{256} - frac{30625}{131072}]Convert to a common denominator:[frac{175}{256} = frac{89600}{131072}, quad frac{30625}{131072} = frac{30625}{131072}][u_3 = frac{89600 - 30625}{131072} = frac{58975}{131072} approx 0.4492]Still increasing. Next, ( u_4 ):[u_4 = 2u_3 - 2u_3^2 = 2 times frac{58975}{131072} - 2 times left(frac{58975}{131072}right)^2]This is getting complicated. Maybe I can find a pattern or a better way to represent the sequence.Looking back at the recurrence relation:[u_{k+1} = 2u_k - 2u_k^2 = 2u_k(1 - u_k)]This looks similar to the logistic map equation, which is a common example in chaos theory, but in this case, with a parameter of 2. The logistic map is ( u_{k+1} = r u_k (1 - u_k) ), and here ( r = 2 ). For ( r = 2 ), the logistic map converges to a fixed point. The fixed points are at ( u = 0 ) and ( u = frac{1}{2} ). Since our initial value is ( frac{1}{8} ), which is between 0 and 1, and the sequence is increasing, it should converge to ( frac{1}{2} ).So, the limit ( L ) is ( frac{1}{2} ). Now, I need to find the smallest ( k ) such that ( |u_k - frac{1}{2}| leq frac{1}{2^{10}} = frac{1}{1024} ).To do this, I can compute each term until the difference from ( frac{1}{2} ) is less than or equal to ( frac{1}{1024} ).Let me compute the terms step by step:- ( u_0 = frac{1}{8} = 0.125 )- ( u_1 = 2 times 0.125 - 2 times (0.125)^2 = 0.25 - 2 times 0.015625 = 0.25 - 0.03125 = 0.21875 )- ( u_2 = 2 times 0.21875 - 2 times (0.21875)^2 = 0.4375 - 2 times 0.0478515625 = 0.4375 - 0.095703125 = 0.341796875 )- ( u_3 = 2 times 0.341796875 - 2 times (0.341796875)^2 ) - First, ( 2 times 0.341796875 = 0.68359375 ) - Then, ( (0.341796875)^2 = 0.1168 ) approximately - So, ( 2 times 0.1168 = 0.2336 ) - Therefore, ( u_3 approx 0.68359375 - 0.2336 = 0.44999375 )- ( u_4 = 2 times 0.44999375 - 2 times (0.44999375)^2 ) - First, ( 2 times 0.44999375 = 0.8999875 ) - Then, ( (0.44999375)^2 approx 0.202494 ) - So, ( 2 times 0.202494 = 0.404988 ) - Therefore, ( u_4 approx 0.8999875 - 0.404988 = 0.4949995 )- ( u_5 = 2 times 0.4949995 - 2 times (0.4949995)^2 ) - First, ( 2 times 0.4949995 = 0.989999 ) - Then, ( (0.4949995)^2 approx 0.245 ) - So, ( 2 times 0.245 = 0.49 ) - Therefore, ( u_5 approx 0.989999 - 0.49 = 0.499999 )- ( u_6 = 2 times 0.499999 - 2 times (0.499999)^2 ) - First, ( 2 times 0.499999 = 0.999998 ) - Then, ( (0.499999)^2 = 0.249999 ) - So, ( 2 times 0.249999 = 0.499998 ) - Therefore, ( u_6 approx 0.999998 - 0.499998 = 0.5 )Wait, that's interesting. So, ( u_6 ) is approximately 0.5, which is the limit. But let's check the differences:- ( |u_0 - 0.5| = |0.125 - 0.5| = 0.375 )- ( |u_1 - 0.5| = |0.21875 - 0.5| = 0.28125 )- ( |u_2 - 0.5| = |0.341796875 - 0.5| = 0.158203125 )- ( |u_3 - 0.5| = |0.44999375 - 0.5| = 0.05000625 )- ( |u_4 - 0.5| = |0.4949995 - 0.5| = 0.0050005 )- ( |u_5 - 0.5| = |0.499999 - 0.5| = 0.000001 )- ( |u_6 - 0.5| = |0.5 - 0.5| = 0 )So, the differences are decreasing rapidly. The question is asking for ( |u_k - 0.5| leq frac{1}{1024} approx 0.0009765625 ).Looking at the differences:- ( |u_4 - 0.5| approx 0.0050005 ) which is larger than ( frac{1}{1024} )- ( |u_5 - 0.5| approx 0.000001 ) which is much smaller than ( frac{1}{1024} )Wait, but according to my calculations, ( u_5 ) is already within the desired bound. However, the options don't include 5 as the answer; the options are up to 6. Wait, no, option D is 5. So, maybe 5 is the answer. But let me double-check my calculations because it seems too quick.Wait, perhaps I made an error in approximating ( u_3 ) and ( u_4 ). Let me compute them more accurately.Starting over with exact fractions:- ( u_0 = frac{1}{8} )- ( u_1 = 2 times frac{1}{8} - 2 times left(frac{1}{8}right)^2 = frac{1}{4} - 2 times frac{1}{64} = frac{1}{4} - frac{1}{32} = frac{8}{32} - frac{1}{32} = frac{7}{32} )- ( u_2 = 2 times frac{7}{32} - 2 times left(frac{7}{32}right)^2 = frac{14}{32} - 2 times frac{49}{1024} = frac{7}{16} - frac{49}{512} = frac{224}{512} - frac{49}{512} = frac{175}{512} )- ( u_3 = 2 times frac{175}{512} - 2 times left(frac{175}{512}right)^2 = frac{350}{512} - 2 times frac{30625}{262144} = frac{175}{256} - frac{30625}{131072} ) - Convert to a common denominator: ( frac{175}{256} = frac{89600}{131072} ) - So, ( u_3 = frac{89600}{131072} - frac{30625}{131072} = frac{58975}{131072} )- ( u_4 = 2 times frac{58975}{131072} - 2 times left(frac{58975}{131072}right)^2 ) - First, ( 2 times frac{58975}{131072} = frac{117950}{131072} = frac{58975}{65536} ) - Then, ( left(frac{58975}{131072}right)^2 = frac{34772250625}{17179869184} ) - So, ( 2 times frac{34772250625}{17179869184} = frac{69544501250}{17179869184} ) - Therefore, ( u_4 = frac{58975}{65536} - frac{69544501250}{17179869184} ) - Convert ( frac{58975}{65536} ) to denominator ( 17179869184 ): - ( frac{58975}{65536} = frac{58975 times 256}{65536 times 256} = frac{15111200}{16777216} ) - Wait, this is getting too messy. Maybe I should use decimal approximations for better understanding.Let me compute ( u_3 ) numerically:- ( u_3 = frac{58975}{131072} approx 0.44921875 )- ( u_4 = 2 times 0.44921875 - 2 times (0.44921875)^2 ) - ( 2 times 0.44921875 = 0.8984375 ) - ( (0.44921875)^2 approx 0.2017578125 ) - ( 2 times 0.2017578125 = 0.403515625 ) - So, ( u_4 approx 0.8984375 - 0.403515625 = 0.494921875 )- ( |u_4 - 0.5| = |0.494921875 - 0.5| = 0.005078125 )- ( u_5 = 2 times 0.494921875 - 2 times (0.494921875)^2 ) - ( 2 times 0.494921875 = 0.98984375 ) - ( (0.494921875)^2 approx 0.2449462890625 ) - ( 2 times 0.2449462890625 = 0.489892578125 ) - So, ( u_5 approx 0.98984375 - 0.489892578125 = 0.499951171875 )- ( |u_5 - 0.5| = |0.499951171875 - 0.5| = 0.000048828125 )- ( u_6 = 2 times 0.499951171875 - 2 times (0.499951171875)^2 ) - ( 2 times 0.499951171875 = 0.99990234375 ) - ( (0.499951171875)^2 approx 0.249951171875 ) - ( 2 times 0.249951171875 = 0.49990234375 ) - So, ( u_6 approx 0.99990234375 - 0.49990234375 = 0.5 )- ( |u_6 - 0.5| = 0 )Now, comparing the differences:- ( |u_4 - 0.5| approx 0.005078125 ) which is ( frac{1}{196.875} ) approximately, which is larger than ( frac{1}{1024} approx 0.0009765625 )- ( |u_5 - 0.5| approx 0.000048828125 ) which is much smaller than ( frac{1}{1024} )So, ( u_5 ) is within the desired bound. But let's check if ( u_4 ) is not. Since ( 0.005078125 > 0.0009765625 ), ( u_4 ) is not within the bound. Therefore, the least ( k ) such that ( |u_k - L| leq frac{1}{2^{10}} ) is ( k = 5 ).But wait, the options are A)2, B)3, C)4, D)5, E)6. So, D)5 is an option. However, I think I might have made a mistake because in my initial approximation, ( u_5 ) is already very close to 0.5, but perhaps the exact calculation shows that ( u_4 ) is still too far.Wait, let me compute ( |u_4 - 0.5| ) exactly:( u_4 = frac{58975}{131072} times 2 - 2 times left(frac{58975}{131072}right)^2 )Wait, no, I think I confused the steps. Let me correct that.Actually, ( u_4 ) was calculated as ( frac{58975}{131072} ) after ( u_3 ). Wait, no, ( u_3 ) was ( frac{58975}{131072} ), and then ( u_4 ) was calculated from that.Wait, perhaps I should use a different approach. Let me consider the transformation ( v_k = 2u_k - 1 ). Then, the recurrence becomes:( v_{k+1} = 2u_k - 1 - 2u_k^2 + 2u_k - 1 ) Wait, no, let me do it properly.Given ( u_{k+1} = 2u_k - 2u_k^2 ), let me express this in terms of ( v_k ):Let ( v_k = 2u_k - 1 ). Then, ( u_k = frac{v_k + 1}{2} ).Substitute into the recurrence:( u_{k+1} = 2 times frac{v_k + 1}{2} - 2 times left(frac{v_k + 1}{2}right)^2 )Simplify:( u_{k+1} = (v_k + 1) - 2 times frac{(v_k + 1)^2}{4} )( = v_k + 1 - frac{(v_k + 1)^2}{2} )Expand ( (v_k + 1)^2 ):( = v_k + 1 - frac{v_k^2 + 2v_k + 1}{2} )( = v_k + 1 - frac{v_k^2}{2} - v_k - frac{1}{2} )Simplify:( = (v_k - v_k) + (1 - frac{1}{2}) - frac{v_k^2}{2} )( = frac{1}{2} - frac{v_k^2}{2} )But ( u_{k+1} = frac{v_{k+1} + 1}{2} ), so:( frac{v_{k+1} + 1}{2} = frac{1}{2} - frac{v_k^2}{2} )Multiply both sides by 2:( v_{k+1} + 1 = 1 - v_k^2 )Subtract 1:( v_{k+1} = -v_k^2 )So, the transformed sequence is ( v_{k+1} = -v_k^2 ). This is a simpler recurrence.Given ( u_0 = frac{1}{8} ), then ( v_0 = 2 times frac{1}{8} - 1 = frac{1}{4} - 1 = -frac{3}{4} ).Now, let's compute the ( v_k ) sequence:- ( v_0 = -frac{3}{4} )- ( v_1 = -(-frac{3}{4})^2 = -frac{9}{16} )- ( v_2 = -(-frac{9}{16})^2 = -frac{81}{256} )- ( v_3 = -(-frac{81}{256})^2 = -frac{6561}{65536} )- ( v_4 = -(-frac{6561}{65536})^2 = -frac{43046721}{4294967296} )Notice that each ( v_k ) is negative and squaring makes it positive, then the next term is negative again. The magnitude is decreasing because each time we square a fraction less than 1 in absolute value, it becomes smaller.Since ( v_k ) approaches 0 as ( k ) increases, ( u_k = frac{v_k + 1}{2} ) approaches ( frac{1}{2} ).Now, we need ( |u_k - frac{1}{2}| leq frac{1}{1024} ). Since ( u_k = frac{v_k + 1}{2} ), then:( |u_k - frac{1}{2}| = |frac{v_k + 1}{2} - frac{1}{2}| = |frac{v_k}{2}| = frac{|v_k|}{2} )So, we need ( frac{|v_k|}{2} leq frac{1}{1024} ), which implies ( |v_k| leq frac{1}{512} ).Looking at the ( v_k ) sequence:- ( |v_0| = frac{3}{4} = 0.75 )- ( |v_1| = frac{9}{16} = 0.5625 )- ( |v_2| = frac{81}{256} approx 0.3164 )- ( |v_3| = frac{6561}{65536} approx 0.1001 )- ( |v_4| = frac{43046721}{4294967296} approx 0.01002 )- ( |v_5| = left(frac{43046721}{4294967296}right)^2 approx (0.01002)^2 approx 0.0001004 )So, ( |v_4| approx 0.01002 ), which is greater than ( frac{1}{512} approx 0.001953125 ). Therefore, ( |v_4| > frac{1}{512} ).Next, ( |v_5| approx 0.0001004 ), which is less than ( frac{1}{512} ). Therefore, ( k = 5 ) is the least value such that ( |u_k - frac{1}{2}| leq frac{1}{1024} ).But wait, according to the options, D)5 is an option, but I thought earlier that ( u_5 ) was already within the bound. However, based on the ( v_k ) sequence, ( |v_5| ) is the first term less than ( frac{1}{512} ), so ( k = 5 ) is the answer.But let me cross-verify with the earlier decimal approximations:- ( |u_4 - 0.5| approx 0.005078125 ) which is ( frac{1}{196.875} ), larger than ( frac{1}{1024} )- ( |u_5 - 0.5| approx 0.000048828125 ), which is much smaller than ( frac{1}{1024} )So, indeed, ( k = 5 ) is the least value. However, the options include E)6, which is higher. But according to both methods, ( k = 5 ) suffices.Wait, but in the transformed sequence, ( v_k ), ( |v_5| ) is the first term less than ( frac{1}{512} ), which corresponds to ( u_5 ). So, ( k = 5 ) is correct.But the options are A)2, B)3, C)4, D)5, E)6. So, D)5 is the correct answer.However, I'm a bit confused because in the initial decimal approximations, ( u_5 ) was already very close, but according to the transformed sequence, ( v_5 ) is the first term below ( frac{1}{512} ). So, yes, ( k = 5 ) is correct.But wait, let me check the exact value of ( |v_4| ):( |v_4| = frac{43046721}{4294967296} approx 0.01002 ), which is approximately ( frac{1}{99.8} ), which is larger than ( frac{1}{512} approx 0.001953 ). Therefore, ( |v_4| > frac{1}{512} ), so ( k = 4 ) is not sufficient.Thus, the least ( k ) is 5, which is option D.

💡First, determine the total number of packs by dividing the total marbles by the number of marbles per pack.Next, calculate the number of packs Leo kept for himself.Subtract the packs kept from the total packs to find out how many packs were given away.Calculate the number of packs Neil received by finding one-eighth of the total packs.Subtract the packs given to Neil from the total given away to find out how many packs Manny received.Finally, determine the fraction of the total packs that Manny received by dividing the packs given to Manny by the total number of packs.

💡Okay, so I have this problem here where I need to find the ratio of the largest element in the set {1, 2, 2², 2³, ..., 2¹⁵} to the sum of all the other elements. Hmm, let's see. The set is a geometric sequence where each term is double the previous one, starting from 1 up to 2¹⁵. First, I need to identify the largest element in the set. Since each term is 2 raised to an increasing power, the largest element is clearly 2¹⁵. That part seems straightforward.Next, I need to find the sum of all the other elements. So, that would be the sum of 1, 2, 2², 2³, ..., up to 2¹⁴. Wait, because 2¹⁵ is the largest and we're excluding it, right? So, the sum is from 2⁰ to 2¹⁴. I remember that the sum of a geometric series can be calculated using the formula S = a(rⁿ - 1)/(r - 1), where 'a' is the first term, 'r' is the common ratio, and 'n' is the number of terms. In this case, the first term 'a' is 1, the common ratio 'r' is 2, and the number of terms is 15 because we're starting from 2⁰ up to 2¹⁴, which is 15 terms in total.So, plugging in the values, the sum S would be (1*(2¹⁵ - 1))/(2 - 1). Simplifying that, the denominator becomes 1, so S is just 2¹⁵ - 1. Now, the ratio we're looking for is the largest element, which is 2¹⁵, divided by the sum of the other elements, which is 2¹⁵ - 1. So, the ratio is 2¹⁵ / (2¹⁵ - 1). Hmm, that looks interesting. Let me compute that. 2¹⁵ is 32768, so 32768 divided by (32768 - 1) is 32768 / 32767. That's approximately 1.0000305. So, it's just a little over 1. But looking at the answer choices, they are 0.5, 1, 2, 3, and 4. So, 1 is one of the options, and since 32768 / 32767 is very close to 1, I think the answer is 1. Wait, let me double-check. Maybe I made a mistake in counting the number of terms. The set starts at 1, which is 2⁰, and goes up to 2¹⁵. So, that's 16 terms in total. But since we're excluding the largest term, 2¹⁵, we're summing 15 terms. So, the sum S should be from 2⁰ to 2¹⁴, which is 15 terms. Using the formula again, S = (2¹⁵ - 1)/(2 - 1) = 2¹⁵ - 1. So, that part is correct. Therefore, the ratio is 2¹⁵ / (2¹⁵ - 1), which is approximately 1.0000305, so it's just over 1. But since the answer choices don't have a value just over 1, and 1 is an option, I think 1 is the correct answer. Alternatively, maybe there's a different way to look at it. If I consider that 2¹⁵ is twice 2¹⁴, and the sum of all the previous terms is 2¹⁵ - 1, which is just one less than 2¹⁵. So, 2¹⁵ is almost equal to the sum of all the previous terms, making the ratio just over 1. But since 1 is an option, and the others are much larger, 1 must be the answer.Yeah, I think that's it. The ratio is approximately 1, so the answer is 1.

💡Okay, so I have this problem where I'm given five real numbers, a1 through a5, and the sum of their squares is 1. I need to prove that the smallest difference between any two of these numbers squared doesn't exceed 1/10. Hmm, that seems interesting. Let me try to break it down.First, I should probably order these numbers to make things simpler. Let me assume without loss of generality that a1 ≤ a2 ≤ a3 ≤ a4 ≤ a5. That way, I can think about the differences between consecutive numbers and so on. Now, the problem is about the least of the numbers (ai - aj)^2 where i ≠ j. So, I need to find the minimum value among all these squared differences and show that it's at most 1/10. Maybe I can define m as the minimum absolute difference between any two of these numbers. So, m = min_{i≠j} |ai - aj|. Since I've ordered them, the differences between consecutive terms will be at least m. That is, a2 - a1 ≥ m, a3 - a2 ≥ m, and so on. If I consider the differences between non-consecutive terms, like a3 - a1, that would be at least 2m, right? Because it's the sum of two consecutive differences, each at least m. Similarly, a4 - a1 would be at least 3m, and a5 - a1 would be at least 4m. So, in general, for any i and j where i > j, ai - aj is at least (i - j)m. That might help me when I consider the sum of all squared differences.Let me think about the sum of all squared differences. There are C(5,2) = 10 pairs of (i, j) where i ≠ j. So, I can write the sum of (ai - aj)^2 for all i < j. Since each difference is at least (i - j)m, the square of each difference is at least (i - j)^2 m^2. Therefore, the sum of all squared differences is at least m^2 times the sum of (i - j)^2 for all i < j.Let me compute that sum. For i=2 to 5 and j=1 to i-1, the differences (i - j) are:- For i=2: j=1, difference=1- For i=3: j=1,2, differences=2,1- For i=4: j=1,2,3, differences=3,2,1- For i=5: j=1,2,3,4, differences=4,3,2,1So, summing up all these squared differences:1^2 + (2^2 + 1^2) + (3^2 + 2^2 + 1^2) + (4^2 + 3^2 + 2^2 + 1^2)Calculating each part:- 1^2 = 1- 2^2 + 1^2 = 4 + 1 = 5- 3^2 + 2^2 + 1^2 = 9 + 4 + 1 = 14- 4^2 + 3^2 + 2^2 + 1^2 = 16 + 9 + 4 + 1 = 30Adding them all together: 1 + 5 + 14 + 30 = 50So, the sum of (i - j)^2 for all i < j is 50. Therefore, the sum of (ai - aj)^2 is at least 50m^2.But wait, I also know from the problem statement that the sum of the squares of the numbers is 1. Maybe I can relate that to the sum of squared differences.I recall that the sum of squared differences can be expressed in terms of the sum of squares and the square of sums. Specifically, the formula is:Sum_{i < j} (ai - aj)^2 = (n - 1) Sum_{i=1}^n a_i^2 - Sum_{i=1}^n a_i^2Wait, no, that doesn't seem right. Let me think again.Actually, the sum of squared differences is related to the variance. The formula is:Sum_{i < j} (ai - aj)^2 = n Sum_{i=1}^n a_i^2 - (Sum_{i=1}^n a_i)^2Where n is the number of terms, which is 5 in this case.So, plugging in n=5:Sum_{i < j} (ai - aj)^2 = 5 * Sum_{i=1}^5 a_i^2 - (Sum_{i=1}^5 a_i)^2Given that Sum_{i=1}^5 a_i^2 = 1, this becomes:5 * 1 - (Sum_{i=1}^5 a_i)^2 = 5 - (Sum_{i=1}^5 a_i)^2Since the square of the sum is always non-negative, the maximum value of Sum_{i < j} (ai - aj)^2 is 5, because (Sum a_i)^2 is subtracted.But earlier, I found that Sum_{i < j} (ai - aj)^2 is at least 50m^2. So, combining these two results:50m^2 ≤ 5Dividing both sides by 50:m^2 ≤ 5 / 50 = 1/10Therefore, m^2 ≤ 1/10, which means that the least of the numbers (ai - aj)^2 does not exceed 1/10.Wait, let me double-check my steps to make sure I didn't make a mistake.1. Ordered the numbers: a1 ≤ a2 ≤ a3 ≤ a4 ≤ a5. That seems fine.2. Defined m as the minimum difference. Correct.3. Noted that consecutive differences are at least m, so non-consecutive differences are multiples of m. That makes sense.4. Calculated the sum of (i - j)^2 for all i < j as 50. Let me verify that:- For i=2: 1^2 = 1- For i=3: 2^2 + 1^2 = 4 + 1 = 5- For i=4: 3^2 + 2^2 + 1^2 = 9 + 4 + 1 = 14- For i=5: 4^2 + 3^2 + 2^2 + 1^2 = 16 + 9 + 4 + 1 = 30- Total: 1 + 5 + 14 + 30 = 50. Yep, that's correct.5. Used the formula for the sum of squared differences: 5 * sum a_i^2 - (sum a_i)^2. Since sum a_i^2 =1, it's 5 - (sum a_i)^2. That seems right.6. Since (sum a_i)^2 is non-negative, the sum of squared differences is at most 5. Therefore, 50m^2 ≤ 5, leading to m^2 ≤ 1/10.Yes, that all checks out. So, I think my reasoning is solid. The key was to relate the sum of squared differences to both the minimum difference and the given condition on the sum of squares. By using the formula that connects these, I could bound the minimum squared difference.I guess another way to think about it is that if all the numbers are too spread out, their squares would sum to more than 1, but since they sum to exactly 1, they can't be too spread out. Hence, at least two of them must be close together, with their squared difference at most 1/10.Yeah, that makes sense. I think I've got it.

💡Okay, so I have this problem here: I need to find how many 15-letter arrangements of 5 A's, 5 B's, and 5 C's have no B's in the first 5 letters, no C's in the next 5 letters, and no A's in the last 5 letters. The options are given as A) through E), with B being a summation involving binomial coefficients cubed. Hmm, okay.Let me break this down. The total arrangement is 15 letters, with 5 of each letter. But there are restrictions on where certain letters can go. Specifically:1. The first 5 letters can't have any B's. So, they can only have A's and C's.2. The next 5 letters can't have any C's. So, they can only have A's and B's.3. The last 5 letters can't have any A's. So, they can only have B's and C's.So, each segment of 5 letters has restrictions on which letters can be in them. I need to figure out how to count the number of ways to arrange the letters under these constraints.Let me think about how to model this. Maybe I can consider each segment separately and then see how the letters are distributed across these segments.Let's denote:- First 5 letters: can have A's and C's.- Middle 5 letters: can have A's and B's.- Last 5 letters: can have B's and C's.Since we have exactly 5 A's, 5 B's, and 5 C's in total, the distribution of these letters across the three segments must account for all of them.Let me define variables to represent the number of each letter in each segment. Let's say:- In the first 5 letters, there are m A's and (5 - m) C's.- In the middle 5 letters, since we can't have C's, we must have (5 - m) A's and m B's. Wait, why? Because we've already used m A's in the first segment, so the remaining A's must be in the middle segment. Since we have 5 A's in total, the middle segment will have (5 - m) A's. Then, the rest of the letters in the middle segment must be B's, which would be m B's.- Similarly, in the last 5 letters, we can't have A's, so they must consist of B's and C's. We've already used m B's in the middle segment, so the remaining B's are (5 - m). Therefore, the last segment will have (5 - m) B's and m C's.Wait, let me verify that. If the first segment has m A's, then the middle segment must have (5 - m) A's because we have a total of 5 A's. Since the middle segment can't have C's, the remaining letters must be B's, so that's m B's. Then, in the last segment, we can't have A's, so we have to place the remaining B's and C's. We've already used m B's in the middle, so we have (5 - m) B's left. Similarly, in the first segment, we've used (5 - m) C's, so we have m C's left. Therefore, the last segment will have (5 - m) B's and m C's.Okay, that seems consistent. So, for each m from 0 to 5, we can have a distribution where the first segment has m A's and (5 - m) C's, the middle segment has (5 - m) A's and m B's, and the last segment has (5 - m) B's and m C's.Now, to find the total number of arrangements, we need to calculate the number of ways to arrange the letters in each segment for each m and then sum over all possible m.So, for each m, the number of ways to arrange the first segment is the number of ways to choose m positions out of 5 for A's, which is C(5, m). Similarly, the number of ways to arrange the middle segment is C(5, 5 - m) for A's, which is the same as C(5, m). And the number of ways to arrange the last segment is C(5, 5 - m) for B's, which is again C(5, m). So, for each m, the number of arrangements is [C(5, m)]^3.Therefore, the total number of arrangements is the sum over m from 0 to 5 of [C(5, m)]^3. That is, the answer is option B.Wait, let me make sure I didn't make a mistake here. So, for each m, we're calculating the number of ways to distribute the letters in each segment, and since each segment's arrangement is independent once we fix m, we multiply the combinations for each segment. Then, since m can vary from 0 to 5, we sum over all possible m.Yes, that makes sense. So, the total number of valid arrangements is the sum from m=0 to 5 of [C(5, m)]^3. Therefore, the correct answer is B.