diff --git "a/HarvardMIT/segmented/en-282-2025-feb-comb-solutions.jsonl" "b/HarvardMIT/segmented/en-282-2025-feb-comb-solutions.jsonl"
--- "a/HarvardMIT/segmented/en-282-2025-feb-comb-solutions.jsonl"
+++ "b/HarvardMIT/segmented/en-282-2025-feb-comb-solutions.jsonl"
@@ -5,9 +5,9 @@
 {"year": "2025", "tier": "T4", "problem_label": "3", "problem_type": "Combinatorics", "exam": "HMMT", "problem": "Ben has 16 balls labeled 1, 2, 3, ..., 16, as well as 4 indistinguishable boxes. Two balls are neighbors if their labels differ by 1. Compute the number of ways for him to put 4 balls in each box such that each ball is in the same box as at least one of its neighbors. (The order in which the balls are placed does not matter.)", "solution": "Answer: 105", "metadata": {"resource_path": "HarvardMIT/segmented/en-282-2025-feb-comb-solutions.jsonl", "problem_match": "\n3. ", "solution_match": "\nProposed by: Benjamin Shimabukuro  \n\n"}}
 {"year": "2025", "tier": "T4", "problem_label": "3", "problem_type": "Combinatorics", "exam": "HMMT", "problem": "Ben has 16 balls labeled 1, 2, 3, ..., 16, as well as 4 indistinguishable boxes. Two balls are neighbors if their labels differ by 1. Compute the number of ways for him to put 4 balls in each box such that each ball is in the same box as at least one of its neighbors. (The order in which the balls are placed does not matter.)", "solution": "Each box must either contain a single group of four consecutive balls (e.g. 5, 6, 7, 8) or two groups of two consecutive balls (e.g. 5, 6, 9, 10). Since all groups have even lengths, this means that 1 and 2 are in the same group, 3 and 4 are in the same group, and so on. We can think of each of these 8 pairs of balls as an individual unit, so the answer is equal to the number of ways to put 8 objects in 4 indistinguishable boxes, where each box has 2 objects without any additional restrictions. The number of ways to do this is \\(\\frac{8!}{2^{4}\\cdot 4!} = \\boxed{105}\\) .", "metadata": {"resource_path": "HarvardMIT/segmented/en-282-2025-feb-comb-solutions.jsonl", "problem_match": "\n3. ", "solution_match": "\nSolution: "}}
 {"year": "2025", "tier": "T4", "problem_label": "4", "problem_type": "Combinatorics", "exam": "HMMT", "problem": "Sophie is at \\((0,0)\\) on a coordinate grid and would like to get to \\((3,3)\\) . If Sophie is at \\((x,y)\\) , in a single step she can move to one of \\((x + 1,y)\\) , \\((x,y + 1)\\) , \\((x - 1,y + 1)\\) , or \\((x + 1,y - 1)\\) . She cannot revisit any points along her path, and neither her \\(x\\) - coordinate nor her \\(y\\) - coordinate can ever be less than 0 or greater than 3. Compute the number of ways for Sophie to reach \\((3,3)\\) .", "solution": "Answer: 2304", "metadata": {"resource_path": "HarvardMIT/segmented/en-282-2025-feb-comb-solutions.jsonl", "problem_match": "\n4. ", "solution_match": "\nProposed by: Derek Liu  \n\n"}}
-{"year": "2025", "tier": "T4", "problem_label": "4", "problem_type": "Combinatorics", "exam": "HMMT", "problem": "Sophie is at \\((0,0)\\) on a coordinate grid and would like to get to \\((3,3)\\) . If Sophie is at \\((x,y)\\) , in a single step she can move to one of \\((x + 1,y)\\) , \\((x,y + 1)\\) , \\((x - 1,y + 1)\\) , or \\((x + 1,y - 1)\\) . She cannot revisit any points along her path, and neither her \\(x\\) - coordinate nor her \\(y\\) - coordinate can ever be less than 0 or greater than 3. Compute the number of ways for Sophie to reach \\((3,3)\\) .", "solution": "Let a lateral move refer to one which is either up or right. Then the lateral moves are the only ones which increase Kelvin's sum of coordinates by 1, while all other moves do not change the sum, so Kelvin must make 6 of them, one to increase this sum from \\(i\\) to \\(i + 1\\) for each \\(i \\in [0, 5]\\) .  \n\n![](data:image/jpeg;base64,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)\n  \n\nWe claim there exists a unique path corresponding to each set of 6 lateral moves chosen in this way. Indeed, the diagonal moves allow Kelvin to get from any point on \\(x + y = i\\) to any second point on \\(x + y = i\\) in exactly one way.  \n\nObserve that when \\(i \\leq 2\\) , the number of lateral moves that increase the sum of coordinates from \\(i\\) to \\(i + 1\\) is \\(2i\\) , as is the number that increase it from \\(5 - i\\) to \\(6 - i\\) . Thus the answer is \\(2 \\cdot 4 \\cdot 6 \\cdot 6 \\cdot 4 \\cdot 2 = \\lfloor 2304 \\rfloor\\) .", "metadata": {"resource_path": "HarvardMIT/segmented/en-282-2025-feb-comb-solutions.jsonl", "problem_match": "\n4. ", "solution_match": "\nSolution: "}}
+{"year": "2025", "tier": "T4", "problem_label": "4", "problem_type": "Combinatorics", "exam": "HMMT", "problem": "Sophie is at \\((0,0)\\) on a coordinate grid and would like to get to \\((3,3)\\) . If Sophie is at \\((x,y)\\) , in a single step she can move to one of \\((x + 1,y)\\) , \\((x,y + 1)\\) , \\((x - 1,y + 1)\\) , or \\((x + 1,y - 1)\\) . She cannot revisit any points along her path, and neither her \\(x\\) - coordinate nor her \\(y\\) - coordinate can ever be less than 0 or greater than 3. Compute the number of ways for Sophie to reach \\((3,3)\\) .", "solution": "Let a lateral move refer to one which is either up or right. Then the lateral moves are the only ones which increase Kelvin's sum of coordinates by 1, while all other moves do not change the sum, so Kelvin must make 6 of them, one to increase this sum from \\(i\\) to \\(i + 1\\) for each \\(i \\in [0, 5]\\) .  \n\n![md5:5de42bbe7b204ff4624cd4bc2816e63f](5de42bbe7b204ff4624cd4bc2816e63f.jpeg)\n  \n\nWe claim there exists a unique path corresponding to each set of 6 lateral moves chosen in this way. Indeed, the diagonal moves allow Kelvin to get from any point on \\(x + y = i\\) to any second point on \\(x + y = i\\) in exactly one way.  \n\nObserve that when \\(i \\leq 2\\) , the number of lateral moves that increase the sum of coordinates from \\(i\\) to \\(i + 1\\) is \\(2i\\) , as is the number that increase it from \\(5 - i\\) to \\(6 - i\\) . Thus the answer is \\(2 \\cdot 4 \\cdot 6 \\cdot 6 \\cdot 4 \\cdot 2 = \\lfloor 2304 \\rfloor\\) .", "metadata": {"resource_path": "HarvardMIT/segmented/en-282-2025-feb-comb-solutions.jsonl", "problem_match": "\n4. ", "solution_match": "\nSolution: "}}
 {"year": "2025", "tier": "T4", "problem_label": "5", "problem_type": "Combinatorics", "exam": "HMMT", "problem": "In an \\(11 \\times 11\\) grid of cells, each pair of edge-adjacent cells is connected by a door. Karthik wants to walk a path in this grid. He can start in any cell, but he must end in the same cell he started in, and he cannot go through any door more than once (not even in opposite directions). Compute the maximum number of doors he can go through in such a path.", "solution": "Answer: 200", "metadata": {"resource_path": "HarvardMIT/segmented/en-282-2025-feb-comb-solutions.jsonl", "problem_match": "\n5. ", "solution_match": "\nProposed by: Derek Liu  \n\n"}}
-{"year": "2025", "tier": "T4", "problem_label": "5", "problem_type": "Combinatorics", "exam": "HMMT", "problem": "In an \\(11 \\times 11\\) grid of cells, each pair of edge-adjacent cells is connected by a door. Karthik wants to walk a path in this grid. He can start in any cell, but he must end in the same cell he started in, and he cannot go through any door more than once (not even in opposite directions). Compute the maximum number of doors he can go through in such a path.", "solution": "![](data:image/jpeg;base64,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)\n  \n\nThis is simply asking for the longest circuit in the adjacency graph of this grid. Note that this grid has \\(4 \\cdot 9 = 36\\) cells of odd degree, 9 along each side. If we color the cells with checkerboard colors so that the corners are black, then 20 of these 36 cells are white. An Eulerian circuit uses an even number of doors from each cell, so at least one door from each of these cells goes unused. No door connects two white cells, so at least 20 doors are unused, leaving at most \\(2 \\cdot 10 \\cdot 11 - 20 = \\lfloor 200 \\rfloor\\) doors crossed.\n\n\n\nTo see this is achievable, we first delete the bottom- right cell and its 2 doors, as well as the top- left cell and its 2 doors. This leaves 8 odd- degree cells along each side of the grid; we can delete 4 doors along each to cover all remaining odd- degree cells, for 20 total doors deleted. The resulting graph is connected and has no odd- degree cells, so it must have an Eulerian circuit. This circuit is our desired path.", "metadata": {"resource_path": "HarvardMIT/segmented/en-282-2025-feb-comb-solutions.jsonl", "problem_match": "\n5. ", "solution_match": "\nSolution:  \n\n"}}
+{"year": "2025", "tier": "T4", "problem_label": "5", "problem_type": "Combinatorics", "exam": "HMMT", "problem": "In an \\(11 \\times 11\\) grid of cells, each pair of edge-adjacent cells is connected by a door. Karthik wants to walk a path in this grid. He can start in any cell, but he must end in the same cell he started in, and he cannot go through any door more than once (not even in opposite directions). Compute the maximum number of doors he can go through in such a path.", "solution": "![md5:7a6cf169c1f9e776c88f599e1de5b658](7a6cf169c1f9e776c88f599e1de5b658.jpeg)\n  \n\nThis is simply asking for the longest circuit in the adjacency graph of this grid. Note that this grid has \\(4 \\cdot 9 = 36\\) cells of odd degree, 9 along each side. If we color the cells with checkerboard colors so that the corners are black, then 20 of these 36 cells are white. An Eulerian circuit uses an even number of doors from each cell, so at least one door from each of these cells goes unused. No door connects two white cells, so at least 20 doors are unused, leaving at most \\(2 \\cdot 10 \\cdot 11 - 20 = \\lfloor 200 \\rfloor\\) doors crossed.\n\n\n\nTo see this is achievable, we first delete the bottom- right cell and its 2 doors, as well as the top- left cell and its 2 doors. This leaves 8 odd- degree cells along each side of the grid; we can delete 4 doors along each to cover all remaining odd- degree cells, for 20 total doors deleted. The resulting graph is connected and has no odd- degree cells, so it must have an Eulerian circuit. This circuit is our desired path.", "metadata": {"resource_path": "HarvardMIT/segmented/en-282-2025-feb-comb-solutions.jsonl", "problem_match": "\n5. ", "solution_match": "\nSolution:  \n\n"}}
 {"year": "2025", "tier": "T4", "problem_label": "6", "problem_type": "Combinatorics", "exam": "HMMT", "problem": "Compute the number of ways to pick two rectangles in a \\(5 \\times 5\\) grid of squares such that the edges of the rectangles lie on the lines of the grid and the rectangles do not overlap at their interiors, edges, or vertices. The order in which the rectangles are chosen does not matter.", "solution": "Answer: 6300", "metadata": {"resource_path": "HarvardMIT/segmented/en-282-2025-feb-comb-solutions.jsonl", "problem_match": "\n6. ", "solution_match": "\nProposed by: Jacob Paltrowitz  \n\n"}}
 {"year": "2025", "tier": "T4", "problem_label": "6", "problem_type": "Combinatorics", "exam": "HMMT", "problem": "Compute the number of ways to pick two rectangles in a \\(5 \\times 5\\) grid of squares such that the edges of the rectangles lie on the lines of the grid and the rectangles do not overlap at their interiors, edges, or vertices. The order in which the rectangles are chosen does not matter.", "solution": "A rectangle can be specified by two intervals, one specifying its horizontal extent ( \\(x\\) - coordinates of left and right sides) and one specifying its vertical extent ( \\(y\\) - coordinates of bottom and top sides). For the rectangles to not overlap, we need either the horizontal intervals or the vertical intervals to be disjoint (possibly both).  \n\nFirst, we will count the number of ways for the horizontal intervals to be disjoint. Let these intervals be \\([a,b]\\) and \\([c,d]\\) . As the order of the rectangles does not matter, we can assume without loss of generality that \\(a< c\\) , so \\(a< b< c< d\\) . Then there are \\(\\binom{6}{4}\\) choices for \\(a\\) , \\(b\\) , \\(c\\) , and \\(d\\) . There are no restrictions on the vertical intervals, so the number of ways to choose them is \\(\\binom{6}{2}^{2}\\) . Thus, the total number of pairs of rectangles for which the horizontal intervals are disjoint is \\(\\binom{6}{4}\\binom{6}{2}^{2}\\) .  \n\nBy symmetry, the total number of pairs of rectangles for which the vertical intervals are disjoint is the same. It remains to count the number of ways for both the horizontal and vertical intervals to be disjoint. Again, let the horizontal intervals be \\([a,b]\\) and \\([c,d]\\) , and let the vertical intervals be \\([e,f]\\) and \\([g,h]\\) . We can assume \\(a< b< c< d\\) without loss of generality, so there are \\(\\binom{6}{4}\\) ways to choose the horizontal intervals. However, the cases \\(e< f< g< h\\) and \\(g< h< e< f\\) are now distinct, so there are \\(2\\binom{6}{4}\\) ways to choose the vertical intervals. Therefore, there are \\(2\\binom{6}{4}^{2}\\) pairs of rectangles for which both the horizontal and vertical intervals are disjoint.  \n\nThrough inclusion- exclusion, we get the final answer of  \n\n\\[2\\cdot \\binom{6}{4}\\binom{6}{2}^{2} - 2\\cdot \\binom{6}{4}^{2} = \\boxed{6300}.\\]", "metadata": {"resource_path": "HarvardMIT/segmented/en-282-2025-feb-comb-solutions.jsonl", "problem_match": "\n6. ", "solution_match": "\nSolution: "}}
 {"year": "2025", "tier": "T4", "problem_label": "7", "problem_type": "Combinatorics", "exam": "HMMT", "problem": "Compute the number of ways to arrange 3 copies of each of the 26 lowercase letters of the English alphabet such that for any two distinct letters \\(x_{1}\\) and \\(x_{2}\\) , the number of \\(x_{2}\\) 's between the first and second occurrences of \\(x_{1}\\) equals the number of \\(x_{2}\\) 's between the second and third occurrences of \\(x_{1}\\) .", "solution": "Answer: 22526!", "metadata": {"resource_path": "HarvardMIT/segmented/en-282-2025-feb-comb-solutions.jsonl", "problem_match": "\n7. ", "solution_match": "\nProposed by: Derek Liu  \n\n"}}
@@ -15,6 +15,6 @@
 {"year": "2025", "tier": "T4", "problem_label": "8", "problem_type": "Combinatorics", "exam": "HMMT", "problem": "Albert writes 2025 numbers \\(a_{1}\\) , ..., \\(a_{2025}\\) in a circle on a blackboard. Initially, each of the numbers is uniformly and independently sampled at random from the interval \\([0,1]\\) . Then, each second, he simultaneously replaces \\(a_{i}\\) with \\(\\max (a_{i - 1}, a_{i}, a_{i + 1})\\) for all \\(i = 1, 2, \\ldots , 2025\\) (where \\(a_{0} = a_{2025}\\) and \\(a_{2026} = a_{1}\\) ). Compute the expected value of the number of distinct values remaining after 100 seconds.", "solution": "Answer: \\(\\boxed{2025}\\)", "metadata": {"resource_path": "HarvardMIT/segmented/en-282-2025-feb-comb-solutions.jsonl", "problem_match": "\n8. ", "solution_match": "\nProposed by: Isaac Zhu  \n\n"}}
 {"year": "2025", "tier": "T4", "problem_label": "8", "problem_type": "Combinatorics", "exam": "HMMT", "problem": "Albert writes 2025 numbers \\(a_{1}\\) , ..., \\(a_{2025}\\) in a circle on a blackboard. Initially, each of the numbers is uniformly and independently sampled at random from the interval \\([0,1]\\) . Then, each second, he simultaneously replaces \\(a_{i}\\) with \\(\\max (a_{i - 1}, a_{i}, a_{i + 1})\\) for all \\(i = 1, 2, \\ldots , 2025\\) (where \\(a_{0} = a_{2025}\\) and \\(a_{2026} = a_{1}\\) ). Compute the expected value of the number of distinct values remaining after 100 seconds.", "solution": "We can assume that the initial numbers are all distinct, since this occurs with probability 1. For clarity, we denote the value of \\(a_{i}\\) after \\(t\\) seconds as \\(a_{i,t}\\) . The index \\(i\\) is taken mod 2025.  \n\nIn general, after \\(k< 1012\\) seconds, we claim the expected number of distinct values remaining is \\(\\frac{2025}{k + 1}\\) . To show this, we first prove that for any remaining value, its appearances are consecutive. Indeed, note that for all \\(i\\) and \\(k\\) ,  \n\n\\[a_{i,k} = \\max (a_{i - 1,k - 1}, a_{i,k - 1}, a_{i + 1,k - 1}) = \\max (a_{i - 2,k - 2}, \\ldots , a_{i + 2,k - 2}) = \\dots = \\max (a_{i - k,0}, \\ldots , a_{i + k,0}).\\]  \n\nGiven an initial number \\(a_{c,0}\\) , let \\(j_{1}\\) and \\(j_{2}\\) be the smallest positive integers such that \\(a_{c - j_{1},0} > a_{c,0}\\) and \\(a_{c + j_{2},0} > a_{c,0}\\) . As the initial numbers are distinct, we conclude \\(a_{i,k} = a_{c,0}\\) if and only if \\(\\{i - k, i - k + 1, \\ldots , i + k\\}\\) contains \\(c\\) but neither \\(c - j_{1}\\) nor \\(c + j_{2}\\) (mod 2025). The indices \\(i\\) that satisfy this are clearly consecutive.\n\n\n\nNow consider the indicator variables  \n\n\\[\\mathbf{1}_{i} = \\left\\{ \\begin{array}{ll}1 & \\mathrm{if~}a_{i,k}\\neq a_{i + 1,k}\\\\ 0 & \\mathrm{otherwise} \\end{array} \\right.\\]  \n\nfor \\(1\\leq i\\leq 2025\\) . Note that the number of distinct values on the board after \\(k\\) seconds is simply \\(\\textstyle \\sum_{i = 1}^{2025}\\mathbf{1}_{i}\\) . By linearity of expectation, it suffices to compute \\(\\mathbb{E}[\\mathbf{1}_{i}]\\) for each \\(i\\) . Recall  \n\n\\[a_{i,k} = \\max (a_{i - k,0},\\ldots ,a_{i + k,0})\\]  \n\nfor each \\(1\\leq i\\leq 2025\\) . Hence, the condition \\(a_{i,k}\\neq a_{i + 1,k}\\) can be written as  \n\n\\[\\max (a_{i - k,0},\\ldots ,a_{i + k,0})\\neq \\max (a_{i + 1 - k,0},\\ldots ,a_{i + 1 + k,0}).\\]  \n\nThis is the case if and only if \\(\\max (a_{i - k,0},\\ldots ,a_{i + k + 1,0})\\) is either \\(a_{i - k,0}\\) or \\(a_{i + k + 1,0}\\) (as \\(a_{i - k,0}\\neq a_{i + k + 1,0}\\) and \\(2k + 2< 2025\\) ). Since the \\(2k + 2\\) values \\(a_{i - k,0}\\) , ..., \\(a_{i + k + 1,0}\\) were sampled independently from the same distribution, each of \\(a_{i - k,0}\\) and \\(a_{i + k + 1,0}\\) has a \\(\\frac{1}{2k + 2}\\) probability of being their maximum. Hence,  \n\n\\[\\mathbb{E}[\\mathbf{1}_{i}] = \\operatorname *{Pr}[a_{i,k}\\neq a_{i + 1,k}] = \\frac{2}{2k + 2} = \\frac{1}{k + 1}.\\]  \n\nThus, the expected number of distinct values is  \n\n\\[\\sum_{i = 1}^{2025}\\mathbb{E}[\\mathbf{1}_{i}] = \\frac{2025}{k + 1}.\\]  \n\nSubstituting \\(k = 100\\) yields the answer \\(\\left[\\frac{2025}{101}\\right]\\)", "metadata": {"resource_path": "HarvardMIT/segmented/en-282-2025-feb-comb-solutions.jsonl", "problem_match": "\n8. ", "solution_match": "\nSolution: "}}
 {"year": "2025", "tier": "T4", "problem_label": "9", "problem_type": "Combinatorics", "exam": "HMMT", "problem": "Two points are selected independently and uniformly at random inside a regular hexagon. Compute the probability that a line passing through both of the points intersects a pair of opposite edges of the hexagon.", "solution": "Answer: \\(\\frac{4}{9}\\)", "metadata": {"resource_path": "HarvardMIT/segmented/en-282-2025-feb-comb-solutions.jsonl", "problem_match": "\n9. ", "solution_match": "\nProposed by: Albert Wang  \n\n"}}
-{"year": "2025", "tier": "T4", "problem_label": "9", "problem_type": "Combinatorics", "exam": "HMMT", "problem": "Two points are selected independently and uniformly at random inside a regular hexagon. Compute the probability that a line passing through both of the points intersects a pair of opposite edges of the hexagon.", "solution": 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 \n\nFirst, we compute the probability that the line through two random points in a triangle \\(A B C\\) passes through segments \\(\\overline{{A B}}\\) and \\(\\overline{{A C}}\\) . We can take an affine transform of the two random points and the triangle such that \\(A B C\\) becomes equilateral. Since the distribution of the two points is still uniform and independent, the probability of the line intersecting any two given sides is \\(\\frac{1}{3}\\) by symmetry.  \n\nNext, we compute the probability that the line through two random points in a rectangle \\(A B C D\\) passes through opposite edges \\(\\overline{{A B}}\\) and \\(\\overline{{C D}}\\) .\n\n\n![](data:image/jpeg;base64,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 \n\nIf the line passes through \\(\\overline{{A B}}\\) and \\(\\overline{{B C}}\\) , the points must both lie in triangle \\(A B C\\) , whose area is half that of \\(A B C D\\) . Given this, the probability the line passes through those two sides is \\(\\frac{1}{3}\\) , as computed before. Thus the probability the line passes through \\(\\overline{{A B}}\\) and \\(\\overline{{B C}}\\) is \\(\\left(\\frac{1}{2}\\right)^{2}\\cdot \\frac{1}{3} = \\frac{1}{12}\\) . The same goes for the other pairs of adjacent edges. By symmetry, the line is equally likely to pass through either pair of opposite edges, each with probability \\(\\frac{1}{2}\\left(1 - 4\\cdot \\frac{1}{12}\\right) = \\frac{1}{3}\\) .  \n\n![](data:image/jpeg;base64,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 \n\nWe now return to the original problem. If the line passes through a pair of opposite edges, then both points must be in the rectangle formed by these edges, which has area \\(\\frac{2}{3}\\) that of the hexagon. Given this, the probability the line passes through those two edges is \\(\\frac{1}{3}\\) as computed before. Thus, the probability that the line passes through the given pair of opposite edges is \\(\\left(\\frac{2}{3}\\right)^{2}\\cdot \\frac{1}{3} = \\frac{4}{27}\\) . Hence, the probability the line passes through any of the three pairs of opposite edges is \\(3\\cdot \\frac{4}{27} = \\left[\\frac{4}{9}\\right]\\) .", "metadata": {"resource_path": "HarvardMIT/segmented/en-282-2025-feb-comb-solutions.jsonl", "problem_match": "\n9. ", "solution_match": "\nSolution:  \n\n"}}
+{"year": "2025", "tier": "T4", "problem_label": "9", "problem_type": "Combinatorics", "exam": "HMMT", "problem": "Two points are selected independently and uniformly at random inside a regular hexagon. Compute the probability that a line passing through both of the points intersects a pair of opposite edges of the hexagon.", "solution": "![md5:8a81ca7569630248d95c5ec1aceea0a0](8a81ca7569630248d95c5ec1aceea0a0.jpeg)\n  \n\nFirst, we compute the probability that the line through two random points in a triangle \\(A B C\\) passes through segments \\(\\overline{{A B}}\\) and \\(\\overline{{A C}}\\) . We can take an affine transform of the two random points and the triangle such that \\(A B C\\) becomes equilateral. Since the distribution of the two points is still uniform and independent, the probability of the line intersecting any two given sides is \\(\\frac{1}{3}\\) by symmetry.  \n\nNext, we compute the probability that the line through two random points in a rectangle \\(A B C D\\) passes through opposite edges \\(\\overline{{A B}}\\) and \\(\\overline{{C D}}\\) .\n\n\n![md5:dfc8a60ea45461cffaf3d402c804534e](dfc8a60ea45461cffaf3d402c804534e.jpeg)\n  \n\nIf the line passes through \\(\\overline{{A B}}\\) and \\(\\overline{{B C}}\\) , the points must both lie in triangle \\(A B C\\) , whose area is half that of \\(A B C D\\) . Given this, the probability the line passes through those two sides is \\(\\frac{1}{3}\\) , as computed before. Thus the probability the line passes through \\(\\overline{{A B}}\\) and \\(\\overline{{B C}}\\) is \\(\\left(\\frac{1}{2}\\right)^{2}\\cdot \\frac{1}{3} = \\frac{1}{12}\\) . The same goes for the other pairs of adjacent edges. By symmetry, the line is equally likely to pass through either pair of opposite edges, each with probability \\(\\frac{1}{2}\\left(1 - 4\\cdot \\frac{1}{12}\\right) = \\frac{1}{3}\\) .  \n\n![md5:58ef8107152d81b6f3825726280a35e3](58ef8107152d81b6f3825726280a35e3.jpeg)\n  \n\nWe now return to the original problem. If the line passes through a pair of opposite edges, then both points must be in the rectangle formed by these edges, which has area \\(\\frac{2}{3}\\) that of the hexagon. Given this, the probability the line passes through those two edges is \\(\\frac{1}{3}\\) as computed before. Thus, the probability that the line passes through the given pair of opposite edges is \\(\\left(\\frac{2}{3}\\right)^{2}\\cdot \\frac{1}{3} = \\frac{4}{27}\\) . Hence, the probability the line passes through any of the three pairs of opposite edges is \\(3\\cdot \\frac{4}{27} = \\left[\\frac{4}{9}\\right]\\) .", "metadata": {"resource_path": "HarvardMIT/segmented/en-282-2025-feb-comb-solutions.jsonl", "problem_match": "\n9. ", "solution_match": "\nSolution:  \n\n"}}
 {"year": "2025", "tier": "T4", "problem_label": "10", "problem_type": "Combinatorics", "exam": "HMMT", "problem": "The circumference of a circle is divided into 45 arcs, each of length 1. Initially, there are 15 snakes, each of length 1, occupying every third arc. Every second, each snake independently moves either one arc left or one arc right, each with probability \\(\\frac{1}{2}\\) . If two snakes ever touch, they merge to form a single snake occupying the arcs of both of the previous snakes, and the merged snake moves as one snake. Compute the expected number of seconds until there is only one snake left.", "solution": "Answer: \\(\\frac{448}{3}\\)", "metadata": {"resource_path": "HarvardMIT/segmented/en-282-2025-feb-comb-solutions.jsonl", "problem_match": "\n10. ", "solution_match": "\nProposed by: Srinivas Arun  \n\n"}}
 {"year": "2025", "tier": "T4", "problem_label": "10", "problem_type": "Combinatorics", "exam": "HMMT", "problem": "The circumference of a circle is divided into 45 arcs, each of length 1. Initially, there are 15 snakes, each of length 1, occupying every third arc. Every second, each snake independently moves either one arc left or one arc right, each with probability \\(\\frac{1}{2}\\) . If two snakes ever touch, they merge to form a single snake occupying the arcs of both of the previous snakes, and the merged snake moves as one snake. Compute the expected number of seconds until there is only one snake left.", "solution": "We solve the problem generally for \\(n\\) snakes and \\(3n\\) arcs. Without loss of generality, fix the two snakes \\(A\\) and \\(B\\) that will eventually form the ends of the last snake. Note that \\(A\\) and \\(B\\) must be consecutive in the initial configuration; assume \\(B\\) lies immediately clockwise from \\(A\\) .  \n\nLet \\(d\\) be the arclength between \\(A\\) and \\(B\\) (measuring clockwise from \\(A\\) to \\(B\\) ). As long as \\(d \\neq 0\\) and \\(d \\neq 2n\\) , we know that \\(A\\) and \\(B\\) lie in different snakes and thus move independently. Therefore, we can consider \\(d\\) to be on a random walk starting at 2, where", "metadata": {"resource_path": "HarvardMIT/segmented/en-282-2025-feb-comb-solutions.jsonl", "problem_match": "\n10. ", "solution_match": "\nSolution: "}}